## 1 Introduction  
Renewable energy systems (RES) are the way forward towards achieving net-zero carbon emission target as mandated by many countries around the globe. This has contributed to a major push for ever increasing integration of RES into the existing electric power grid. This has attracted a lot of research attention into the control of grid-connected converters (GCC) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Grid-connected converters can transfer maximum power to the grid only when the operation is in-phase with the grid voltage signal. Moreover, operation of GCC in unbalanced grid also require real-time extraction of grid voltage sequences. This motivated the researchers to devote significant attention to grid-synchronization based sequence extraction techniques [12, 13, 14, 15]. Some of the recent and most popular grid synchronization techniques are: Kalman filter [16, 17, 18], demodulation [19, 20], phase-locked loop (PLL) [21, 22, 23, 24, 25, 26], adaptive observer [27, 28, 29], second-order generalized integrator (SOCI) [30, 31, 32], open-loop techniques [33], etc. Kalman filter [16] and least square-based techniques [34] typically work by considering a linear-parametric model of the grid voltage. Real-time matrix inversion are required for these techniques. This adds computational complexity for real-time low-cost embedded systems. Similar problems are to some extent true for the maximum likelihood estimator [18]. Demodulation [19] has several low-pass filters to tune, requires the use of trigonometric function, and can only be used for frequency estimation and not for sequence extraction. Phase-locked loop (PLL) [21, 22, 23, 24, 25, 26] has gained lot of attention for grid-synchronization application due to simplicity. Classical PLL generates double frequency oscillation in the presence of unbalanced grid. To avoid this issue, several solutions are proposed in the literature that require additional filtering. Moreover, tuning the PLL in the presence of additional filtering stage can be complicated. PLL is a closed-loop system where the stability is dependent on various parameters. Open-loop grid-synchronization technique [33] has a significant advantage in this regard as unconditional stability can be guaranteed. However, this type of technique often require calculating the derivative. Real-time estimation of signal derivatives are often challenging when the base signal is noisy. Many nonlinear techniques are presented in the literature in recent times, such as adaptive observer [27; 28; 29; 35], adaptive notch filter (ANF) [4], self-tuning filter (STF) [5], second-order generalized integrator (SOGI) [30; 31; 32], etc. Unlike PLL, these techniques perform the task of grid-synchronization through the help of frequency-locked loop. ANF, SOGI, and STF are similar in nature. All these techniques have band-pass filter property. SOGI has a limited complex conjugate closed-loop poles tuning range [36]. Adaptive observers [27; 28; 29] do not have any such limitation. This made adaptive observers-based technique a suitable choice for grid-synchronization in the literature. [PERSON] and [PERSON], in Reference [28], proposed the application of adaptive observer for grid-synchronization and sequence extraction. Using Lyapunov function and [PERSON]'s invariance principle, this observer provides global asymptotic stability of the state and parameter estimation error dynamics. Unfortunately, the frequency update law used in this work do not use any gain normalization. This makes the frequency estimation dynamics dependent on the grid voltage amplitude. This can slow down the convergence of the unknown frequency estimation when the grid voltages suffer significant sag. This issue has been resolved in Reference [27] by introducing gain normalized frequency estimation law. Both [27; 28] consider the grid voltage and its derivative as the state variables to obtain the dynamic model. SOGI-type adaptive observers as introduced in Reference [29; 36] use the same dynamical model as used by SOGI.