Frequency is an index of the operating practices of a power system and is one of its most important power quality parameters (Zhao et al., 2019). The frequency of a power system can deviate from its nominal value by an amount within a certain tolerance level; any deviation beyond the predetermined tolerance level is detrimental to the system and is a sign of abnormal system operating conditions that necessitate remedial actions. These problems include generation–load imbalances and improper functioning of the power system. An inaccurate frequency parameter in a power network can lead to insufficient load shedding via frequency relays, which may ultimately result in a catastrophic breakdown of the grid (Bose, 2020; Alhelou et al., 2019); moreover, the power signal would no longer be static owing to the presence of noise, voltage imbalances, and harmonics, among others. Therefore, accurate frequency estimations of dynamic signals are necessary in power system operations and have garnered considerable attention from researchers.

In reality, numerous strategies have been proposed over the years to estimate the power system frequency. The time between two zero crossings is the basis of the traditional approach (Friedman, 1994; Zhang et al., 2017), which assumes that the power system voltage is entirely sinusoidal and that the interval between two zero crossings is a measure of the grid frequency. These techniques are ineffective when dealing with dynamic changes and in the presence of harmonics and excessive noise. Several approaches have been proposed on the basis of phase-locked loops (Han et al., 2015; Xu and Song, 2020; Li et al., 2019), least-squares adaptive filtering (Zhang et al., 2019; Martinek et al., 2019), recursive total least-squares approach (Avalos et al., 2021), adaptive notch filters (Man et al., 2021; Pan et al., 2021), and extended Kalman filters (Mojiri et al., 2007; Xie et al., 2023), which have been investigated extensively for estimation purposes; such approaches have the potential to overcome the aforementioned restrictions. However, most of the currently used techniques focus on voltage measurements in single-phase systems or make assumptions about balanced-power system states. Owing to the possibility of imbalance, single-phase signals cannot be used to accurately measure the system frequencies in three-phase systems (Wold and Wilches-Bernal, 2021); hence, knowledge of all three phases must be considered for reliable and robust frequency estimations.

Furthermore, advancements in fractional calculus approaches have enabled more research activities in numerous domains, such as biomedical engineering, fluid mechanics, optics, computer vision, electrochemistry, control systems, and signal processing (Tan et al., 2015). In this context, newer fractional adaptive algorithms have inherited their concepts from fractional calculus and incorporated adaptive algorithms (Naik et al., 2024). For instance, the fractional least mean squares (FLMS) identification method that adds a percentage of each gradient based on the value of a forgetting factor was established by applying the ideas of fractional calculus; consequently, better convergence is obtained compared to the original least mean squares (LMS) algorithm. By using a sliding window that considers the past input parameters in addition to the present parameters, the convergence qualities of the FLMS approach are further enhanced (Aslam et al., 2017). Chaudhary et al. (2015) presented a scheme that simply incorporates a partial component of the gradient in the weight vector equation to decrease the computing complexity. The total convergence is unaffected by the elimination of the integer-order gradient and retention of only the fractional component, but the computational complexity is decreased by the integer-order gradient; as the fractional order approaches unity, the convergence rate increases. The fractional orders employed in algorithms thus far are in the range of (0,1), where a higher fractional order may also increase the steady-state error, as reported by Cheng et al. (2017); researchers have discovered that the rapidity and accuracy have the same characteristics as those of the original FLMS (Chaudhary et al., 2021) and modified LMS (Bershad et al., 2017). The above discussion demonstrates that several LMS variations have been extended to fractional order and that their attributes have been investigated. These works also include the momentum term in FLMS (mFLMS) in parameter estimation modeling of sinusoidal signals (Sharma et al., 1998). The convergence rate of the mFLMS increases as the weight update equation is revised to include a percentage change of the previously calculated gradients and an additional learning rate (Zubair et al., 2018). Hence, the momentum-learning-rate-based FLMS technique is a perfect candidate for identification and estimation studies; the widespread advantages of this algorithm warrant further investigations, particularly in the context of power frequency estimation. In the present work, the concept of the momentum-based FLMS algorithm is used for power system frequency estimation.

Several constants are also known to be associated with the weight update equation. For instance, the mFLMS algorithm considered in this study has three parameters, namely the step size, fractional-order step constants, and momentum learning control terms. Moreover, the values of these parameters are in the range of 0–1 and considered manually or empirically, which may result in drastic errors in the final outcome. There are no available roadmaps that provide robust methods to consider appropriate values for the variables. Therefore, optimization algorithms can be employed in these scenarios to identify the optimal values of each of the constants. The optimization algorithms are computational in nature and search for constant values in global as well as local space vectors to achieve the least possible errors (Kanoongo and Giri, 2023). Such algorithms have also been shown to produce encouraging results for noise cancellation, adaptive filtering, controller tuning, computer vision, and other applications (Alhussan et al., 2023). Some of the commonly used optimization algorithms include gray wolf optimization (GWO), genetic algorithm (GA), bat optimization, teaching-learning-based optimization (TLBO), fuzzy rules, cuckoo search algorithm (CSA), and flower pollination algorithm. A more effective approach called the Harris hawks optimization (HHO) algorithm was suggested recently and has been shown to be very effective for multivariable optimization problems (Heidari et al., 2019). The innovative HHO algorithm was developed in response to the chasing behaviors of Harris hawks; by simulating the pursuit of prey, sudden attacks, and various attack strategies used by Harris hawks, the algorithm has been applied to six well-known benchmark engineering problems, including the formulation of the 3-bar truss, tension/compression spring design, pressure vessel requirement, welded beam design, rolling element design, and multiplate disc clutch brake problems. The HHO is prone to local optimum and slow convergence problems, just like any other natural heuristic algorithm (Alabool et al., 2021). Therefore, there is still scope for development in the HHO, where the solution involves incorporating chaotic-based functions (Shehab et al., 2022). In addition, it has been demonstrated that adding a chaotic function to the optimizer can significantly affect the algorithm’s ability to perform local and global searches (Menesy et al., 2019). Consequently, the chaotic map-based HHO (iHHO) method was considered in this study along with momentum-learning-rate-based FLMS to achieve better accuracy and faster convergence than existing mFLMS algorithms; named as improved momentum-learning- rate-based FLMS (imFLMS) estimation algorithm.

As discussed above, the fractional-order LMS algorithm provides significant benefits over the integer-order LMS algorithm for power frequency estimation, with faster convergence rates and improved accuracies, especially in environments with high degrees of noise. Furthermore, considering that the momentum learning control term in the FLMS algorithm enables more precise adjustments, enhanced resistance to noise and capacity to adapt to dynamic signal conditions are obtained. This adaptability leads to reduced steady-state errors and improved stability across a wider spectrum of operating conditions. Furthermore, the momentum learning control terms in the FLMS algorithm entail multiple parameters such as step size, fractional-order step constants, and momentum learning control terms, all of which range from 0 to 1 and are chosen manually or empirically, which can potentially cause significant errors. Optimization algorithms are used to determine the optimal values for these parameters computationally, thereby minimizing errors by exploring the global and local space vectors (Chao et al., 2024). Therefore, the momentum-learning-rate-based FLMS method tuned by optimization ensures more precise and reliable estimates, making it a preferable option for real-time power frequency applications.

The main contributions of this work are as follows:

• Design of an innovative power frequency estimation scheme considering the momentum-based fractional-order LMS approach.

The remainder of this paper is organized as follows. Section 2 provides a brief description of the momentum-based FLMS algorithm. Section 3 describes the HHO optimizer and its improved version, i.e., chaotic-based HHO with OBL. Section 4 outlines the power system frequency model. Section 5 presents the validation of the estimation scheme by considering various cases. Finally, Section 6 provides some concluding remarks and scope for future work.

Two constants ( μ 1 a n d μ F ) are associated with the imFLMS algorithm weight update equation and have values between 0 and 1. It is observed from literature that there is no established procedure for selecting the values of these constants. In other words, these constants are assigned values either manually or through the trial and error method, which may impact the system performance parameters such as convergence speed severely. A lot of time is also needed to identify the optimal combination of values that produces an effective outcome. Hence, the selection of constants is achieved with the help of an optimization algorithm in this study. The square of the error signal is considered the cost function, and the optimal value of the constant is set as a constraint described by Equation 13. The upper and lower bounds of both constraints are considered 1 and 0, respectively. Once the optimization is formulated, the optimal values of the constants are searched so that the errors are minimal, as reported below. J k = M i n . e 2 k = M i n . d k − e k 2 S u b j e c t t o : 0 < μ 1 < 1 0 < μ F < 1 . (13)

The HHO belongs to a family of modern metaheuristic algorithms based on the collective hunting behaviors of Harris hawks. Rabbits are the principal food items of Harris hawks and often flee from the hawks (Heidari et al., 2019). As a result, two dynamic factors, namely surprise attack by the hawk and the rabbit’s capacity to flee, form the basis for the HHO algorithm. The two steps of the optimizer are exploration and exploitation; there are four substages under exploitation: soft and hard besiege as well as soft and hard besiege with successive fast dives. Figure 1 depicts the methods of exploration and exploitation (Alabool et al., 2021).

Most of the metaheuristic algorithms start with a random variable for initialization and corresponding distribution. Recently, chaotic mapping has been used in such metaheuristic techniques as it allows similar properties and characteristics as randomness, with improved dynamic and statistical performances. Therefore, chaotic mapping is added to the unique swarm intelligence approach in place of a random variable. As added advantages to the optimization strategy, this mapping offers accurate initialization, arbitrariness, and ergodicity (Baliarsingh and Vipsita, 2020). By producing non-repeated numbers with periodicity, chaotic mapping enhances the efficiency with respect to a random number. These characteristics resulting from chaotic mapping provide the means to escape local optima and preserve distinction while increasing the efficacy of the global search. Appropriate initialization is achieved when the preliminary points are adjusted such that a meaningful result is generated, assuming that the outcome is wisely dependent on the preliminary points. Throughout the phase of arbitrariness, chaotic mapping takes on the roles of the random variables. Ergodicity describes the capacity of a chaotic variable to identify non-repeated values given a specific range.

Different metaheuristic methods have been used to implement various chaotic functions. Recently, Zahmoul et al. (2017) reported a new approach to the family of chaotic functions; Equation 25 provides the mathematical representation of this mapping: β x : p , q , x 1 , x 2 = x − x 1 x c − x 1 0 p x 2 − x x 2 − x c q , i f x ∈ x 1 , x 2 , o t h e r w i s e , (25) where x 1 , x 2 , p , q ∈ R a n d x 1 > x 2 are calculated using the following Equations 26–29. x c = p x 1 + q x 2 p + q (26) p = a 11 + b 11 c (27) q = a 22 + b 22 c (28) X t + 1 = k β x t : p , q , x 1 , x 2 (29) where c is the bifurcation parameter; a 11 , a 22 , b 11 , and b 22 are constants. The β -chaotic map exhibits initialization and variation sensitivity, and its bifurcation diagram is shown in Figure 2. The bifurcation diagram is often used to demonstrate the values that approach asymptoticity to a given system state as a function of the bifurcation parameter, where the abscissa represents the bifurcation parameter and ordinate indicates the range of values of the map that approaches asymptoticity subject to the initial conditions.

Exploration and exploitation are appropriately balanced when chaotic mapping is applied to the traditional HHO algorithm. Chaotic mapping replaces the random functions employed in traditional HHO techniques. Then, the constants q, r 1 , r 2 , r 3 , r 4 , Lb, and Ub (from Equation 14) are all considered to be 0 if x t > 0.5 or 1 if x t < 0.5, where x t is the chaotic mapping of the t t h iteration.

A fresh and practical theory for improving the performances of diverse metaheuristic approaches from an optimization standpoint is OBL. The idea of OBL is to take the opposite candidate as an alternative approach to reach the optimal solution, which could be closer to the global optimum solution. The objective of OBL in optimization is to improve the solution’s effectiveness by evaluating the contender for the solution of the relevant pair. The ideal response candidate is taken into account for the subsequent individual solution (Barisal and Prusty, 2015). Suppose that Z i denotes the candidate solution; then, its corresponding opposite candidate solution Z i ′ can be calculated as per Equation 30, where L and U represent the lower and upper bounds of the search space of the constraints, which are 0 and 1 in our case, respectively. Z i ′ = L + U − Z i (30)

The global searchability is enhanced with OBL, and the solution using this method with chaotic mapping is shown in Equation 31. In this instance, Z min and Z max are the lower and upper bounds, respectively, while Z hho ′ is the opposing candidate hawk; Z hho displays the hawk’s position vector, while Z best displays the hawk’s optimal solution; x t denotes the chaotic mapping. Z hho ′ = Z max + Z min − Z best + x t Z best + Z hho (31)

The improved HHO algorithm was used to obtain the optimal step size that minimizes the cost function of the proposed imFLMS estimation algorithm. Several distinct benchmark functions were used to verify the reliability and effectiveness of the proposed technique. The improved HHO showed greater optimal performance after 100 trials. The enhanced version is more accurate and resilient while requiring less processing time; the ideal solution demonstrates higher search process efficiency and flawless balance between exploration and exploitation while handling the benchmark functions. Figure 3 shows the convergence graph for comparison, where the optimization approach offers a more optimal value than the traditional HHO, SCA, and PSO techniques. The graphs show that the suggested iHHO algorithm maintains its ideal outcome in terms of fewer iterations and shorter execution time. The proposed optimizer was also applied to several common restricted benchmark functions, such as Mishra’s bird, Simionescu, and Townsend functions, as well as unconstrained benchmark functions like the Ackley, Levi, and Rastrigin functions (Pati and Subudhi, 2023). Their results show that the iHHO algorithm outperforms the other algorithms in terms of the best, worst, and mean values as well as execution time.

The temporal complexity of the proposed algorithm can be computed as follows and stated in Equation 37.

• Time required for population initialization.

where O() represents the order; D m is the problem size; L and U are the lower and upper bounds, respectively; M a x iter is the maximum number of iterations. The execution times based on the suggested technique for different estimation algorithms are presented in Tables 2, 3.

Using filtering methods based on the LMS, mFLMS, and the newly proposed imFLMS algorithms, several examples of balanced power signals were simulated as follows.

The control parameters of the weight update expression ( μ 1 and μ F ) were tuned using the improved HHO technique, and the remaining variables like the momentum learning term and fractional order were assigned based on system performance. The values of all constants and variables were evaluated using MATLAB/Simulink. For the purpose of the simulations, the voltage equations for a balanced three-phase system were considered according to Equation 32; the phase signal amplitudes V _an , V _bn , and V _cn were each considered to be 1 p.u. in this study. The signal frequency was considered to be 50 Hz, and AWGN with a signal-to-noise ratio (SNR) of 30 dB was introduced to corrupt the complex phasor produced by the transform. The comparisons were performed to determine whether the proposed adaptive filtering technique provides the most accurate estimation. The frequency estimation plot of the proposed technique was compared with those of the mFLMS and standard LMS techniques, and it was observed that the proposed method estimated the frequency well before the other techniques, as displayed in Figure 4. A comparison of the mean-squared errors of the various frequency estimation techniques is depicted in Figure 5; here, the imFLMS method was observed to have faster error convergence over the other adaptive algorithms. The absolute frequency inaccuracy was found by determining the difference between the desired value and estimated output from the adaptive algorithm. A comparative assessment of the estimated errors for different SNRs is further shown in the bar chart of Figure 6. Although the imFLMS, mFLMS, and LMS algorithms can provide frequency estimations, their effectiveness reduces in noisier environments.

The power system signal with a fundamental frequency of 50 Hz was then applied with a step shift in frequency between 0.02 and 0.04 s, for which it was observed that the imFLMS estimation technique followed the frequency variation more closely than the other algorithms (Figure 7). Next, a ramp shift in frequency from 50 Hz to 46 Hz was applied between 0.03 and 0.05 s, following which the frequency was maintained constant at 46 Hz for the remainder of the period of the three-phase signal. As seen in Figure 8, the suggested technique successfully tracks the signal ramp change with a convergence time of less than half a cycle and a high degree of estimation accuracy. With the FLMS algorithm, the tracking began at about 0.033 s and leading to an inconsistent result. Similarly, as seen in the figure, the LMS algorithm followed the ramp variation from 0.038 s.

The step size variation μ 1 with respect to number of iterations for the proposed iHHO algorithm is displayed in Figure 9. Here, the adaptive algorithm approach suggests that the optimal step size selection should satisfy the condition of 0 < step size < 1/(maximum eigenvalue of the input autocorrection matrix). Thus, the highest allowed value is obtained initially, and the step size is reduced thereafter until the change in error is stabilized. A similar behavior was also observed with the proposed model. However, the main feature here is that the selection process automatically addresses the aforementioned condition. The improved optimization is so effective that the optimal value (0.031) is identified well before the 200th iteration. Similarly, the fractional-order step size variation based on the iterations of the optimizer algorithm is shown in Figure 10 and is found to be 0.0012.

Table 2 displays the performance evaluation results for the suggested scheme in terms of the initial convergence rates for different momentum learning rates ( γ ) and fractional orders (F). The findings reveal that the initial convergence of the imFLMS algorithm is much faster than those of typical adaptive approaches and that the convergence rate increases with increases in the preceding gradients. Estimated frequency error validation was then used to examine the performance of the suggested method further; the results of the imFLMS approach were compared with those of the other methods for various values of γ and F. It is noted that all techniques were sufficiently accurate and convergent; however, the accuracy of any approach decreases as the noise variance increases. It was also found that fractional adaptive algorithms, such as the imFLMS and mFLMS methods, were steady for all possible fractional order variations and that there were no significant variations in accuracy between various fractional orders. Additionally, a larger fractional order value (F = 0.75) produced a considerably better outcome. According to the findings presented in Table 2, the imFLMS algorithm achieves faster convergence when the proportion of prior gradients is relatively high ( γ = 0.9) but better steady-state results when this proportion is lower ( γ = 0.2). Therefore, a larger value of the learning parameter ( γ = 0.7) would be an excellent choice for achieving good balance between faster convergence and better steady-state results. Table 3 shows the performances of the imFLMS, mFLMS, and LMS estimators for various noise levels. As discussed earlier, the proposed imFLMS technique has higher efficacy in noisy environments than other estimation techniques, but its execution time is slight higher than that of the traditional LMS algorithm at different noise levels.

To examine the effectiveness of the proposed scheme, it was tested with a signal derived from a renewable-based two-area-load frequency control model. This power system was earlier modeled and simulated in MATLAB as well as evaluated by Pati and Subudhi (2021). Figure 11 shows the block diagram model of the test system under consideration (Pati and Subudhi, 2021).

To validate the proposed scheme, a signal generated with a step load perturbation of 2% was considered and compared for the suggested imFLMS and existing estimation techniques. Figure 12 displays the relevant comparative plots for these estimation methods. Furthermore, the dynamic load signal shown in Figure 13 was applied to the test system to verify the algorithmic efficacy. The imFLMS, mFLMS, and LMS algorithms were used to estimate the test system frequency, and their results are shown in Figure 14. As verified earlier, the imFLMS algorithm outperforms the other estimation techniques; it estimates the frequency more precisely in less time than the other methods even with the random load, thus proving its robustness.

To validate the performance of the proposed algorithm for power system frequency estimation, a simple experimental setup was established in the laboratory. In this setup, an induction motor was connected as a load to a single-phase power supply, as suggested by Daw (2016). A variable AC supply was then employed to regulate the input voltage. A voltmeter, an ammeter, and a wattmeter were used to measure the output voltage, current, and power, respectively. The voltage signals were captured using a digital storage oscilloscope from the load side, where the induction motor was connected, as illustrated in Figure 15. The detailed specifications of the equipment used are as follows:

• Variable AC supply: 0–230 V/0–270 V, 10 A

The input voltage waveform across the induction motor was recorded with a digital storage oscilloscope, and the data were subsequently transferred to a computer through communication software. Figure 16 illustrates the voltage signal estimation using the proposed imFLMS algorithm and the data obtained from the experiment. The results demonstrate that the proposed algorithm yields an estimation significantly closer to the actual signal and with a nearly identical period. Consequently, the fundamental frequency of 50 Hz was accurately estimated with minimal mean-squared error of 0.0741 Hz, indicating that the proposed estimator was highly effective for applications involving real-world data.

The percentage improvement was measured in the comparative state-of-the-art study using Equations 38, 39, where A P 1 and A P 2 stand for “amelioration (in percentage)” and represent the percentage enhancements obtained as the performance measures (namely convergence rate and estimated error) compared to the reference schemes noted in Subudhi et al. (2020) and Nefabas et al. (2020). The convergence rate and estimated error have their own amelioration values and are denoted by the notations APi(Convergence) and APi(Error), respectively. A P 1 % C o n v e r g e n c e = R a t e o f C o n v e r g e n c e R e f R a t e o f C o n v e r g e n c e Estimated × 100 (38) A P 2 % E r r o r = E s t i m a t e d E r r o r R e f E s t i m a t e d E r r o r obtained × 100 (39)

The results for the suggested adaptive scheme tuned by iHHO for power system frequency estimation were compared with those reported by Subudhi et al. (2020) and Nefabas et al. (2020). These comparisons are summarized in Table 4. Additionally, it has been demonstrated beyond reasonable doubt that the proposed method greatly enhances the reaction time of the system. Thus, we conclude that the suggested system configuration has better stability compared to those discussed in Subudhi et al. (2020) and Nefabas et al. (2020). From A P 1 and A P 2 , it is observed that the system responsiveness is greatly improved with the iHHO algorithm and proposed technique. The adoption of the imFLMS algorithm results in a convergence rate improvement of 150%–375% and estimated error improvement of 114.28%–128.57% compared to the findings reported by Subudhi et al. (2020) and Nefabas et al. (2020). The sparse H-infinity filtering approach was reported to produce absolute frequency errors of 0.031, 0.008, and 0.0010 at SNR levels of 20 dB, 30 dB, and 40 dB, respectively. In contrast, the proposed imFLMS estimation technique demonstrates superior accuracy at these SNR levels, as detailed in Table 3. Similarly, the linear LMS kurtosis estimator was reported to provide approximate frequency biases of 0.196, 0.0075, and 0.0038 Hz at SNR levels of 20 dB, 30 dB, and 40 dB, respectively.