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For an uninterrupted power supply, renewable energy promises to be a suitable alternative compared to the conventional sources. System delays or communication delays may cause significant synchronization imbalances between various components in big electrical grids. Since the properties of solar and wind generation constantly change with climatic circumstances, engineers encounter many difficulties when substituting sustainable power with conventional electricity. The computation delay margin may be leveraged to handle a time-delayed automatic generation control (AGC) system. In order to regulate a distributed hybrid renewable energy system in a three-area AGC configuration, this paper investigates the influence of the fractional integral order on the stable system's delay parameter region. By changing the fractional order range, the delay margin can be increased, potentially broadening the time-delayed system's stability region. The controller's stability region has dependency on the order of fraction and the time delay. For this purpose, the asymptotic Bode diagram of the time-delayed fractional proportional-integral controller is determined. The gain and phase margins are used to calculate the delay margin for the application in discussion. The Honey Badger algorithm helps to adjust the controller parameters. It is also confirmed that the suggested controller is resilient to random load perturbations, nonlinearities, and parameter variations. [ABSTRACT FROM AUTHOR]

This paper proposes a further generalization of the fractional-order filters whose limiting form is that of the second-order filter. This new filter class can also be regarded as a superset of the recently reported power-law filters. An optimal approach incorporating constraints that restricts the real part of the roots of the numerator and denominator polynomials of the proposed rational approximant to negative values is formulated. Consequently, stable inverse filter characteristics can also be achieved using the suggested method. Accuracy of the proposed low-pass, high-pass, band-pass, and band-stop filters for various combinations of design parameters is evaluated using the absolute relative magnitude/phase error metrics. Current feedback operational amplifier-based circuit simulations validate the efficacy of the four types of designed filters and their inverse functions. Experimental results for the frequency and time-domain performances of the proposed fractional-order band-pass filter and its inverse counterpart are also presented. [ABSTRACT FROM AUTHOR]

Summary: This paper deals with the optimal analog‐to‐digital transformation of fractional‐order Butterworth filter (FOBF) in terms of infinite impulse response templates. The fractional‐order transfer function of the analog FOBF is transformed into its digital counterpart by employing the Binomial series expansion of different truncation orders, based on the Al‐Alaoui operator. This nonoptimal solution is then treated as an initial point for a local search optimizer such as the Nelder–Mead simplex (NMS) algorithm and also injected as a super‐fit individual in the initial population of a global search constrained evolutionary optimization algorithm (CEOA). Design stability and minimum‐phase response constraints are formulated for the super‐fit scheme. Both the techniques demonstrate good modeling performance; however, the super‐fit CEOA can markedly outperform the NMS method as the problem dimensionality increases. [ABSTRACT FROM AUTHOR]

Summary: This letter presents the design of digital fractional‐order Butterworth filter (DFOBF) of order (n+α), where n is an integer, and α ∈ (0,1), from the perspective of optimal realization. The magnitude–frequency characteristic of the DFOBF is optimally modeled using the computationally efficient lattice wave digital filters (LWDFs). Design examples for the third‐ and fifth‐order LWDF‐DFOBFs with various values of n, α, and cut‐off frequencies are presented. [ABSTRACT FROM AUTHOR]

Summary: In this letter, the optimal rational approximation of fractional‐order bandpass Butterworth filter (FOBBF) is presented. The transfer function of the FOBBF is decomposed into a multiplication of first‐order and second‐order terms. As a result, the design stability conditions can be easily satisfied using only the variable boundary constraints. The proposed technique generalizes the symmetric fractional‐order roll‐off characteristic as only a special case of the asymmetric one. Several examples are presented to validate the modeling efficacy. [ABSTRACT FROM AUTHOR]

Optimal integer‐order transfer function approximation of linear time‐invariant fractional‐order systems (FOS) using a straight‐forward approach is proposed here. The error between the rational approximant and the FOS (commensurate/non‐commensurate/oscillatory types etc.) is minimised directly in the frequency domain, subject to design stability constraints, by using the colliding bodies optimisation (CBO) algorithm. Comparisons with other well‐known metaheuristics regarding the computational time and the design effort expended in parameter tuning justify the efficiency of the proposed method. The CBO‐optimised models achieve a superior solution quality regarding both the magnitude and phase responses in comparison to the reported literature. Analogue circuit implementations of the models using the current feedback operational amplifier are also presented. Simulations carried out in OrCAD PSPICE environment validate the practical efficacy of the proposed models. [ABSTRACT FROM AUTHOR]

A new approach to determine the integer‐order transfer function approximation for α and α + β, where 0 < α, β < 1, fractional‐order analogue filters is presented. Reported works deal with this design problem by using techniques which are non‐optimal. The proposed designs with guaranteed stability are obtained by employing a global search optimisation algorithm called colliding bodies optimisation (CBO). The modelling error is iteratively minimised by CBO in the least‐squares sense directly in the frequency domain. The improved accuracy in the magnitude response of the proposed fractional‐order low pass filters, fractional‐order high pass filters, and fractional‐order band pass filters as compared to the designs based on the Matsuda and the continued fractions expansion‐based techniques is demonstrated. The efficacies of the proposed filters are also validated in SIMULINK platform. [ABSTRACT FROM AUTHOR]

Optimal integer‐order transfer function approximations to model the single fractance element‐based fractional‐order low‐pass filter (FLF) for any arbitrary order α, where, 0 < α < 1, is proposed here. First of all, the integer‐order filter coefficients for FLFs, with α varying from 0.01 to 0.99 in steps of 0.01, are directly obtained by using a metaheuristic algorithm called colliding bodies optimisation. For practical usability, the approximated FLF coefficients are explicitly provided in the form of analytical equations by employing a curve fitting on the optimised coefficients in the second step. The proposed approach provides a simpler design procedure in comparison to the reported literature which approximates the FLF by substituting an integer‐order rational approximation of the sα operator in the transfer function of the ideal FLF. Simulations confirm the superior modelling accuracy of the proposed design over the recent literature. [ABSTRACT FROM AUTHOR]

This paper presents an efficient approach to design wideband, accurate, stable, and minimum-phase fractional-order digital differentiators (FODDs) in terms of the infinite impulse response (IIR) filters using an evolutionary optimization technique called flower pollination algorithm (FPA). The efficiency comparisons of FPA with real-coded genetic algorithm (RGA), particle swarm optimization (PSO), and differential evolution (DE)-based designs are conducted with respect to different magnitude and phase response error metrics, parametric and nonparametric statistical hypotheses tests, computational time, and fitness convergence. Exhaustive simulation results clearly demonstrate that FPA significantly outperforms RGA, PSO, and DE in attaining the best solution quality consistently. Extensive analysis is also conducted in order to determine the role of control parameters of FPA on the performance of the designed FODDs. The proposed FPA-based FODDs outperform all the designs published in the recent literature with respect to the magnitude responses and also achieve a competitive performance in terms of the phase response. [ABSTRACT FROM AUTHOR]

This study presents a new approach to design fractional‐order low‐pass Butterworth filters (FOLPBF) in terms of integer‐order rational approximations meeting an accurate magnitude response. A parameter‐independent, metaheuristic optimisation algorithm called colliding bodies optimisation (CBO) is used for this purpose. The CBO‐based optimisation routine determines the optimal values of the coefficients for the proposed integer‐order models for the (1 + α), where, 0 < α < 1, order FOLPBFs. The performance of the proposed filter is examined concerning the passband and the stopband characteristics, solution quality robustness, and the convergence rate. The generic nature of the proposed design approach is also demonstrated. The roll‐off characteristics of the proposed higher orders of FOLPBFs exhibit accurate stopband attenuation behaviour. The proposed designs also achieve the best magnitude responses as compared with state‐of‐the‐art designs published in the recent literature. The proposed models can be practically implemented without using any fractance devices. [ABSTRACT FROM AUTHOR]

Optimal design of digital rational approximations with α‐dependant coefficients to model the fractional order integrator of any arbitrary order α, where α ε (0, 1), is presented in this work. The analytical expressions of the coefficients for the proposed fractional order digital integrators (FODIs) are derived by a two‐step method: (a) the coefficients of FODIs for α varying from 0.01 to 0.99 in steps of 0.01 are determined by a meta‐heuristic optimisation algorithm called symbiotic organisms search (SOS) and (b) curve fitting is applied on the SOS‐optimised coefficients to obtain their generalised expressions. Previous works dealing with the design of FODI based on various meta‐heuristic optimisers have considered only a few specific fractional orders; hence, the practical usability of such designs is restricted. This gap provides the motivation for conducting this research. Design quality robustness and convergence consistency of SOS are extensively compared with three other well‐known meta‐heuristic algorithms. The superior modelling accuracy of the proposed designs is justified by comparing with the recent literature. Simulation results validate the effectiveness of the proposed models as a fractional order proportional–integral controller. [ABSTRACT FROM AUTHOR]

Design of wideband infinite impulse response (IIR) digital fractional order integrators (DFOIs) based on a bio-inspired metaheuristic optimization approach called the cat swarm optimization (CSO) algorithm is presented in this paper. To investigate the efficiency of the proposed approach, the CSO-based DFOIs are evaluated against those of the approximations designed using real-coded genetic algorithm (RGA), standard particle swarm optimization (PSO), and differential evolution (DE) by different magnitude and phase response error metrics. Simulation results reveal the better frequency response of the CSO-based DFOIs in comparison with the competing designs. Both parametric and nonparametric statistical hypothesis tests validate the performance consistency of CSO. Comparisons with the cited literature confirm the efficacy of the proposed models. [ABSTRACT FROM AUTHOR]

This paper presents an efficient approach to design stable, wideband, and infinite impulse response digital integrators (DIs) and digital differentiators (DDs) of first, second, third, and fourth order using an evolutionary optimization algorithm called harmony search (HS). In recent years, although wideband DIs and DDs have been designed using metaheuristic optimization techniques such as simulated annealing, genetic algorithm, and particle swarm optimization (PSO), these algorithms lead to sub-optimal solutions because of stagnation and premature convergence. HS algorithm, however, promises an enhanced frequency response for DIs and DDs because of the better exploration and exploitation of the search space. Simulation results demonstrate the superiority of HS-based designs as compared with three well-known benchmark evolutionary optimization algorithms, namely real coded genetic algorithm (RGA), PSO, and differential evolution (DE) based designs by yielding the least values of different magnitude response error metrics. Parametric and non-parametric statistical hypothesis tests are also conducted to compare the consistency in the performance of HS-based DIs and DDs with those of the designs based on RGA, PSO, and DE. The proposed HS-based designs also outperform those of the designs based on both classical and evolutionary optimization approaches reported in recent literature in terms of the maximum absolute magnitude error metric. [ABSTRACT FROM AUTHOR]

This study deals with the implementation of highly accurate, stable, minimum phase, and wideband fractional‐order digital differentiators (FODDs) in terms of infinite impulse response filters using an efficient evolutionary optimisation algorithm called adaptive Gbest‐guided gravitational search algorithm (GGSA). Performance evaluation of GGSA as compared with real coded genetic algorithm (RGA), particle swarm optimisation (PSO), and differential evolution (DE) based designs are carried out in terms of different magnitude and phase response error metrics, solution quality reliability, and convergence speed. Simulation results clearly demonstrate that GGSA significantly outperforms RGA, PSO, and DE in consistently achieving the most accurate FODDs in a computationally efficient manner. The proposed FODDs also significantly outperform all state‐of‐the‐art designs in terms of magnitude responses. [ABSTRACT FROM AUTHOR]

This study presents a novel approach to design wideband infinite impulse response fractional order digital integrators (FODIs) for the half and one‐fourth order integrators based on a parameter independent metaheuristic algorithm called colliding bodies optimisation (CBO). The performance of CBO‐based FODIs have been compared with the designs based on three well‐known benchmark evolutionary optimisation algorithms namely, real coded genetic algorithm (RGA), particle swarm optimisation (PSO), and differential evolution (DE) in terms of robustness, consistency, parameter sensitivity, convergence speed, and computational time. Simulations results confirm that the proposed CBO‐based designed FODIs achieve consistently superior magnitude responses in a computationally efficient manner as compared with the designs based on RGA, PSO, and DE. The proposed CBO‐based FODIs also significantly outperform all state‐of‐the‐art designs reported in literature in terms of two different magnitude response error metrics. This research work underlines the potential of CBO as an efficient optimisation tool for the design of accurate digital approximations to the fractional order integrators. [ABSTRACT FROM AUTHOR]

This paper presents the optimal modeling of Power Law Filters (PLFs) with the low-pass (LP), high-pass (HP), band-pass (BP), and band-stop (BS) responses by means of rational approximants. The optimization is performed for three different objective functions and second-order filter mother functions. The formulated design constraints help avoid placement of the zeros and poles on the right-half s-plane, thus, yielding stable PLF and inverse PLF (IPLF) models. The performances of the approximants exhibiting the fractional-step magnitude and phase responses are evaluated using various statistical indices. At the cost of higher computational complexity, the proposed approach achieved improved accuracy with guaranteed stability when compared to the published literature. The four types of optimal PLFs and IPLFs with an exponent α of 0.5 are implemented using the follow-the-leader feedback topology employing AD844AN current feedback operational amplifiers. The experimental results demonstrate that the Total Harmonic Distortion achieved for all the practical PLF and IPLF circuits was equal or lower than 0.21%, whereas the Spurious-Free Dynamic Range also exceeded 57.23 and 54.72 dBc, respectively. [ABSTRACT FROM AUTHOR]