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This article deals with the fractional‐order modeling of a complex four‐dimensional energy supply‐demand system (FOESDS). First, the fractional calculus techniques are adopted to describe the dynamics of the energy supply‐demand system. Then the complex behavior of the proposed fractional‐order FOESDS is studied using numerical simulations. It is shown that the FOESDS can exhibit stable, chaotic, and unstable states. When it exhibits chaos, the FOESDS's strange attractors are plotted to validate the chaotic behavior of the system. Moreover, we calculate the maximal Lyapunov exponents of the system to confirm the existence of chaos. Accordingly, to stabilize the system, a finite‐time active fractional‐order controller is proposed. The effects of model uncertainties and external disturbances are also taken into account. An estimation of the stabilization time is given. Based on the latest version of the fractional Lyapunov stability theory, the finite‐time stability and robustness of the proposed method are proved. Finally, two illustrative examples are provided to illustrate the usefulness and applicability of the proposed control scheme. © 2014 Wiley Periodicals, Inc. Complexity 20: 74–86, 2015

Modern power systems are increasingly challenged by frequency stability issues due to dynamic load variations and the growing complexity of interconnected networks. Traditional PID controllers, while widely utilized, struggle to address the rapid fluctuations and uncertainties inherent in contemporary multi‐area interconnected power systems (MAIPS). This paper introduces an innovative approach to Load Frequency Control (LFC) using a Fractional‐Order PID (FOPID) controller, optimized by a Neural Network Algorithm (NNA). The proposed NNA‐FOPID framework leverages the biological principles of neural networks to dynamically tune controller parameters, significantly enhancing system performance. The solution is tested under various scenarios involving step load changes across multi‐area systems. The proposed method demonstrates marked improvements over traditional PID controllers and advanced optimization techniques such as Differential Evolution (DE) and Artificial Rabbits Algorithm (ARA). The comparisons show that the FOPID controller's NNA‐based design effectively and successfully handles LFC in MAIPSs for ITAE minimizations, and statistical evaluation supports its superiority. Motivated by artificial neural networks, this paper develops the NNA to optimize a Fractional‐Order PID controller for LFC in multi‐area interconnected power systems. It improves frequency regulating capabilities and reduces overshoots/undershoots. It achieves a 27.66%–93.78% improvement in Integral of Time Absolute Error compared to other techniques.

This paper is devoted to Bode’s integrals based design of fractional order [PI], [PD] and [PI][PD] controllers for integer order systems. The gain crossover frequency, the phase margin and the Nyquist plot slope adjustment in gain crossover frequency are considered as the design specifications for FO[PI] and FO[PD] controllers. To calculate the Nyquist plot slope, the Bode’s integrals are utilized. The FO[PI][PD] controller is constructed by series connection of the FO[PI] and FO[PD] controllers. To design the FO[PI][PD] controller, the measurement noise rejection and the integral time absolute error minimization are employed as the additional design requirements. Simulation results are presented to show the performance of the Bode’s integral based fractional order controllers.

Summary Nonlinear dynamical system exhibit bifurcation, chaos, and instability due to initial conditions and parametric variation. The power system is also nonlinear dynamical system, which exhibits these behaviors due to different initial operating conditions and uncertain variation in parameter like reactive power load demand. The fractional‐order controller is introduced as the better choice for several control systems as compared to conventional proportional‐integral‐derivative controllers owing to its merits. This paper proposes a first ever numerical study of novel use of optimized fractional‐order controller for control of chaos in power system. This paper discusses the steps to optimize the order of fractional controller. The role of fractional‐order controller is to inject or withdraw the precise amount of reactive power to control chaotic nature of voltage. The proposed controller will perturb the dynamics of the nonlinear power system and push it to nonchaotic and bounded stable state because of precise state feedback control. Precise perturbation in reactive‐power value inhibits voltage bifurcation point, chaos, and instability thereafter. Fourth order benchmark model of the power system is used as a case study in the paper.

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PI λ D μ controllers. It is based on plotting the global stability region in the ( k p , k i )-plane for the PI λ controller and in the ( k p , k i , k d )-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.

Owing to integrating the dense range of distinct electric power sources, high volume of power generation units, abrupt and continuous changes in load demand, and rising utilization of power electronics, the electric power system (EPS) is striving for high-performance control schemes to counterwork the concerns depicted above. Additionally, it is highly creditable to have the controller structure as simple as possible from a viewpoint of practical implementation. Thus, this paper describes a virgin application of fractional order proportional integral–fractional order proportional derivative (FOPI–FOPD) cascade controller for load frequency control (LFC) of electric power generating systems. The proposed controller includes fractional order PI and fractional order PD controllers connected in cascade wherein orders of integrator ( λ ) and differentiator ( μ ) may be fractional. The gains and fractional order parameters of the controller are concurrently tuned using recently proposed dragonfly search algorithm (DSA) by minimizing the integral time absolute error (ITAE) of frequency and tie-line power deviations. DSA is the mathematical model and computer simulation of static and dynamic swarming behaviors of dragonflies in nature, and its implementation in LFC studies is very rare, unveiling additional research gap to be bridged. Performance of the advocated approach is first explored on popular two-area thermal PS with/without governor dead band (GDB) nonlinearity and then on three-area hydrothermal PS with suitable generation rate constraints. To highlight the prominence and universality of our proposal, the work is extended to single-/multi-area multi-source EPSs. Several comparisons with DSA optimized FOPID controller and the relevant recent works for each test system indicate the contribution of proposed DSA optimized FOPI–FOPD cascade controller in alleviating settling time/undershoot/overshoot of frequency and tie-line power oscillations.

This paper introduces a novel multi-stage FOPD(1 + PI) controller for DC motor speed control, optimized using the Pelican Optimization Algorithm (POA). Traditional PID controllers often fall short in handling the complex dynamics of DC motors, leading to suboptimal performance. Our proposed controller integrates fractional-order proportional-derivative (FOPD) and proportional-integral (PI) control actions, optimized via POA to achieve superior control performance. The effectiveness of the proposed controller is validated through rigorous simulations and experimental evaluations. Comparative analysis is conducted against conventional PID and fractional-order PID (FOPID) controllers, fine-tuned using metaheuristic algorithms such as atom search optimization (ASO), stochastic fractal search (SFS), grey wolf optimization (GWO), and sine-cosine algorithm (SCA). Quantitative results demonstrate that the FOPD(1 + PI) controller optimized by POA significantly enhances the dynamic response and stability of the DC motor. Key performance metrics show a reduction in rise time by 28%, settling time by 35%, and overshoot by 22%, while the steady-state error is minimized to 0.3%. The comparative analysis highlights the superior performance, faster response time, high accuracy, and robustness of the proposed controller in various operating conditions, consistently outperforming the PID and FOPID controllers optimized by other metaheuristic algorithms. In conclusion, the POA-optimized multi-stage FOPD(1 + PI) controller presents a significant advancement in DC motor speed control, offering a robust and efficient solution with substantial improvements in performance metrics. This innovative approach has the potential to enhance the efficiency and reliability of DC motor applications in industrial and automotive sectors.

Automatic Generation Control (AGC) delivers a high quality electrical energy to energy consumers using efficient and intelligent control systems ensuring nominal operating frequency and organized tie-line power deviation. Subsequently, for the AGC analysis of a two-area interconnected hydro-gas-thermal-wind generating unit, a novel Fractional Order Integral-Tilt Derivative (FOI-TD) controller is proposed that is fine-tuned by a powerful meta-heuristic optimization technique referred as Improved-Fitness Dependent Optimizer (I-FDO) algorithm. For more realistic analysis, various constraints, such as Boiler Dynamics (BD), Time Delay (TD), Generation Rate Constraint (GRC), and Governor Dead Zone (GDZ) having non-linear features are incorporated in the specified system model. Moreover, a comparative analysis of I-FDO algorithm is performed with state-of-the-art approaches, such as FDO, teaching learning based optimization, and particle swarm optimization algorithms. Further, the proposed I-FDO tuned controller is compared with Fractional Order Tilt Integral Derivative (FOTID), PID, and Integral-Tilt Derivative (I-TD) controllers. The performance analysis demonstrates that proposed FOI-TD controller provides better performance and show strong robustness by changing system parameters and load condition in the range of  ± 50%, compared to other controllers.

This study focuses on shaping the transient response of special case of fractional-order systems by using fractional-order PD (FOPD) and fractional-order PID (FOPID) controllers. For a plant with a fractional-order pole, a FOPID controller is designed in which the orders of its derivative and integral terms are considered equal to the commensurate order of the plant while for an integrating plant with a fractional-order pole, a FOPD controller is designed. The region of controller parameters is extracted to obtain a closed-loop system with a monotonically decreasing magnitude–frequency response. This leads to a non-overshooting or low-overshoot step response. The gain crossover frequency and phase isodamping conditions are employed to select appropriate controller parameters among the mentioned region. The numerical examples are provided to show the efficiency of the proposed FOPD and FOPID controllers.

Active power filters (APFs) are used to mitigate the harmonics generated by nonlinear loads in distribution networks. Therefore, due to the increase of nonlinear loads in power systems, it is necessary to reduce current harmonics. One typical method is utilizing Shunt Active Power Filters (SAPFs). This paper proposes an outstanding controller to improve the performance of the three-phase 25-kVA SAPF. This controller can reduce the current total harmonic distortion (THD), and is called fractional order PI-fractional order PD (FOPI-FOPD) cascade controller. In this study, another qualified controller was applied, called multistage fractional order PID controller, to show the superiority of the FOPI-FOPD cascade controller to the multistage FOPID controller. Both controllers were designed based on a non-dominated sorting genetic algorithm (NSGA-II). The obtained results demonstrate that the steady-state response and transient characteristics achieved by the FO (PI + PD) cascade controller are superior to the ones obtained by the multistage FOPID controller. The proposed controller was able to significantly reduce the source current THD to less than 2%, which is about a 52% reduction compared to the previous work in the introduction. Finally, the studied SAPF system with the proposed cascade controller was developed in the hardware-In-the Loop (HiL) simulation for real-time examinations.