Fractional Order PI Research Articles

Complex Dynamics and PID Control Strategies for a Fractional Three-Population Model

In recent decades, there have been many studies on Hopf bifurcation and population stability with time delay. However, the stability and Hopf bifurcation of fractional-order population systems with time delay are lower. In this paper, we discuss the dynamic behavior of a fractional-order three-population model with pregnancy delay using Laplace transform of fractional differential equations, stability and bifurcation theory, and MATLAB software. The specific conditions of local asymptotic stability and Hopf bifurcation for fractional-order time-delay systems are determined. A fractional-order proportional–integral–derivative (PID) controller is applied to the three-population food chain system for the first time. The convergent speed and vibration amplitude of the system can be changed by PID control. For example, after fixing the values of the integral control gain ki and the differential control gain kd, the amplitude of the system decreases and the convergence speed changes as the proportional control gain kp decreases. The effectiveness of the PID control strategy in complex ecosystem is proved. The numerical simulation results are in good agreement with the theoretical analysis. The research in this paper has potential application values concerning the management of complex population systems. The bifurcation theory of fractional-order time-delay systems is also enriched.

Application of a Fractional Order PI Controller for a Speed Servo Drive Control

This paper deals with the tuning of the parameters of a fractional-order PI controller for the speed control of an electric servo drive in which the torque is set by a torque generator. The controller parameters are tuned using the multiple dominant pole method (MDPM), while the fractional order integrator is approximated by the Oustaloup method. The input parameters required for tuning the controller using MDPM are calculated using the optimization algorithm presented in this paper. This algorithm selects the optimal parameters from a set of points in three-dimensional space, based on the symmetry around a central point. The controller tuning is performed for the normalized control loop model. The obtained optimized normalized fractional order PI controller can then be applied to a real servo drive with specific parameters. The proposed tuning was also verified experimentally, comparing the obtained closed-loop responses with those of the integer-order PI controller. Both simulation and experimental results showed a significant reduction in the integral of the absolute error at the disturbance step compared to a control loop using an integer-order PI controller. This results in a faster output response to load torque steps and a smaller control error in a real servo drive.

Optimizing PIλDμ controller parameters using tree seed optimization for enhanced control performance and stability

This research article exploits the tree seed optimization algorithm to optimize the parameters of the fractional-order proportional–integral–derivative (FOPID) controller. The study focuses on designing an FOPID controller by imposing constraints on both the performance metrics and closed-loop amplitude ratio to achieve enhanced control system performance. The tree seed algorithm is inspired by the growth and reproductive behavior of trees and demonstrates its efficacy in finding optimal solutions within constrained search spaces. The optimization process involves formulating an objective function subject to the constraints on performance metrics and closed-loop amplitude ratio, reflecting both tracking performance and stability. Besides, inverted decoupling is used to reduce the interaction between the control loops and enable individual tuning of each loop for improved performance. Besides, the smallest disc perturbation is employed to determine the system’s stability. The proposed approach is implemented in various industrial processes to demonstrate its effectiveness in achieving desirable performance. The tree seed algorithm efficiently navigates the parameter space, thereby converging to ideal FOPID controller configurations that meet the specified constraints. Comparative studies with other methods highlight the improved performance and robustness obtained by the tree seed–based FOPID controller.

Application of an Optimal Fractional-Order Controller for a Standalone (Wind/Photovoltaic) Microgrid Utilizing Hybrid Storage (Battery/Ultracapacitor) System

Nowadays, standalone microgrids that make use of renewable energy sources have gained great interest. They provide a viable solution for rural electrification and decrease the burden on the utility grid. However, because standalone microgrids are nonlinear and time-varying, controlling and managing their energy can be difficult. A fractional-order proportional integral (FOPI) controller was proposed in this study to enhance a standalone microgrid’s energy management and performance. An ultra-capacitor (UC) and a battery, called a hybrid energy storage scheme, were employed as the microgrid’s energy storage system. The microgrid was primarily powered by solar and wind power. To achieve optimal performance, the FOPI’s parameters were ideally generated using the gorilla troop optimization (GTO) technique. The FOPI controller’s performance was contrasted with a conventional PI controller in terms of variations in load power, wind speed, and solar insolation. The microgrid was modeled and simulated using MATLAB/Simulink software R2023a 23.1. The results indicate that, in comparison to the traditional PI controller, the proposed FOPI controller significantly improved the microgrid’s transient performance. The load voltage and frequency were maintained constant against the least amount of disturbance despite variations in wind speed, photovoltaic intensity, and load power. In contrast, the storage battery precisely stores and releases energy to counteract variations in wind and photovoltaic power. The outcomes validate that in the presence of the UC, the microgrid performance is improved. However, the improvement is very close to that gained when using the proposed controller without UC. Hence, the proposed controller can reduce the cost, weight, and space of the system. Moreover, a Hardware-in-the-Loop (HIL) emulator was implemented using a C2000™ microcontroller LaunchPad™ TMS320F28379D kit (Texas Instruments, Dallas, TX, USA) to evaluate the proposed system and validate the simulation results.

A self-adaptive fractional-order PID controller for the particle velocity regulation in a pneumatic conveying system

This paper presents the methodical development of a state-error-driven self-adaptive fractional-order proportional–integral–derivative (AFOPID) control algorithm to efficiently regulate the velocity of pneumatically conveyed particles at the desired set point and to prevent the blockage of particles in a pipe due to an imbalanced combination of their velocity and corresponding mass flow rate. The proposed fractional control law is constituted by adaptively modulating fractional orders of the integral and differential operators in the control based on the state-error variations in the velocity of solid particles. The particle’s velocity is measured and updated via electrostatic sensors in conjunction with cross-correlation signal processing algorithms. All the other fixed hyper-parameters associated with the proportional–integral–derivative (PID) control scheme are meta-heuristically optimized by using a genetic algorithm. The proposed AFOPID is benchmarked against conventional integer-order PID and the fractional-order proportional–integral–derivative (FOPID) controllers. Experiments are performed on a laboratory-scale test rig to comparatively analyze the aforesaid control schemes, where each controller is examined for three velocity set points and three disturbance levels. The experimental results validate the superior time optimality and robustness of the proposed AFOPID controller against bounded disturbances and abrupt velocity set-point variations by manifesting relatively faster settling time, low overshoots (and undershoots), and smaller steady-state fluctuations.

Spider Wasp Optimizer-Optimized Cascaded Fractional-Order Controller for Load Frequency Control in a Photovoltaic-Integrated Two-Area System

The integration of photovoltaic (PV) systems into traditional power grids introduces significant challenges in maintaining system stability, particularly in multi-area power systems. This study proposes a novel approach to load frequency control (LFC) in a two-area power system, where one area is powered by a PV grid and the other by a thermal generator. To enhance system performance, a cascaded control strategy combining a fractional-order proportional–integral (FOPI) controller and a proportional–derivative with filter (PDN) controller, FOPI(1+PDN), is introduced. The controller parameters are optimized using the spider wasp optimizer (SWO). Extensive simulations are conducted to validate the effectiveness of the SWO-tuned FOPI(1+PDN) controller. The proposed method demonstrates superior performance in reducing frequency deviations and tie-line power fluctuations under various disturbances. The results are compared against other advanced optimization algorithms, each applied to the FOPI(1+PDN) controller. Additionally, this study benchmarks the SWO-tuned controller against recently reported control strategies that were optimized using different algorithms. The SWO-tuned FOPI(1+PDN) controller demonstrates superior performance in terms of faster response, reduced overshoot and undershoot, and better error minimization.

Golden eagle optimized fractional-order PI controller design for a PFC SEPIC converter in EV charging

The power factor correction converter is the function of the front-end converter, followed by the DC–DC converter of the electric vehicle charger. It improves the power factor and regulates the output voltage and current. This research article proposes the Golden Eagle optimization for fractional order PI (FOPI) controller for Single Ended Primary Inductor Converter (SEPIC) power factor correction. The Golden eagle optimization is based on its knowledge of hunting tactics at various degrees of spiral trajectories to catch the prey. The FOPI controller has a broad range of controller parameters that provide better control and performance of the converter. The tuning of the parameters of the FOPI controller is optimized in Golden Eagle Optimization, and the Integral Absolute error with Integral Square error is used for the objective function. The optimized parameters of FOPI compare with the conventional PI controller performance. The SEPIC converter is designed and derived from the state space model by state space averaging, and the reduced model is obtained through the moment matching method. This system is tested under MATLAB/SIMULINK, and simulation results show improved settling time, fast dynamic response, reduction of inrush current, less harmonic distortion, and stability.

Enhancing Transient Response in a DC-DC Converter for Electric Vehicle DC Fast Charging Applications Using Fractional-Order PI Control

DC fast charging is critical for the widespread adoption of electric vehicles (EVs) due to its impact on convenience, economics, and environmental sustainability. Due to the critical role of DC fast charging in EV adoption, improving its efficiency and performance is paramount. This paper presents a Fractional-Order PI (FOPI) controller for obtaining a well-regulated output voltage of a Dual Active Bridge (DAB) converter, a widely used topology in EV applications. The proposed FOPI is validated to the conventional PI controller in a simulated DAB model that is relevant to DC fast charging applications. The evaluation is performed, and various time-domain parameters and performance indices to evaluate the dynamic response of the model are considered. The results are expected to demonstrate significant improvement in the converter’s transient and steady-state response using the proposed FOPI controller compared to the conventional PI controller, contributing to a more efficient and robust DC fast charging system. This improvement can translate to faster charging times, better stability, and potentially reduced stress on the EV battery during DC fast charging.

Design of a novel intelligent adaptive fractional-order proportional-integral-derivative controller for mitigation of seismic vibrations of a building equipped with an active tuned mass damper

The primary objective of this study is to introduce a novel adaptive fractional order proportional–integral–derivative (FOPID) controller. The adaptive FOPID controller’s parameters are dynamically adjusted in real-time using five distinct multilayer perceptron neural networks. The extended Kalman filter (EKF) is employed to facilitate the parameter-tuning process. A multilayer perceptron neural network, trained using the error Backpropagation algorithm, is employed to identify the structural system and estimate the plant. The real-time estimated Jacobian is applied to the controller to control the model. The stability and robustness of the adaptive interval type-2 fuzzy neural networks controller are enhanced by utilizing the EKF and the feedback error learning strategy for compensator tuning. This improvement increases resilience against estimation errors, seismic disturbances, and unknown nonlinear functions. The primary objective is to address the challenges posed by maximum displacement, acceleration, and drift, as well as the uncertainties arising from variations in stiffness and mass. In order to validate the reliability of the proposed controller, the performance investigation is carried out on an 11-story building equipped with an active tuned mass damper under far and near-field earthquakes. Numerical findings show the remarkable effectiveness of the proposed controllers compared to their predecessors. In addition, it is revealed that the inclusion of the adaptive interval type-2 fuzzy neural networks compensator has increased the performance of the proposed controller and shows significant capabilities in reducing the seismic responses of structures during severe earthquake events.

Experimentally validated fractional-order PI with anti-windup for fractional-order plus time delay processes

Experimentally validated fractional-order PI with anti-windup for fractional-order plus time delay processes

Robust fractional-order PID controller assisted by active disturbance rejection control for the first-order plus time-delay systems

Time-delay characteristics of various industrial processes may degrade the stability and dynamic performance of the control systems. Aiming at the problems of the existing methods in dealing with the time delay plant, a modified fractional-order proportional–integral–derivative (FOPID) controller for the first-order plus time-delay (FOPTD) system is developed. Assisted by a modified active disturbance rejection control (ADRC) scheme with increased observer bandwidth, the proposed FOPID controller inherently obtains good robustness to time-delay uncertainties and external disturbances. In addition, taking advantage of the fractional-order operator, the proposed controller can provide larger stability margin over the proportional–integral–derivative (PID) controller. By suitably establishing the relation between ADRC and FOPID controller parameters, the proposed controller can be analytically tuned based on the common design indices. A practical tuning guideline is developed according to frequency-domain characteristic analysis, making the proposed controller more acceptable to industrial application. The performance of the ADRC-based FOPID controller is tested by the control simulation of some typical FOPTD systems and a diesel engine speed regulation system. The efficiency of the ADRC-based FOPID controller is demonstrated by the comparisons with some existing controllers.

Application of DPC to improve the integration of DFIG into wind energy conversion systems using FOPI controller

While the direct power control (DPC) approach has proven effective in improving the efficiency of wind energy conversion systems (WECS) using doubly fed induction generators (DFIG), its applicability is currently confined to a single usage and has not been extended to meet numerous applications. This work aimed to modify the implementation of DPC in WECS-DFIG for several objectives. This is accomplished by updating the reference power of the conventional DPC method into an adapted one to achieve two goals independently. The first objective is to track the maximum power during wind speed variations. This tracking is performed by updating the reference power to match the maximum available power at the current wind speed. The second purpose is to ensure that the WECS remains connected to the grid and continues to operate smoothly even in the event of faults; supporting fault-ride through (FRT) capability. That is achieved by reducing the reference power during these faults. The discrimination between these two objectives is based on the voltage level at the point of connecting WECS to the grid. The controller provided is an improved fractional order PI controller developed using arithmetic optimization technique (AOA). A comparison between the AOA and cuckoo search is presented. The results demonstrate the efficacy of the suggested configuration and regulator in enhancing the performance of integrating DFIG into the WECS in the presence of wind fluctuations and short circuit faults occurring. It is worth noting that AOA is better than cuckoo search in fine-tuning the settings of the FOPI controller.

Power quality improvement of grid-tied PV systems with continuous-time adaptive LMS and FOPI control-based DSTATCOM

A control technique for the single-stage solar photovoltaic-distribution static compensator (PV-DSTATCOM) integrated to a three-phase grid is presented in this article. The continuous-time adaptive Least Mean Square (CLMS) algorithm with Fractional Order Proportional Integral (FOPI) controller has been proposed to control the DSTATCOM. Perturb and Observe-based Maximum Power Point Tracking yields peak power production from the solar photovoltaic array. The precise control of PV-DSTATCOM by the proposed CLMS with FOPI controller has improved the power quality of the grid. This is achieved by utilizing the proposed algorithm's capability to effectively track the time-varying nature of irradiance and load changes, adjust weighting factors, and minimize errors. Simulations in MATLAB/Simulink demonstrate its superior performance in optimizing system behavior under intermittent sunlight, nonlinearity, and load imbalances. The grid current has been sinusoidally balanced with reduced total harmonic distortion and meets IEEE-519 standards in both steady-state and dynamic situations. The proposed CLMS with FOPI controller performs better than other methods in the literature.

Layered learning in a quadrotor drone: Simultaneous controlling and path planning using optimal fuzzy fractional order proportional integral derivative and proximal policy optimization

Unmanned aerial vehicles (UAVs), particularly quadrotors, have seen a surge in applications ranging from surveillance to delivery services. However, autonomous control remains a challenge due to the complexity of their dynamics and the need for real-time responsiveness. This paper addresses the significant problem of autonomous quadrotor control by introducing a novel learning method that integrates a deep neural network with advanced control techniques. Novelty of the work is distribution of different learning tasks in different layers of learning and combining the total learning system in a applicable UAV. The purpose is to enhance the quadrotor’s ability to perform complex maneuvers autonomously, which is crucial for expanding its application scope. The proposed method begins with the acquisition of direction control using a fractional order proportional–integral–derivative (FOPID) controller. The learning parameters are refined through a fuzzy method, ensuring adaptability and precision. For the optimization of the FOPID parameters, we apply two stochastic algorithms: Genetic Algorithm and Particle Swarm Optimization, comparing their efficacy. Our approach successfully teaches the quadrotor to generate and control a flight path autonomously. The second stage of learning involves proximal policy optimization, enabling the execution of intricate maneuvers. The experimental setup includes a quadrotor on a test stand with three degrees of freedom, where we validate the method through rigorous simulation and physical tests. The results demonstrate that our method not only teaches the quadrotor complex maneuvers but also ensures robust and precise execution, marking a significant advancement in UAV control systems. The implications of this research extend to improving UAV reliability for critical applications, potentially transforming the landscape of autonomous aerial operations.

Optimal fractional PI controller-based VSC-HVDC system with LCL filter design

For decades, high voltage direct current (HVDC) has been an essential means in designing smart microgrids. Though, a control approach for Voltage Source Converter (VSC)-based HVDC, like dampening oscillations, must be established for improved operational execution, as these oscillations can cause stability issues. Considering its versatility, the Fractional order Proportional Integral (FPI) controller is used in this study's control approach instead of the PI controller. With frequency domain analysis, comprehensive modelling and design of LCL filters for harmonic reduction to improve power quality are presented. The dynamic characteristics of VSC-HVDC are comprehensively presented based on dq axis synchronous frame of reference. The VSC-HVDC improved control outcome is validated by the result analysis using PI and FPI controller using ITLBO. It is found in both frequency and time-domain simulations and analysis that the AFPI-VSC-HVDC system is capable of effectively dampening oscillations to enhance the small-signal stability.

Enhanced grid integration through advanced predictive control of a permanent magnet synchronous generator - Superconducting magnetic energy storage wind energy system

In this study, the use of an Unscented Kalman Filter as an indicator in predictive current control (PCC) for a wind energy conversion system (WECS) that employs a permanent magnetic synchronous generator (PMSG) and a superconducting magnetic energy storage (SMES) system connected to the main power grid is presented. The suggested UKF indication in the hybrid WECS-SMES arrangement is in charge of estimating vital metrics such as stator currents, electromagnetic torque, rotor angle, and rotor angular speed. To optimize control strategies, PCCs use these projected properties rather than direct observations. To control the unpredictable wind energy's nature, SMES must be regulated to minimize fluctuations in the DC-link voltage and power output to the main grid. Fractional order-PI (FOPI) controllers are used in a novel control structure for the SMES system to regulate the output power and DC-link voltage. An artificial bee colony optimization approach is employed to optimize the FOPI controllers. Three commonly utilized indicators, including sliding-mode, EKF, and Luenberger, were evaluated using “MATLAB" to evaluate the performance of the UKF estimate. Assessment criteria such as mean absolute percentage error and root mean squared error were used to gauge the accuracy of the estimates. Simulation findings showed the efficiency of fractional order-PI controllers for SMES and the proposed UKF indication for predictive current control, especially in the presence of measurement noise and over a variety of wind speeds. An improvement in estimation accuracy of up to 99.9 % was demonstrated by the UKF indicator. Moreover, the stability of the suggested UKF-based PCC control for the hybrid WECS-SMES combination was confirmed using Lyapunov stability criteria."

Modeling and Control Research of Fractional-Order Cascaded H-Bridge Multilevel STATCOM

Abstract: This paper introduces fractional-order capacitors and fractional-order inductors into the conventional integer-order cascaded H-bridge multilevel static compensator (ICHM-STATCOM), thereby constructing the main circuit of the fractional-order cascaded H-bridge multilevel static compensator (FCHM-STATCOM). Mechanism-based modeling is employed to establish switching function models and low-frequency dynamic models for the FCHM-STATCOM in the three-phase stationary coordinate system (a-b-c). Subsequently, fractional-order rotating coordinate transformation is introduced to establish the mathematical model of the FCHM-STATCOM in the synchronous rotating coordinate system (d-q). Additionally, a fractional-order proportional-integral (FOPI)-based fractional-order dual closed-loop current decoupling control strategy is proposed. Finally, this paper validates the correctness of the established mathematical models through digital simulation. Moreover, the simulation results demonstrate that by appropriately selecting the order of fractional-order capacitors and fractional-order inductors, the FCHM-STATCOM exhibits superior dynamic and static characteristics compared to the conventional ICHM-STATCOM, and the FCHM-STATCOM provides a more flexible reactive power compensation solution for power systems.

Comparative Analysis: Fractional PID vs. PID Controllers for Robotic Arm Using Genetic Algorithm Optimization

This paper presents a comparative analysis of a fractional-order proportional–integral–derivative (FO-PID) controller against the standard proportional–integral–derivative (PID) controller, applied to a nonlinear robotic arm manipulator systems. The genetic algorithm (GA) optimization method was implemented to tune the gain parameters of the FO-PID and PID controllers. The performance of the FO-PID and PID controllers were evaluated though different cost functions, including integral of squared error (ISE), integral of absolute error (IAE), integral of time-weighted absolute error (ITAE), and integral of time-weighted squared error (ITSE). The performance of these controllers was examined via extensive simulations by using MATLAB/SIMULINK for different operating scenarios of the robotic arm manipulator system. Based on the obtained results, a comparative performance matrix is proposed, wherein cost functions ISE, IAE, ITAE, and ITSE are represented as columns while characteristic parameters (overshoot, rising time, and settling time) are represented as rows. The proposed performance matrix facilitates the selection between the PID and FO-PID controllers.

Global MPPT controllers for enhancing dynamic performance of photovoltaic systems under partial shading condition

This research focuses on the development and evaluation of global maximum power point tracking MPPT controllers for enhancing the dynamic performance of Photovoltaic (PV) systems under partial shading conditions. Three different controllers, namely Proportional-Integral (PI), Fractional-Order PI (FOPI), and Fuzzy Logic Controllers (FLC), are studied in conjunction with a modified Incremental Conductance (IC) and Fractional Open-Circuit Voltage (FOCV) algorithms. The proposed controllers with a novel optimization algorithm called Atom Search Optimization Algorithm (ASOA) are compared with classical control. The objective is to investigate the effectiveness of the global MPPT controllers in achieving higher tracking accuracy, faster convergence, and improved stability under varying shading conditions. The study involves extensive simulations using a representative PV system model subjected to different shading scenarios. The results demonstrate that the proposed global MPPT controllers with the ASOA algorithm, when integrated with the modified MPPT algorithms, outperform the classical control techniques in terms of dynamic performance and response to partial shading conditions. The suggested ASOA-based fuzzy IC MPPT controllers can reach the MPP more quickly with a shorter settling time and a better tracking response. Furthermore, with a reduced settling time of about 1.9 % and an average power harvesting efficiency of over 97.9 %, ASOA-based fuzzy IC MPPT controllers offer the best harvesting characteristics. The controller exhibits superior tracking accuracy and stability, mitigating the effects of shading-induced multiple peaks in the P-V curve.

Hybrid FOT2F-FOPD controller for permanent magnet synchronization motor based on ILA optimization with SRF-PLL

Permanent magnet synchronous motor (PMSM) systems have gained popularity in various fields due to their advantages such as high speed, high accuracy, low maintenance, and high reliability. This paper presents the speed tracking control of a permanent magnet synchronous motor (PMSM) using a hybrid fractional order PI and type 2 fuzzy control with fractional order PD control (FOT2F-FOPD). The SRF-PLL observes the motor speed and estimates the rotor's position by interpreting the input voltages of the motor instead of using a sensor. Then, the controller parameters (gain, μ and λ) are tuned based on a novel optimization algorithm called Incomprehensible but Intelligible-in-time (IbI) Logics algorithm (ILA). The proposed controller enhances the performance of the system and regulates the speed of the motor under parameter variations such as the speed and the load. So, the proposed ILA (FOT2F-FOPD) controller is assessed using MATLAB/Simulink simulation and compared with other controller techniques. The proposed technique reduces the settling time, steady state error and overshoot by at least 65%, 54% and 53% respectively under load conditions compared with (PSO, optimized FOPD, FOPI and PI). While at no load condition, the settling time and the error are reduced by 31% and 12.5% respectively with no overshoot in output response. The results show a significant improvement in the performance of motors used with the application of the proposed controller and the employment of the (ILA) optimization compared with FOPI and PI controllers.