Integral Of Time Multiplied Squared Error Research Articles

Investigation of reciprocal rank method for automatic generation control in two-area interconnected power system

In this article, the design and performance analysis of Jaya-assisted reciprocal rank-based proportional–integral–derivative controller with derivative filter (PIDn) is proposed for the automatic generation control (AGC) problem of two-area interconnected power systems. The filter with a derivative gain is used to nullify the impact of noise in the input signal. The integral of time multiplied square error (ITSE) of frequency deviations, tie-line power deviations, and area-control errors is the foundation of the goal function developed for tuning controller parameters (ACEs). A single overall objective function is created by combining these sub-objectives. ITSEs of two areas, ITSEs of tie-line power deviation, and ITSEs of ACEs of two areas are weighted and summarized to form the overall goal function. Each sub-relative goal’s importance in the control design is evaluated using the weights in the overall objective function. The weights in this article are determined using the reciprocal rank approach in a methodical manner, in contrast to earlier techniques where weights were either assumed equally by ignoring the relative relevance of sub-objectives or chosen at random. The resultant total objective function is minimized using the Jaya method. Six different test cases involving various load disturbances in the interconnected areas are used to evaluate the performance of the suggested Jaya optimizer-assisted reciprocal rank technique-based controller. Additionally, the performance of various controllers tuned with the sine cosine algorithm (SCA), salp swarm algorithm (SSA), symbiotic organisms search (SOS), Nelder–Mead simplex (NMS), and Luus–Jaakkola (LJ) algorithms is compared with that of the Jaya-tuned controller. The time domain specifications are provided for each of the six test scenarios. To display the differences in frequencies and tie-line power, the results are also shown. Statistical analysis is also provided to assess the suggested controller’s overall efficacy.

Optimal tuning proportional integral derivative controller on direct current motor using reptile search algorithm

<span lang="EN-US">This paper presents the reptile search algorithm (RSA) method to optimize the proportional integral derivative (PID) parameters on direct current (DC) motors. RSA was adopted from crocodile hunting behavior. Crocodile behavior is modeled in two important steps: surrounding and attacking prey. The RSA method was applied using twenty-three classical test functions. The search method of the proposed RSA method with other existing algorithms such as particle swarm optimization (PSO), and differential evolution (DE). Integral multiplied by absolute error (ITAE) and integral of time multiplied squared error (ITSE) were used as comparisons in measuring the performance of the RSA method. The results show that the proposed method, namely RSA, has better efficiency. Optimization of PID parameters with RSA on DC motor control shows superior performance. From the experiment, the ITSE average value of the RSA method is 4.17% better than the conventional PID method.</span>

Frequency Support Studies of a Diesel–Wind Generation System Using Snake Optimizer-Oriented PID with UC and RFB

The present paper discusses the modeling and analysis of a diesel–wind generating system capable enough to cater to the electrical power requirements of a small consumer group or society. Due to high variations of the load demand or due to changes in the wind speed, the frequency of the diesel–wind system will be highly disturbed, and hence to regulate the frequency and power deviations of the wind turbine system, an effective controller design is a necessary requirement, and therefore this paper proposes a novel controller design based on PID scheme. The parameters of this controller is effectively optimized through a new snake optimizer (SO) in an offline manner to minimize frequency and power deviations of an isolated diesel–wind system. The performance of SO-PID for the diesel–wind system is evaluated by considering the integral of time multiplied absolute error (ITAE), integral absolute error (IAE), and integral of time multiplied square error (ITSE). The results were calculated for a step change in load, step change in wind speed, load change at different instants of time with diverse magnitude, and for random load patterns, and they were compared with some of the recently published results under similar working conditions. In addition, the effect of an ultracapacitor (UC) and redox flow battery (RFB) on SO-PID was investigated for the considered system, and the application results demonstrated the advantages of our proposal over other studied designs.

Vertical channel stabilization of barometer‐aided inertial navigation systems by optimal control

AbstractThis study investigates the problem of inertial navigation system's (INS's) vertical channel instability, which primarily results from the positive gravity feedback that takes place in the numerical integration step of the accelerometer measurements. Traditional solutions to the problem include using information from auxiliary sensors, typically barometers and global navigation satellite systems (GNSSs), integrated into INSs through feedback control loops (referred to as mechanizations) and/or Kalman filters (KFs). Most studies on this subject lack clarity regarding the methodology used for the mechanization tuning (by showing excessive empiricism) and/or disregard the benefits of using optimal control techniques for the purpose of the latter. This study, therefore, reviews the main mechanization used for vertical channel stabilization of baro‐inertial integrations and presents, as its main contribution, the performance analysis of new tuning techniques based on optimal control theory, namely, by linear quadratic regulator (LQR) and by minimization of performance indices such as the integral absolute error (IAE), integral squared error (ISE), integral of time multiplied absolute error (ITAE), and integral of time multiplied squared error (ITSE). To validate the proposed tunings, the integrated altitude and vertical velocity mean errors (MEs) and standard deviations (SDs) are used as metrics, which are compared with the corresponding MEs/SDs of the empirical tuning with highest performance/acceptance in the literature. We also evaluate the ability of the mechanizations to fulfill their ultimate goal, namely, to track the reference barometer input, by also attenuating its noise. Based on simulated and experimental results, the present study demonstrates and validates the effectiveness/robustness of the proposed optimal tunings, particularly those based on LQR and ITAE minimization.

Rank-Sum-Weight Method Based Systematic Determination of Weights for Controller Tuning for Automatic Generation Control

The design and performance evaluation of a grey wolf optimizer (GWO) aided rank-sum-weight method based proportional-integral-derivative regulator with derivative filter (PIDn) for automatic generation control (AGC) of two-area interconnected power systems are presented in this research. The derivative gain filter is used to lessen the impacts of noise in the input signal. Sub-objectives based on integral of time multiplied square error (ITSE) of frequency deviations, tie-line power deviation, and area-control errors (ACEs) are used to formulate the objective function for adjusting regulator settings. A single overall objective function is formed by combining these sub-objectives. ITSEs of two areas, ITSEs of tie-line power deviation, and ITSEs of ACEs of two areas comprise up the overall objective function. In the control design, the weights in the overall objective function are used to evaluate relative significance of each sub-objective. In contrast to previous techniques, where weights are either considered equal by ignoring the relative relevance of sub-objectives or selected randomly, the weights in this article are obtained using the rank-sum-weight method systematically. Using the GWO algorithm, the overall objective function is minimized. For six different circumstances including different load disturbances in interconnected areas, the effectiveness of the proposed GWO aided rank-sum-weight method based controller is examined. The performance of the GWO-tuned controller is also compared to those of other controllers tuned using the differential evolution, elephant herding optimization, Nelder-Mead simplex, membrane computing, and Luus-Jaakola algorithms. Time domain specifications are tabulated for each of the six circumstances. The findings are also plotted to demonstrate the frequency and tie line power fluctuations. A statistical analysis is also performed in order to assess the overall efficacy of the suggested controller.

An optimal second‐order sliding mode based inter‐area oscillation suppressor using chaotic whale optimization algorithm for doubly fed induction generator

AbstractThis paper suggests an inter‐area oscillation suppressor based on second‐order sliding mode controller (SOSMC) for doubly fed induction generator (DFIG) to suppress the inter‐area power system oscillations. SOSMC‐based suppressor targets at improving the reactive power control capability of DFIG to retrieve the stability of affected power system caused by fault occurrence into the stable condition. This controller has been initially designed for two‐area power system, after that it has been extended for multi‐area power systems. SOSMC‐based suppressor has no sensitivity to the uncertainty and variation of the parameter in all system operation conditions as compared with the conventional suppressors. Due to exploration and exploitation capability of chaotic whale optimization algorithm (CWOA), it is applied as main optimizer to optimally tune the suppressor's parameters and merge with the dynamic stability scheme. This optimization algorithm is beneficial for complex dynamic stability problem, because during the iterative procedure, it can enhance the exploration capability to find the best solution. The optimization scheme based on CWOA has been compared with the conventional whale optimization algorithm (WOA) and particle swarm optimization (PSO). The suppressor damping capability has been evaluated with consideration of some possible disturbance scenarios. Furthermore, the objective function has been carried out based on the prominent dynamic performance criteria such as integral of time multiplied absolute error (ITAE), integral of absolute error (IAE), integral of squared error (ISE), and integral of time multiplied squared error (ITSE). Considering the various operating points, the SOSMC‐based suppressor ameliorates the system robustness as compared to the conventional suppressor. At last, the simulation results extracted from both interconnected power systems confirm the enhancement of power system operability and performance in suppression of the inter‐area oscillations, and also validate the robustness of the suggested control strategy against the model uncertainties and unmodeled dynamics.

A novel design of optimal intelligent fuzzy TID controller employing GA for nonlinear level control problem subject to actuator and system component fault

PurposeThe purpose of this article is about the design of controllers for conical two-tank noninteracting level (CTTNL) system in simulation. Local linearization around the equilibrium point has been done for the nonlinear CTTNL system to obtain a linearized model transfer function.Design/methodology/approachThis article deals with the design of novel optimal fractional-order tilt-integral-derivative (TID) controller using type-1 fuzzy set for the CTTNL prototype system. In this study, type-1 fuzzy TID controller parameters have been optimized through genetic algorithm (GA) and those set of values have been employed for the design of proportional-integral-derivative (PID) controller.FindingsA performance comparison between FTID and PID controller is then investigated. The analysis shows the superiority of FTID controller over PID controller in terms of integral absolute error (IAE), integral square error (ISE), integral of time multiplied absolute error (ITAE) and integral of time multiplied squared error (ITSE) integral errors. The transient and steady state performance of the FTID controller are superior as compared to conventional PID controller. In future, the FTID controller fault-tolerance capability tested on CTTNL system subject to actuator and system component (leak) faults. The detailed study of robustness in presence of model uncertainties will be incorporated as a scope of further research.Originality/valueA performance comparison between FTID and PID controller is then investigated. The analysis shows the superiority of FTID controller over PID controller in terms of IAE, ISE, ITAE and ITSE integral errors. Additionally, fault-tolerant performance of the proposed controller evaluated with fault-recovery time (Frt) parameter. The transient and steady state performance of the FTID controller are superior as compared to conventional PID controller.

Modified PID controller for automatic generation control of multi-source interconnected power system using fitness dependent optimizer algorithm.

In this paper, a modified form of the Proportional Integral Derivative (PID) controller known as the Integral- Proportional Derivative (I-PD) controller is developed for Automatic Generation Control (AGC) of the two-area multi-source Interconnected Power System (IPS). Fitness Dependent Optimizer (FDO) algorithm is employed for the optimization of proposed controller with various performance criteria including Integral of Absolute Error (IAE), Integral of Time multiplied Absolute Error (ITAE), Integral of Time multiplied Square Error (ITSE), and Integral Square Error (ISE). The effectiveness of the proposed approach has been assessed on a two-area network with individual source including gas, hydro and reheat thermal unit and then collectively with all three sources. Further, to validate the efficacy of the proposed FDO based PID and I-PD controllers, comprehensive comparative performance is carried and compared with other controllers including Differential Evolution based PID (DE-PID) controller and Teaching Learning Based Optimization (TLBO) hybridized with Local Unimodal Sampling (LUS-PID) controller. The comparison of outcomes reveal that the proposed FDO based I-PD (FDO-I-PD) controller provides a significant improvement in respect of Overshoot (Osh), Settling time (Ts), and Undershoot (Ush). The robustness of an I-PD controller is also verified by varying parameter of the system and load variation.

Automatic generation controller for multi area multisource regulated power system using grasshopper optimization algorithm with fuzzy predictive PID controller

AbstractThis paper presents a fuzzy predictive‐proportional integral derivative (FP‐PID) controller approach for automatic generation controller (AGC). This new AGC approach aims to balance the total generating system without any power losses and load changes in keeping constant system frequency per tie‐lie power flow. But, a sudden load variation in multi‐area Interconnected power system (MIPS) creates nonlinearities (frequency deviation & tie‐line) in all control areas. Because, penetrating of renewable sources to the power system, the sluggish control action may cause inefficiency in migrating frequency and tie‐line power flow. Hence, an accurate and fast‐acting controller is required to maintain the nominal value also the quality and stability of power system, because traditional AGC is not feasible for further process. Here, we propose an FP‐PID controller in MIPS for controlling large parametric uncertainties. Modeling error is taken into account to reduce the maximum deviation and time of oscillation. The main purpose of FP‐PID based AGC is to ensure stable and reliable power system operation. A grasshopper optimization algorithm (GOA) is used to tune the parameters of FP‐PID controller with Integral of time multiplied squared error (ITSE) as the objective function. The proposed method is compared with conventional two‐area and three‐area system, the result are built‐in Simulink/MATLAB show that the proposed method has a good dynamic response, fast operation reduced magnitude error and minimized frequency transients for three power areas. The robustness of the controller is, it need not be retuned for wide variations in system parameters and random step load power.

Optimal design of CDM controller to frequency control of a realistic power system equipped with storage devices using grasshopper optimization algorithm

This paper presents a new robust load frequency control (LFC) technique based on an optimal design of the coefficient diagram method (CDM) in a three area thermal power system equipped with redox flow batteries (RFB). In order to emphasize on a realistic power system and obtain an accurate insight, important nonlinearities due to generation rate constraint (GRC), governor dead band (GDB) and time delay (TD) were considered. The innovation of the proposed controller in this paper is the use of a hybrid intelligent combination of a decentralized CDM technique and optimization throughout its algebraic equations. In addition, a new algorithm namely grasshopper optimization algorithm (GOA) was used to find the key parameters of the proposed controller in solving the LFC problem for the first time. Furthermore, this study applied a powerful modified objective function by considering the integral of time multiplied squared error (ITSE) criteria for both controller input signal (ACE), to minimize the area control error and output signal (Δu) to reduce the size of the actuator, settling time (Ts) to have a faster response and a function to increase the minimum damping ratio (MDR) among all eigen values. To demonstrate the effectiveness of the proposed scheme, the studied simulated power system was tested through different cases including a large step and sinusoidal load perturbations and wide uncertainty in dynamic parameters of a nonlinear power system. Comparative results have revealed the superiority of the optimal CDM technique, especially when it is equipped with RFB. In addition to graphical results, by taking into account the MDR of different control strategies, the preference of the proposed controller has become more validated. This newly developed strategy leads to a flexible and accurate controller with a powerful mathematical back up which can successfully cope with GRC, GDB and TD nonlinearities in perturbed uncertain power systems and provide fast, stable and robust dynamic responses. Thus, the proposed control strategy can be constructive and successfully applied to real world power system application.

Tuning Techniques Evaluation for Satellite Launch Vehicle Attitude Controllers

This communication presents a comparative analysis of tuning techniques for satellite launch vehicle attitude controllers. The investigated tuning techniques consist in the minimization of specific performance indexes, namely the Integral Absolute Error (IAE) index, the Integral of Time Multiplied Absolute Error (ITAE) index, the Integral Squared Error (ISE) index, and the Integral of Time Multiplied Squared Error (ITSE) index, being hence, termed optimal. By defining adequate figures of merit, relevant for evaluating the overall performance of satellite launch vehicles, and also taking into account requirements related to the physical limitations of the latter, the performance of attitude controllers tuned by the investigated techniques is compared to the one tuned by the methodology currently employed in the Brazilian Satellite Launch Vehicle (VLS), namely, the Linear Quadratic (LQ) methodology. Through simulation results, it is demonstrated that, despite sparse benefits produced by the alternative tuning techniques, in particular ​ITAE and ISE, the LQ methodology remains globally superior.

Optimal Tuning of Fractional Order PID Controller for DC Motor Speed Control via Chaotic Atom Search Optimization Algorithm

In this paper, atom search optimization (ASO) algorithm and a novel chaotic version of it [chaotic ASO (ChASO)] are proposed to determine the optimal parameters of the fractional-order proportional+integral+derivative (FOPID) controller for dc motor speed control. The ASO algorithm is simple and easy to implement, which mathematically models and mimics the atomic motion model in nature, and is developed to address a diverse set of optimization problems. The proposed ChASO algorithm, on the other hand, is based on logistic map chaotic sequences, which makes the original algorithm be able to escape from local minima stagnation and improve its convergence rate and resulting precision. First, the proposed ChASO algorithm is applied to six unimodal and multimodal benchmark optimization problems and the results are compared with other algorithms. Second, the proposed ChASO-FOPID, ASO-FOPID, and ASO-PID controllers are compared with GWO-FOPID, GWO-PID, IWO-PID, and SFS-PID controllers using the integral of time multiplied absolute error (ITAE) objective function for a fair comparison. Comparisons were also made for the integral of time multiplied squared error (ITSE) and Zwe-Lee Gaing's (ZLG) objective function as the most commonly used objective functions in the literature. Transient response analysis, frequency response (Bode) analysis, and robustness analysis were all carried out. The simulation results are promising and validate the effectiveness of the proposed approaches. The numerical simulations of the proposed ChASO-FOPID and ASO-FOPID controllers for the dc motor speed control system demonstrated the superior performance of both the chaotic ASO and the original ASO, respectively.

Power System Rotor Angle Stability Improvement via Coordinated Design of AVR, PSS2B, and TCSC-Based Damping Controller

The current study is dedicated to design a novel coordinated controller to effectively increase power system rotor angle stability. In doing so, the coordinated design of an AVR (automatic voltage regulator), PSS2B, and TCSC (thyristor controlled series capacitor)-based POD (power oscillation damping) controller is proposed. Although the recently employed coordination between a CPSS (conventional power system stabilizer) and a TCSC-based POD controller has been shown to improve power system damping characteristics, neglecting the negative impact of existing high-gain AVR on the damping torque by considering its parameters as given values, may reduce the effectiveness of a CPSS-POD controller. Thus, using a technologically viable stabilizer such as PSS2B rather than the CPSS in a coordinated scheme with an AVR and POD controller can constitute a well-established design with a structure that as a high potential to significantly improve the rotor angle stability. The design procedure is formulated as an optimization problem in which the ITSE (integral of time multiplied squared error) performance index as an objective function is minimized by employing an IPSO (improved particle swarm optimization) algorithm to tune adjustable parameters. The robustness of the coordinated designs is guaranteed by concurrently considering some operating conditions in the optimization process. To evaluate the performance of the proposed controllers, eigenvalue analysis and time domain simulations were performed for different operating points and perturbations simulated on 2A4M (two-area four-machine) power systems in MATLAB/Simulink. The results reveal that surpassing improvement in damping of oscillations is achieved in comparison with the CPSS-TCSC coordination.

Performance comparison of TCSC with TCPS and SSSC controllers in AGC of realistic interconnected multi-source power system

Abstract The primary goals of employing series flexible ac transmission system (FACTS) in automatic generation control (AGC) studies of interconnected power systems are mitigating area frequency and tie-line power oscillations. This paper compares dynamic performance of thyristor controlled series capacitor (TCSC) as damping controller with thyristor controlled phase shifter (TCPS) and static synchronous series compensator (SSSC) which are series FACTS damping controllers. Commonly used lead-lag controllers are used in structure of damping controllers. The effect of TCSC in tie-line power exchange is modeled mathematically based on the Taylor series expansion for small-signal load disturbance. The performance of the proposed TCSC controller in coordination with integral AGC is compared with cases of TCPS–AGC and SSSC–AGC. An improved particle swarm optimization (IPSO) algorithm and integral of time multiplied squared error (ITSE) performance index are used to design the damping controllers. A two-area power system having generations from reheat thermal, hydro, and gas units in each area is evaluated regarding nonlinearity effects of generation rate constraint (GRC) and governor dead band (GDB). The simulations results in MATLAB/SIMULINK environment show that the proposed TCSC–AGC yields superior performance than others in damping of area frequencies and tie-line oscillations. Furthermore, sensitivity analyses are performed to show greater robustness of TCSC–AGC.

A new adaptive configuration of PID type fuzzy logic controller

In this paper, an adaptive configuration for PID type fuzzy logic controller (FLC) is proposed to improve the performances of both conventional PID (C-PID) controller and conventional PID type FLC (C-PID-FLC). The proposed configuration is called adaptive because its output scaling factors (SFs) are dynamically tuned while the controller is functioning. The initial values of SFs are calculated based on its well-tuned counterpart while the proceeding values are generated using a proposed stochastic hybrid bacterial foraging particle swarm optimization (h-BF-PSO) algorithm. The performance of the proposed configuration is evaluated through extensive simulations for different operating conditions (changes in reference, load disturbance and noise signals). The results reveal that the proposed scheme performs significantly better over the C-PID controller and the C-PID-FLC in terms of several performance indices (integral absolute error (IAE), integral-of-time-multiplied absolute error (ITAE) and integral-of-time-multiplied squared error (ITSE)), overshoot and settling time for plants with and without dead time.

Effective oscillation damping of an interconnected multi-source power system with automatic generation control and TCSC

Abstract Stabilizing area frequency and tie-line power oscillations in interconnected power systems are main concerns that have received significant attention in automatic generation control (AGC) studies. This paper deals with modeling and simulation of thyristor controlled series capacitor (TCSC) based damping controller in coordination with AGC to damp the oscillations and thereby, improve the dynamic stability. The contribution of TCSC in tie-line power exchange is formulated analytically for small perturbation and a systematic method based on the Taylor series expansion is proposed for modeling of TCSC based damping controller. The integral gains of AGC and TCSC parameters are optimized simultaneously using an improved particle swarm optimization (IPSO) algorithm through minimizing integral of time multiplied squared error (ITSE) performance index. The performance of the proposed TCSC–AGC coordinated controller is compared with case of AGC alone. A two-area interconnected multi-source power system, including TCSC located in series with the tie-line, is studied considering nonlinearity effects of generation rate constraint (GRC) and governor dead band (GDB). Simulation results show that proposed controller shows greater performance in damping of the oscillations and enhancing the frequency stability. Furthermore, sensitivity analyses are carried out against system loading condition, parametric uncertainties, and different perturbation patterns to show the robustness of TCSC–AGC.

LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled.

Design a pre-compensator to guarantee the finiteness of integral performance indices in fractional-order control systems

It is known that due to the existence of non-exponential modes in the step response of a fractional-order control system having zero steady-state error, integral performance indices of such a system may be unbounded. In the present study, a pre-compensator is designed to guarantee the boundedness of different integral performance indices (such as the integral of absolute error (IAE), the integral of squared error (ISE), the integral of time multiplied absolute error (ITAE), the integral of time multiplied squared error (ITSE), and the integral of squared of time multiplied error (ISTE)) in fractional-order control systems. Using this pre-compensator can improve the convergence rate of the step responses of fractional-order control systems. Numerical simulation results are presented to show the usefulness of the proposed method.