Fractional Differential Equations

The appropriate control and management of reactive power is of great relevance in the electrical reliability, stability, and security of power grids. This issue is considered in order to increase system efficiency and to maintain voltage under the acceptable value range. In this regard, novel technologies as FACTS, renewable energies, among others, are varying conventional grid behavior leading to unexpected limit capacity reaching due to large reactive power flow. Thus, optimal planning of this must be considered. This paper proposes a new application for a simple and easy implementation optimization algorithm, called Rao-3, to solve the constrained non-linear optimal reactive power dispatch problem. Moreover, the integration of solar and wind energy as the most applied technologies in electric power systems are exploited. Due to the continuous variation and the natural intermittence of wind speed and solar irradiance as well as load demand fluctuation, the uncertainties which have a global concern are investigated and considered in this paper. The proposed single-objective and multi-objective deterministic/stochastic optimal reactive power dispatch algorithms are validated using three standard test power systems, namely IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus. The simulation results show that the proposed optimal reactive power dispatch algorithms are superior compared with two recent algorithms (Artificial electric field algorithm (AEFA) and artificial Jellyfish Search (JS) algorithm) and other optimization algorithms used for solving the same problem.

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Optimal Reactive Power Dispatch With Time-Varying Demand and Renewable Energy Uncertainty Using Rao-3 Algorithm

In this paper, a novel nature inspired meta heuristic optimization approach of Grey Wolf Optimization (GWO) algorithm is employed to solved the optimal reactive power dispatch (ORPD) problems. Essentially, it is the sub and non-linear optimization problem of optimal power flow (OPF) in which the control parameters of the power networks are optimized. The Grey wolf optimizer (GWO) which is inspired from grey wolves’ leadership and hunting behaviors to solve the ORPD problems. For which, the optimizer is tested on two test cases of IEEE30 standards specially, for 13 and 19 variables in order to get three fitness objectives for instance; transmission line losses ( $\text{P}_{\mathrm {losses}}$ , MW), voltage deviation (VD), voltage stability index (VSI) and cost of energy in ($). During computing all fitness objectives, the minimum fitness values are possibly achieved by the finest settings of control variables. The simulation results are compared with other artificial intelligence methods in previous literature to ensure the superior performance of the GWO for ORPD problem. The consistency of GWO will further be validated through detailed statistical analysis including histogram illustrations, boxplots, empirical CDF plot, probability plot and plot of minimum fitness during each independent trial.

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A Novel Nature Inspired Meta-Heuristic Optimization Approach of GWO Optimizer for Optimal Reactive Power Dispatch Problems

The aim of this study is to design a layer structure of feed-forward artificial neural networks using the Morlet wavelet activation function for solving a class of pantograph differential Lane-Emden models. The Lane-Emden pantograph differential equation is one of the important kind of singular functional differential model. The numerical solutions of the singular pantograph differential model are presented by the approximation capability of the Morlet wavelet neural networks (MWNNs) accomplished with the strength of global and local search terminologies of genetic algorithm (GA) and interior-point algorithm (IPA), i.e., MWNN-GAIPA. Three different problems of the singular pantograph differential models have been numerically solved by using the optimization procedures of MWNN-GAIPA. The correctness of the designed MWNN-GAIPA is observed by comparing the obtained results with the exact solutions. The analysis for 3, 6 and 60 neurons are also presented to check the stability and performance of the designed scheme. Moreover, different statistical analysis using forty number of trials is presented to check the convergence and accuracy of the proposed MWNN-GAIPA scheme.

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Design of Morlet Wavelet Neural Network for Solving a Class of Singular Pantograph Nonlinear Differential Models

Integrating renewable energy resources (RERs) has become the head of concern of the modern power system to diminish the dependence of using conventional energy resources. However, intermittent, weather dependent, and stochastic natural are the main features of RESs which lead to increasing the uncertainty of the power system. This paper addresses the optimal reactive power dispatch (ORPD) problem using an improved version of the lightning attachment procedure optimization (LAPO), considering the uncertainties of the wind and solar RERs as well as load demand. The improved lightning attachment procedure optimization (ILAPO) is proposed to boost the searching capability and avoid stagnation of the traditional LAPO. ILAPO is based on two improvements: i) Levy flight to enhance the exploration process, ii) Spiral movement of the particles to improve the exploitation process of the LAPO. The scenario-based method is used to generate a set of scenarios captured from the uncertainties of solar irradiance and wind speed as well as load demand. The proposed ILAPO algorithm is employed to, optimally, dispatch the reactive power in the presence of RERs. The power losses and the total voltage deviations are used as objective functions to be minimized. The proposed algorithm is validated using IEEE 30-bus system under deterministic and probabilistic conditions. The obtained results verified the efficacy of the proposed ILAPO for ORPD solution compared with the traditional LAPO and other reported optimization algorithms.

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An Improved Lightning Attachment Procedure Optimizer for Optimal Reactive Power Dispatch With Uncertainty in Renewable Energy Resources

In fact, optimal RPD is one of the most critical optimization matters related to electrical power stability and operation. The minimization of overall real power losses is obtained by adjusting the power systems control variables, for instance; generator voltage, compensated reactive power and tap changing of the transformer. In this search, a new heuristic computing method named as fractional particle swarm optimization gravitational search algorithm (FPSOGSA) is presented by introducing fractional derivative of velocity term in standard optimization mechanism. The designed FPSOGSA is implemented for the optimal RPD problems with IEEE-30 and IEEE-57 standards by attaining the near finest outcome sets of control variables along with minimization of two fitness objectives; active power transmission line losses ($P_{loss,}$ MW) and voltage deviation ($\text{V}_{\mathrm {D}}$ ). The superior performance of the proposed FPSOGSA is verified for both single and multiple runs through comparative study with state of art counterparts for each scenario of optimal RPD problems.

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Design of Fractional Particle Swarm Optimization Gravitational Search Algorithm for Optimal Reactive Power Dispatch Problems

The Flexible alternating current transmission systems (FACTS) is considered one of the most developed technologies applied for enhancing the performance of system due to their ability of adjusting different parameters in the transmission systems such as the buses voltage, the transmission line impedance, the active and reactive powers flow in transmission lines. The static synchronous series compensator (SSSC) is an effective member of the FACTS which is connected in series with the transmission lines and it consists of a solid-state voltage source inverter coupled with a transformer which aims to control and secure the operation of the power system. The main function of SSSC is inserting a controllable voltage in series with the transmission line to control the active and reactive powers flow in transmission lines. Solving the optimal reactive power dispatch (ORPD) problem is nonlinear, non-convex and it becomes a complex problem with integration of the SSSC. The contributions of article include, 1) an efficient and reliable optimization algorithm is developed to solve the ORPD problem and identify the optimal location and ratings of the SSSC, 2) The proposed algorithm is based on modifying the salp swarm algorithm (SSSA) using Levy Flight Distribution and spiral movement of particles to enhance the searching capabilities of the SSA, 3) an efficient model of SSSC based on power injection approach is used for representation the SSSC in ORPD. The ORPD is solved with and without the SSSC controller to minimize power losses and voltage deviations as well as improve the voltage stability. The proposed algorithm for ORPD is tested on the standards IEEE 30-bus and 57-bus systems. The simulation results demonstrate that MSSA is more effective and superior for solving the ORPD compared with some other reported meta-heuristic techniques. Moreover, the system performance is enhanced considerably with optimal inclusion the SSSC.

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Optimal Siting and Sizing of SSSC Using Modified Salp Swarm Algorithm Considering Optimal Reactive Power Dispatch Problem

The optimal reactive power dispatch (ORPD) is a major tool, and it plays a vital role for enhancement of the power system performance. ORPD is one multimodal, non-convex, and non-linear problem. Many elegant benefits can be obtained by using the renewable energy resources (RERs), but many technical issues related to the RERs including the stochastic characteristics of these resources due to continuous variations of solar irradiance and the wind speed lead to increasing the uncertainties of system. Thus, solving the ORPD problem with RERs is a crucial task. The contribution of the paper includes application a modified hybrid algorithm for solving the ORPD considering the uncertainties of the RERs and the load demand. The proposed algorithm is based on Fractional Calculus with Particle Swarm Optimization Gravitational Search Algorithm (FPSOGSA) which aims to enhance the searching capabilities of the conventional PSOGSA algorithm and overcome its tendency to stagnation. The proposed algorithm is tested on IEEE 30-bus system for reducing power losses and voltage deviation as well as enhancing voltage stability. The scenario-based method is employed to produce a set of scenarios from the uncertainties of load, wind speed and solar irradiance. The simulation results verify the effectiveness of the proposed algorithm for solving the ORPD problem with and without considering the uncertainties in the system. Furthermore, the proposed algorithm is superior compared with the state-of-the-art techniques in terms of the reduction of power losses and voltage deviations as well as the stability enhancement.

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Fractional PSOGSA Algorithm Approach to Solve Optimal Reactive Power Dispatch Problems With Uncertainty of Renewable Energy Resources

The Flexible AC Transmission Systems (FACTS) are the static power electronic devices that are installed and used in AC transmission networks to enhance the capability of transferring the power for providing the controllability and stability. However, the optimization of site and size of these devices is a crucial task due to their high capital cost and practical capabilities. In this paper, the optimal reactive power dispatch (ORPD) embedded with two effective controllers including the Static VAR Compensator (SVC) and Thyristor Control Series Capacitor (TCSC) is solved using a Modified Lightning Attachment Procedure Optimizer (MLAPO). The searching capability of basic LAPO is enhanced and the stagnation of basic LAPO is avoided by application of levy flight distribution and spiral orientation motion. The optimization for these devices is for reducing the power losses, voltage deviations, and the operating cost. MLAPO is examined and tested on modified IEEE30 and IEEE57 bus-standards considering SVC and TCSC. The effectiveness of MLAPO is further analyzed and compared with the outcomes of the well-known optimization techniques namely LAPO, PSO, ALO, EO, MPA, WOA and SCA with and without FACTS devices.

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De Tian

Papers

Design of Fractional Particle Swarm Optimization Gravitational Search Algorithm for Optimal Reactive Power Dispatch Problems

Optimal Siting and Sizing of SSSC Using Modified Salp Swarm Algorithm Considering Optimal Reactive Power Dispatch Problem

Fractional PSOGSA Algorithm Approach to Solve Optimal Reactive Power Dispatch Problems With Uncertainty of Renewable Energy Resources

A Novel Modified Lightning Attachment Procedure Optimization Technique for Optimal Allocation of the FACTS Devices in Power Systems