A Comparative Analysis of the Fractional‐Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

Aljahdaly, Noufe H.; Akgül, Ali; Shah, Rasool; Mahariq, İbrahim; Kafle, Jeevan

doi:10.1155/2022/8876149

Cited by 48 publications

(16 citation statements)

References 42 publications

(47 reference statements)

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“…This study uses GAMS software to code and simulate the presented scheme [132], in which the IPOPT solver [132] has been utilized to find a solution to the problem. This problem is a mathematical model, including objective function and constraints [133][134][135][136][137][138][139][140][141][142].…”

Section: Resultsmentioning

confidence: 99%

Energy management of distribution network with inverter‐based renewable virtual power plant considering voltage security index

Azarhooshang,

Rezazadeh

2023

IET Renewable Power Gen

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Virtual power plants (VPP) with resources and storages are able to control the active power of the network. They are also connected to the network through an inverter, which is capable of controlling reactive power. Therefore, it is expected that the optimal use of inverter‐based VPP can play an effective role in improving the economic and technical status of the distribution network. So, the operation of a smart distribution system is presented in this paper by considering inverter‐based VPPs constrained to the operator's measures. The weighted sum of expected energy loss (EEL) and voltage security index (VSI) is minimized while considering AC optimal power flow equations, restrictions of network's security, and operating model of the inverter‐based VPPs. Uncertainties with an origin of the amount of demand, renewable energy, and parameters of mobile energy storage are also discussed. The uncertainties are modelled using a stochastic optimization approach relying on the unscented transformation (UT). Evaluating inverter‐based VPP performance, providing models of flexible resources such as responsive loads and mobile storages, checking network voltage security status, and modelling uncertainties using the UT method are among the innovations of this study. According to the results, it is demonstrated that the technical situation of the distribution system is improved with the help of optimal management of the VPP. With energy management of the inverter‐based VPP, the suggested design has succeeded to enhance the operating status (voltage security) of the system by approximately 33–73% (12%) in comparison to power flow studies.

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Section: Resultsmentioning

confidence: 99%

Energy management of distribution network with inverter‐based renewable virtual power plant considering voltage security index

Azarhooshang,

Rezazadeh

2023

IET Renewable Power Gen

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“…The vehicle equation (Equation ( 8)) and the bridge equation (Equation ( 9)) are independent equations, while the vehicle-bridge coupled vibration equation system (Equation ( 11)) becomes non-independent due to the introduction of mutual interaction force (Equation ( 10)). The analytical solution cannot be obtained because Equation ( 11) is a higher-order inhomogeneous system of differential equations, which could be solved by numerical solution methods [28][29][30][31][32]. The Newmark-β in the direct integration methods (step-by-step integration) was used in this paper [5,33].…”

Section: Vibration Equationmentioning

confidence: 99%

Assessing Dynamic Load Allowance of the Negative Bending Moment in Continuous Girder Bridges by Weighted Average Method

Wang

Tian²,

Zhou

et al. 2022

Coatings

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Accurate acquisition of dynamic load allowance (DLA) based on measurement data is essential to the safety assessment of a bridge. When static load tests cannot be achieved, and filtering fails, the estimated DLAs from the experimental method vary widely due to the choice of a left or right band. In this paper, the proposed weighted average method (WAM) is used to possibly solve the above problem in continuous gird bridges. Two-span and three-span precast concrete box-gird bridges were selected to optimize intercepted segments of WAM for the first time with the assistance of standard deviation and coefficient of variation in statistics. Then, a DLA measurement case of the negative bending moment was utilized to verify the validity of the WAM. The results show that the intercepted segments of 10L/16 to L were suitable for the WAM to calculate the DLA of the negative bending moment due to small offset moments and stable variation coefficients. The WAM had a strong anti-interference ability of outliers filtering in “bad data,” which differed significantly from the experimental method. In three measurements of a field bridge, DLAs obtained by the WAM had less dispersion than the experimental and low-pass filtering methods.

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“…Fractional differential equations have various applications in widespread fields of science, such as engineering [1], chemistry [2][3][4], biology [5,6], physics [7][8][9], numerical analysis [10,11], and others [12,13]. In addition, Caputo fractional differential equations have the same initial and boundary conditions as the corresponding integer order dynamic equations.…”

Section: Introductionmentioning

confidence: 99%

Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions

Denton

Vatsala

2023

Foundations

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One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard’s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard’s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag–Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag–Leffler function that suffices to establish the generalized Picard’s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard’s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence.

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A Comparative Analysis of the Fractional‐Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

Cited by 48 publications

References 42 publications

Energy management of distribution network with inverter‐based renewable virtual power plant considering voltage security index

Energy management of distribution network with inverter‐based renewable virtual power plant considering voltage security index

Assessing Dynamic Load Allowance of the Negative Bending Moment in Continuous Girder Bridges by Weighted Average Method

Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions

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