Robust complex-valued Levenberg-Marquardt algorithm as applied to power flow analysis

Pires, Robson Celso; Mili, Lamine; Chagas, Guilherme

doi:10.1016/j.ijepes.2019.05.032

Cited by 20 publications

(18 citation statements)

References 27 publications

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“…As a further advantage provided by the Wirtinger calculus [2,3], the Jacobian matrix which emerged in Cartesian coordinates needs lesser algebra task as well as minor implementation effort (encoding) than the former procedure in real domain [14]. Thereby, the Jacobian matrix in expanded form may be represented through four partitions matrix, yielding…”

Section: Complex-valued Newton-raphson Methodsmentioning

confidence: 99%

“…Firstly, as the former, it occurs over the corrections to be applied to the state variables and simultaneously over the mismatches vector. This latter is included, aiming to be aware against ill-conditioned systems [14], yielding…”

Section: Jacobian Matrix Factorizationmentioning

confidence: 99%

“…This latter task can be mandatory or not once the Jacobian matrix may be kept constant throughout the iterative process (approximate, instead of full gain) which is a decision very often adopted after the second iteration aiming to lighten the computational burden. Further details can be found in [14], but in the sequence, a small example is presented forwarded of simulations carried out on large systems. However, as any other application in the industry, the power flow model lies in the solution of a system of linear algebraic equations which is summarized thereafter.…”

Section: Jacobian Matrix Factorizationmentioning

confidence: 99%

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Solution Methods of Large Complex-Valued Nonlinear System of Equations

Pires¹

2020

Advances in Complex Analysis and Applications

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Nonlinear systems of equations in complex plane are frequently encountered in applied mathematics, e.g., power systems, signal processing, control theory, neural networks, and biomedicine, to name a few. The solution of these problems often requires a first-or second-order approximation of nonlinear functions to generate a new step or descent direction to meet the solution iteratively. However, such methods cannot be applied to functions of complex and complex conjugate variables because they are necessarily nonanalytic. To overcome this problem, the Wirtinger calculus allows an expansion of nonlinear functions in its original complex and complex conjugate variables once they are analytic in their argument as a whole. Thus, the goal is to apply this methodology for solving nonlinear systems of equations emerged from applications in the industry. For instances, the complexvalued Jacobian matrix emerged from the power flow analysis model which is solved by Newton-Raphson method can be exactly determined. Similarly, overdetermined Jacobian matrices can be dealt, e.g., through the Gauss-Newton method in complex plane aimed to solve power system state estimation problems. Finally, the factorization method of the aforementioned Jacobian matrices is addressed through the fast Givens transformation algorithm which means the square root-free Givens rotations method in complex plane.

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Section: Complex-valued Newton-raphson Methodsmentioning

confidence: 99%

Section: Jacobian Matrix Factorizationmentioning

confidence: 99%

Section: Jacobian Matrix Factorizationmentioning

confidence: 99%

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Solution Methods of Large Complex-Valued Nonlinear System of Equations

Pires¹

2020

Advances in Complex Analysis and Applications

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“…The Levenberg-Marquardt algorithm applies the Jacobian and Hesse matrices to solve multidimensional optimization problems [52,53], whose principle is as follows:…”

Section: Construction Of the Neural Networkmentioning

confidence: 99%

Compressive Strength Prediction of PVA Fiber-Reinforced Cementitious Composites Containing Nano-SiO2 Using BP Neural Network

et al. 2020

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In this study, a method to optimize the mixing proportion of polyvinyl alcohol (PVA) fiber-reinforced cementitious composites and improve its compressive strength based on the Levenberg-Marquardt backpropagation (BP) neural network algorithm and genetic algorithm is proposed by adopting a three-layer neural network (TLNN) as a model and the genetic algorithm as an optimization tool. A TLNN was established to implement the complicated nonlinear relationship between the input (factors affecting the compressive strength of cementitious composite) and output (compressive strength). An orthogonal experiment was conducted to optimize the parameters of the BP neural network. Subsequently, the optimal BP neural network model was obtained. The genetic algorithm was used to obtain the optimum mix proportion of the cementitious composite. The optimization results were predicted by the trained neural network and verified. Mathematical calculations indicated that the BP neural network can precisely and practically demonstrate the nonlinear relationship between the cementitious composite and its mixture proportion and predict the compressive strength. The optimal mixing proportion of the PVA fiber-reinforced cementitious composites containing nano-SiO2 was obtained. The results indicate that the method used in this study can effectively predict and optimize the compressive strength of PVA fiber-reinforced cementitious composites containing nano-SiO2.

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“…The use of Wirtinger calculus has been recently gaining popularity in applications of power flow computation, as the underlying variables are naturally complex-valued. The applications are mainly in power flow [31] - [35] and transmission network state estimation [36]. The paper describes the formulation of multi-phase DSSE in the complex domain for the first time, and unravels the advantages of loop unrolling in its computer implementation.…”

Section: Introductionmentioning

confidence: 99%

Complex Variable Multi-phase Distribution System State Estimation Using Vectorized Code

Džafić

Jabr

Hrnjic

2020

Journal of Modern Power Systems and Clean Energy

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With the advent of advanced energy management systems in distribution systems, there is a growing interest in rapid and reliable code for distribution system state estimation (DSSE) in large-scale systems. Fast DSSE methods employed in the industry are based on load scaling as they are well suited to the abundance of pseudo-measurements. Due to the paucity of real-time measurements in DSSE, phasor measurement units (PMUs) have been proposed as a potential solution to increase the estimation accuracy. However, load scaling methodologies are not extendable for exploiting PMUs. This paper proposes a high-performance DSSE method that can handle the PMUs together with all common measurement types in industrial DSSE. By using Wirtinger calculus, the method operates entirely in complex variables and employs the latest version of advanced vector extensions (AVX-2) to reap the maximum potential of computer processing units. The paper highlights the derivation of complex DSSE in matrix form, from which one can infer the implications on code reliability and maintenance. Numerical results are reported on large-scale multi-phase distribution systems, and they are contrasted with a publicly available code for DSSE in real variables. The simulation results show that loop unrolling in AVX-2 contributes about a twofold increase in the solving speed.

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Robust complex-valued Levenberg-Marquardt algorithm as applied to power flow analysis

Cited by 20 publications

References 27 publications

Solution Methods of Large Complex-Valued Nonlinear System of Equations

Solution Methods of Large Complex-Valued Nonlinear System of Equations

Compressive Strength Prediction of PVA Fiber-Reinforced Cementitious Composites Containing Nano-SiO2 Using BP Neural Network

Complex Variable Multi-phase Distribution System State Estimation Using Vectorized Code

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