Choosing starting values for certain Newton–Raphson iterations

Kornerup, Peter; Muller, Jean-Michel

doi:10.1016/j.tcs.2005.09.056

Cited by 32 publications

(22 citation statements)

References 15 publications

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“…e Householder method [22] is a numerical algorithm for solving the nonlinear equation such as Colebrook's. During the Householder procedure, in successive calculation, that is, in iterative cycles, the original assumed value of the unknown quantity (the initial starting point [50]) needs to be brought as much as possible close to the real value of the quantity using the least possible number of iterations. e same situation is with the three-point methods [28].…”

Section: Iterative Methods Adopted For the Colebrook Equationmentioning

confidence: 99%

“…e starting point is a significant factor in convergence speed in the three-point and the Householder methods [50], and there are the different methods to choose a good start, but here we examine (1) starting point as function of the input parameters and (2) initial starting point with the fixed value.…”

Section: Initial Estimate Of Starting Point For the Iterative Proceduresmentioning

confidence: 99%

“…In this paper, we show the three-point and the Householder iterative procedures (the 3rd order, the 2nd order: Halley's [49] and Schröder's method, and the 1st order: Newton-Raphson) with the additional recommendations in order to solve the empirical Colebrook equation implicitly given in respect of the flow friction factor λ. e goal of this paper is to show the improved iterative solutions which can obtain the value of the unknown friction factor λ accurately after the least possible number of iterations. Additionally, we developed a strategy how to choose the best starting point [50] for the iterative procedure in the domain of interest of the Colebrook equation, how to generate required symbolic derivatives to the Colebrook equation in MATLAB, and how to avoid use of the derivatives (secant method). Finally, we use findings from our paper to present a novel explicit approximation of the Colebrook equation, which would be interesting for engineering practice.…”

Section: Introductionmentioning

confidence: 99%

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Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Praks

Brkić

2018

Advances in Civil Engineering

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The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 < Re < 108) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

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Section: Iterative Methods Adopted For the Colebrook Equationmentioning

confidence: 99%

Section: Initial Estimate Of Starting Point For the Iterative Proceduresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Praks

Brkić

2018

Advances in Civil Engineering

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“…Choosing starting points for the NR method that deviate too far from the sought solution may cause the iterative method to diverge [Kornerup and Muller 2006]. Even when choosing appropriate starting values, the iterative method may be insufficient for timely detection of congestions in highly dynamic scenarios due to the number of (possibly computational complex and time consuming) iterations until the algorithm converges.…”

Section: Introductionmentioning

confidence: 99%

“…This is a sharp contrast to conventional power flow calculations, which precisely determine the state of the network on the basis of correct data for the (deterministic) behavior of every conventional generator and load connected to the network. Conventional power flow calculation by design is not able to cope with fuzzy input data and is very sensitive to misguesses, in the sense that a poorly chosen value in a scenario-based congestion analysis may lead to false results and ultimately to congestions not being detected [Kornerup and Muller 2006]. Probabilistic load flow calculation can tolerate this up to a certain extend without the result becoming useless.…”

Section: Introductionmentioning

confidence: 99%

Load Flow with Uncertain Loading and Generation in Future Smart Grids

Krause

Lehnhoff²

2011

Soft Computing in Green and Renewable Energy Systems

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Abstract. The growing amount of renewable and fluctuating energy sources for the production of electrical energy increases the volatility and level of uncertainty in the operation of power systems. Whether it is the growing number of photovoltaic installations harnessing solar energy or large-scale wind farms, these new class of environmentally dependent appliances increase the unpredictability of load situations hitherto known only from consumer behavior. One of the mayor concerns in grid operation under increasing feed-in from unpredictable generation and consumption is the detection of peaks in network strain. In order to limit investments into grid infrastructure to a reasonable level node-specific limitations for power injections are introduced to reduce the probability of such peaks that may pose a threat to a stable operation of the power system. In order to support the ongoing integration of renewable generation into the grid, a trade-off has to be found between investment costs and imposed operational constraints. In order to determine the probability of congestions under these unpredictable conditions, mathematical algorithms are employed that are able to estimate the probability of certain line loading levels from the probabilistic data derived from the appliances' behavior. This chapter will cover a variety of approaches to solve (probabilistic) load flow problems, ranging from currently deployed state-of-the-art procedures to the newest advances in probabilistic load flow calculation and determination. Advantages and drawbacks of those methods are discussed in detail.

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