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Comparative Implementation of numerical integration methods for Transient Stability Constrained Optimal Power Flow

Comparative Implementation of Numerical Integration Methods ...

IEEE International Conference on Environment and Electrical Engineering (EEEIC) / IEEE Industrial and Commercial Power Systems Europe (ICPS Europe)

作者： Arredondo, Francisco Daniel Castronuovo, Edgardo Ledesma, Pablo Leonowicz, Zbigniew Univ Carlos III Madrid Dept Elect Engn Leganes Spain Wroclaw Univ Sci & Technol Fac Elect Engn Wroclaw Poland

ISBN: (纸本)9781538651865

Transient Stability Constrained Optimal Power Flow (TSCOPF) is a non-linear mathematical programming problem that optimizes the operation of power systems considering steady state and dynamic constrains. TSCOPF models extend the OPF solution by considering the dynamic behavior of the system. One of the main approaches in TSCOPF studies discretizes the differential equations representing the dynamics of the system and includes them in the optimization problem. This paper evaluates the impact of numerical integration methods and time steps on TSCOPF. The results show that: 1) the forward Euler method obtains the most conservative representation for all cases evaluated;2) as the integration time step increases, the backward Euler method and the trapezoidal rule tend to dampen electromechanical oscillations;and 3) the adequate use of a variable integration time step achieves a reduction in the number of equations, variables and computation times, while maintaining the accuracy on the results.

关键词： Power system transient stability numerical integration methods non-linear programming optimal power flow

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On numerical improvement of the Second Kind of Gauss-Chebyshev quadrature rules

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APPLIED MATHEMATICS AND COMPUTATION 2005年第1期168卷 431-446页

作者： Dehghan, M Masied-Jamei, M Eslahchi, MR Amirkabir Univ Technol Dept Appl Math Fac Math & Comp Sci Tehran Iran KN Toosi Univ Technol Dept Math Tehran Iran Minist Sci Res & Technol Ctr Res & Studies Sanjesh Org Tehran Iran

One of the integration methods is the Second Kind of Gauss-Chebyshev quadrature rule, denoted by: integral(1)(-1) f(x)root 1-x(2)dx = pi/n+1 Sigma(n)(k=1)sin(2)(k pi/n+1)f(cos(k pi/n+1)) + n/2(2n+1)(2n)! f((2n))(eta), -1 < eta < 1. According to Gauss quadrature rules, the precision degree of above formula is the highest, i.e. 2n - 1. Hence, it is not possible to increase the precision degree of Second Kind of Gauss-Chebyshev integration formulas anymore. But, on the other hand, we claim that we can improve the above formula numerically. To do this, we consider the integral bounds as two unknown variables. This causes to numerically be extended the monomial space f(x) = x(j) from j = 0, 1,...,2n - 1 to j = 0, 1,...,2n + 1. This means that we have two monomials more than Second Kind Gauss-Chebyshev integration method. In other words, we give an approximate formula as: integral(b)(a) f(x)root 1-x(2)dx similar or equal to Sigma(n)(i=1)w(i)f(x(i)), in which a,b and w(1),w(1),,...,w(n) and x(1),x(2),..,x(n) are all unknowns and the formula is almost exact for the monomial basis f(x) = x(i), j = 0, 1,...,2n + 1. Some important examples are finally given to show the excellent superiority of the proposed nodes and coefficients with respect to the Second Kind Gauss-Chebyshev nodes and coefficients. Let us add that in this part we give also some wonderful 2-point, 3-point and 4-point formulas that are respectively comparable with 103-point, 261-point and 108-point formulas of Second Kind Gauss-Chebyshev quadrature rules in average. (c) 2004 Elsevier Inc. All rights reserved.

关键词： Second Kind Gauss-Chebyshev formula numerical integration methods degree of accuracy the method of undetermined coefficient the method of solving nonlinear systems

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A note on equal coefficient quadrature rules

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APPLIED MATHEMATICS AND COMPUTATION 2006年第1期180卷 153-159页

作者： Hashemiparast, S. M. Eslahchi, M. R. Dehghan, Mehdi Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran Iran KN Toosi Univ Technol Dept Math Tehran Iran Management & Planning Org Tehran Iran

In this paper we introduce an integration method with equal coefficient in the following form: integral(b)(a) w(x)f(x)dx similar or equal to C-n Sigma(n)(i=1) f (x(i)). Then by using the formulas of Newton's equations and degree of precision we introduce a method which express the nodes and coefficients in this integration formulas. Finally some examples are presented to illustrate the procedure. (c) 2006 Published by Elsevier Inc.

关键词： equal coefficients quadrature rules numerical integration methods degree of precision the method of undetermined coefficient

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Modeling and analysis of fit clearance between rolling bearing outer ring and housing

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JOURNAL OF SOUND AND VIBRATION 2019年 438卷 419-440页

作者： Chen, Guo Qu, Meijiao Nanjing Univ Aeronaut & Astronaut Coll Civil Aviat Nanjing Jiangsu Peoples R China

In the assembly and using the process of the rolling bearings in high-speed rotating machines, including aero-engines. Because the change of the bearing temperature and prestressing force of installation, there will inevitably appear fit clearance between the bearing outer ring and the bearing housing. The fit clearance fault may lead to the impact and friction between bearing and bearing housing, a strong non-linear vibration be caused, the rolling bearing and the entire rotor system vibrate strongly, which leads to the service life of the bearing and rotor system reliability reduces greatly. In this study, the rotor-rolling bearing tester is considered to be the research object, and a rotor-rolling bearing dynamic model with the bearing fit clearance fault is established. The time-domain numerical integration method is used to study the influence of factors such as fit clearance, rotor unbalance, and bearing outer ring tightening torque on the vibration response. Finally, the bearing fit clearance fault test is performed to verify the simulation results. The research work in this study is of great significance for understanding, diagnosing and preventing the bearing fit clearance fault between rolling bearing outer ring and housing. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Rolling bearing Bearing fit clearance fault numerical integration methods Dynamic modeling Whole vibration Fault analysis

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Comparison of the trapezoidal rule error with the spline rule error

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期183卷 1256-1261页

作者： Ivaz, K. Taghvafard, H. Univ Tabriz Dept Math Tabriz Iran

关键词： numerical integration methods cubic spline function

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On numerical improvement of the first kind Gauss-Chebyshev quadrature rules

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APPLIED MATHEMATICS AND COMPUTATION 2005年第1期165卷 5-21页

作者： Eslahchi, MR Dehghan, M Masied-Jamei, M Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran 15914 Iran Minist Sci Res & Technol Sanjesh Org Ctr Res & Studies Tehran Iran

One of the integration methods of the equality type is Gauss-Chebyshev quadrature rule, which is in the following form: integral(-1)(1) f(x)/root 1-x(2) dx = pi/n (n)Sigma(k=1)f (cos ((2k - 1)pi/2n)) + 2 pi/2(2n)(2n)!f(eta), -1 < eta < 1. According to Gauss quadrature rules, the precision degree of above formula is the highest, i.e. 2n - 1. Hence, it is not possible to increase the precision degree of Gauss-Chebyshev integration formulas anymore. In this way, we present a matrix proof for this matter. But, on the other hand, we claim that we can improve the above formula numerically. To do this, we consider the integral bounds as two unknown variables. This causes to numerically be extended the monomial space f(x) = X from j = 0, 1,...,2n - 1 to j = 0, 1,...,2n + 1. This means that we have two monornials more than Gauss-Chebyshev integration method. In other words, we give an approximate formula as: integral(h)(a) f(x)/root 1-x(2) dx similar or equal to (n)Sigma(i=l) w(if)(x(i)), in which a,b and and w(1), w(2),..,w(n) and x(1,) x(2),...,x(n) are all unknowns and the formula is almost exact for the monomial basis f(x) = x(i), j = 0, 1,.., 2n + 1. Some important examples are finally given to show the excellent superiority of the proposed nodes and weights with respect to the usual Gauss-Chebyshev nodes and weights. Let us add that in this part we have also some wonderful 2-point formulas that are comparable with 71-point formulas of Gauss-Chebyshev quadrature rules in average. (c) 2004 Elsevier Inc. All rights reserved.

关键词： Gauss-Chebyshev formula numerical integration methods degree of accuracy the method of undetermined coefficient the method of solving nonlinear systems

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On numerical improvement of Open Newton-Cotes quadrature rules

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APPLIED MATHEMATICS AND COMPUTATION 2006年第1期175卷 618-627页

作者： Dehghan, Mehdi Masjed-Jamei, M. Eslahchi, M. R. Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran Iran KN Toosi Univ Technol Dept Math Tehran Iran Management & Planning Org Tehran Iran

In this paper, we discuss about numerical improvement of the Open Newton-Cotes integration rules that are in forms of: integral(b=xn+1=x-1+(n+2)h)(a=x-1) f(x)dx similar or equal to Sigma(n)(k=0)B(k)((n))f(x-1 + (k + 1)h). It is known that the precision degree of above formula is n + 1 for even n's and is n for odd n's. However, if the integral bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of the above integration formula up to degree n + 2. In this way, some numerical tests are given to show the numerical superiority of our idea with respect to the usual Open Newton-Cotes integration rules. (c) 2005 Elsevier Inc. All rights reserved.

关键词： Open Newton-Cotes formula numerical integration methods degree of accuracy the method of undetermined coefficient the method of solving nonlinear systems midpoint rule

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Frequency warping in time-domain circuit simulation

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS 2003年第7期50卷 904-913页

作者： Brambilla, A Storti-Gajani, G Politecn Milan Dipartimento Elettr & Informaz I-20133 Milan Italy

Time-domain simulation of dynamic circuits and, in general, of any physical model characterized by ordinary differential equations or differential algebraic equations, implies the use of so me numerical integration method to find an approximate solution in a discrete set of time points. Among these methods,,the class known as linear multistep includes many well-known formulas such as the backward Euler method, the trapezoid method, and the implicit backward differentiation formulas used in most circuit simulators. All these methods introduce a very subtle effect that-is, here called the warping error. As shown, it is equivalent to a perturbation of the eigenvalues of the linearized ordinary differential problem. The perturbation introduced depends on the integration time step;it is often very small and in most cases irrelevant or even not noticeable. Nevertheless an exception to this situation is found when simulating high-quality factor circuits where even very small warping errors can lead to qualitatively wrong solutions. In this paper, we demonstrate that higher order linear multistep methods, while characterized by weaker stability properties, introduce less of a warping error and are well suited to the simulation of high-quality factor circuits.

关键词： frequency warping linear multistep methods numerical integration methods

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Stability preserving integration of index-2 DAEs

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APPLIED numerical MATHEMATICS 2003年第2-3期45卷 201-229页

作者： Higueras, I März, R Tischendorf, C Humboldt Univ Inst Math D-10099 Berlin Germany Univ Publ Navarra Dept Matemat & Informat Pamplona 31006 Spain

For linear index-2 DAEs with properly stated leading term we characterize contractive and dissipative flows. We study under which conditions the qualitative properties of the DAE solutions are reflected by the numerical approximations. This is the case if the discretization and the decoupling processes commute. Commutativity is achieved if two subspaces associated with the index-2 DAE are constant;in this case we say that the index-2 DAE is numerically qualified. If both subspaces are time dependent, the problem should be reformulated;in, order to avoid numerically equivalent reformulations, a criterion is given. If only one of these subspaces is constant, the problem is in some sense close to a numerically qualified one and thus depending on the problem context, no reformulations are needed. Contractive and dissipative flows induced by the DAE are characterized and results on qualitative properties of the numerical solution are given. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

关键词： differential algebraic equations numerical integration methods global stability BDF Runge-Kutta

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Grid Size Adapted Multistep methods for High Q Oscillators

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IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2013年第11期32卷 1682-1693页

作者： Brachtendorf, Hans Georg Bittner, Kai University of Applied Sciences of Upper Austria Austria

The numerical calculation of the limit cycle of oscillators with resonators exhibiting a high-quality factor Q such as quartz crystals is a difficult task in the time domain. Time domain integration formulas, when not carefully selected, introduce numerical damping that leads to erroneous limit cycles or spurious oscillations. A novel class of adaptive multistep integration formulas based on finite difference (FD) schemes is derived, which circumvent the aforementioned problems. The results are compared with the well-known harmonic balance (HB) technique. Moreover, the range of absolute stability is derived for these methods. The resulting discretized system by FD methods is sparser than that of HB and, therefore, easier to solve and easier to implement.

关键词： High Q oscillators modified backward differentiation formulas (BDF) numerical damping numerical integration methods

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