Simplified analytic stability test for systems with commensurate time delays

“…A number of authors [1][2][3][4][5][6] have discussed tests for stability independent of delay for retarded delay-differential equations containing several delay terms, each delay being an integer multiple of a single delay T (commensurate delays):…”

Section: Introductionmentioning

confidence: 99%

An interference effect of independent delays

MacDonald

1987

IEE Proc. D Control Theory Appl. UK

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For a second-order linear retarded delay-differential equation with two independent delays, two methods are used to demonstrate an interference effect that can reduce the likelihood of instability. Both methods rely on the basic result that stability changes only take place at pure imaginary values of the complex eigenvalue. One method uses the eliminant, which is zero if and only if both real and imaginary parts of the characteristic equation are satisfied. The other is an extension of a method used in problems with one delay, in which a certain curve is plotted to find whether it intersects the unit circle. When this interference effect is present, equal delays are particularly unlikely to destabilise. IntroductionA number of authors [1-6] have discussed tests for stability independent of delay for retarded delay-differential equations containing several delay terms, each delay being an integer multiple of a single delay T (commensurate delays):Here the characteristic equation for stability takes the form of a polynomial equation in two complex variables, the eigenvalue z and u = e~z T :The methods used in general rely on the result [1-4] that any change of sign of the real part of a root z of eqn. 2 must take place at z = + ico, co real (and not at z = 0 or \z\ = oo), and so with the delay appearing in a phase factor u = e~i e , 9 = ±coT. Stability independence of delay is ensured if (a) Q(z, 1) = 0 has no root with Re z > 0 (b) Q(ico, u) = 0 has no root with \u\ = 1. Now if the physical system modelled by eqn. 1 derives all its delay features from a single type of component or a single physical effect operating in several parts of the system, then the assumption of commensurate delays is reasonable. Otherwise, it is made only because this last result is needed to carry out the analysis. We shall show that this assumption can lead to an incomplete picture of the delay effects. Recently Hale et al. [7] have proved that, for the more Paper 5009D (C8), first received 7th February and in revised form 13th June 1986 The author is with the Department of Natural Philosophy, The University, Glasgow G12 8QQ, United Kingdom.38 general characteristic equation, Q(z, u, v, ..) = 0 (3) with u = e zT , v = e zS , ..., it again suffices to examine the requirement that Q{ico, u, v, ...) = 0 for | u | = | v \ = • • • = 1 (4) where there are now two or more independent phase factors: u = e~i e , v = e~i 4> , ... with 9 = coT, cf> = coS,... So, if a convenient method can be found for studying commensurate delays, it should be possible to extend it to accommodate independent delays. Hertz et al. [4] provide such a method, using the eliminant of two polynomials in z and u. These authors point out that, for z = ico and \u\ = 1, implies Q*(z,u) = Q(-z,u-1 ) = 0 which gives a second polynomial equation: u m Q(-z,u' 1 ) = 0 (5)The eliminant of z from eqns. 5 and 6 is a function of u, which is zero if and only if both these equations are satisfied. [Hertz et al. [4] use the eliminant of u (in their notation z) from eqns. 5 and 6, which...

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“…The characteristic equation of such a system is given by Equation (7). Similar to the single-delay case, if the characteristic equation (7) 22) Recall that in the single-delay case, the transcendental term and the time delay t are eliminated. In this case, the purpose is the same.…”

Section: Commensurate-delay Casementioning

confidence: 97%

Computation of time delay margin for power system small‐signal stability

Ayasun¹

2008

Int Trans Elec Energy Syst

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SUMMARYWith the extensive use of phasor measurement units (PMU) in the wide-area measurement/monitoring systems (WAMS), time delays have become unavoidable in power systems. This paper presents a direct and exact method to compute the delay margin of power systems with single and commensurate time delays. The delay margin is the maximum amount of time delay that the system can tolerate before it becomes unstable for a given operating point. First, without using any approximation or substitution, the transcendental characteristic equation is converted into a polynomial without the transcendentality such that its real roots coincide with the imaginary roots of the characteristic equation exactly. The resulting polynomial also enables us to easily determine the delay dependency of the system stability and the sensitivities of crossing roots with respect to time delay. Then, an expression in terms of system parameters and imaginary root of the characteristic equation is derived for computing the delay margin. The proposed method is applied to a single-machine-infinite bus (SMIB) power system with an exciter. Delay margins are computed for a wide range of system parameters including generator mechanical power, damping and transient reactance, exciter gain, and transmission line reactance. The results indicate that the delay margin decreases as the mechanical power, exciter gain and line reactance increase while it increases with increasing generator transient reactance Additionally, the relationship between the delay margin and generator damping is found be relatively complex. Finally, the theoretical delay margin results are validated using the time-domain simulations of Matlab.

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Simplified analytic stability test for systems with commensurate time delays

Cited by 61 publications

References 1 publication

Direct method for TDS stability analysis

Direct method for TDS stability analysis

An interference effect of independent delays

Computation of time delay margin for power system small‐signal stability

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