A First Course in the Numerical Analysis of Differential Equations: Gaussian elimination for sparse linear equations

We come now to the connecting link between linear transformations and matrices. Our approach will be to consider first the case of a linear transformation 

$$ {\text{T}}:{R^3} \to {R^3} $$

in some detail and then abstract the salient features to the general case. Let us therefore suppose given a fixed linear transformation $ {\text{T}}:{R^3} \to {R^3} $. As usual we will denote by E1, E2, E3 the standard basis vectors (1,0,0), (0, 1, 0), (0, 0, 1) in ℝ3. Let

$${\text{T}}({{\text{E}}_1}){\text{ = }}({a_{11}},{a_{21}},{a_{31}}){\text{T}}({{\text{E}}_2}){\text{ = }}({a_{12}},{a_{22}},{a_{32}}){\text{T}}({{\text{E}}_3}){\text{ = }}({a_{13}},{a_{23}},{a_{33}})$$

.

Matrices and linear transformations

The Universal Line Model (ULM) is among the most accurate frequency-dependent transmission line models available in Electromagnetic Transients Program-type simulation tools. One major drawback of this line model is that it sometimes gives unstable simulation results. The instability is related to the occurrence of close poles in the rational model of the propagation function which leads to large residue-pole ratios. In a time domain simulation, these large ratios give a magnification of the error associated with the interpolation of the reflected current wave which acts as the stimulus of the propagation function. An approach is described for avoiding the instability problem by introducing a two-segment interpolation scheme for the extraction of the current wave. The approach gives zero interpolation error when used together with the integration scheme known as recursive convolution and so the error magnification becomes inconsequential. The new approach is demonstrated for pertinent examples, including one case with residue-pole ratios exceeding one-million.

Avoiding numerical instabilities in the universal line model by a two-segment interpolation scheme

This paper proposes applying the 2-Stage Diagonally Implicit Runge-Kutta (2S-DIRK) method of numerical integration to the calculation of electromagnetic transients (EMTs) in a power system. The accuracy and the numerical stability of 2S-DIRK are almost the same as those of the trapezoidal method, while 2S-DIRK does not produce sustained numerical oscillation due to a sudden change of an inductor current or a capacitor voltage unlike the trapezoidal method. Firstly, this paper reviews the 2S-DIRK integration scheme and derives the 2S-DIRK formulas of inductors and capacitors for both linear and nonlinear cases. Then, analytical comparisons of 2S-DIRK with the trapezoidal, backward Euler, and Gear-Shichman methods are carried out, and numerical examples which verify the analytical comparisons are shown. Finally, 2S-DIRK is compared with CDA (Critical Damping Adjustment) implemented in EMTP (Electro-Magnetic Transients Program) for some simulation cases.

Numerical integration by the 2-stage diagonally implicit Runge-Kutta method for electromagnetic transient simulations

This paper describes the latest innovations and experimental results for assured time and location based on Low Earth Orbit (LEO) satellites.
Interest is growing worldwide to identify additional sources of PNT to augment existing GNSS sources. Industry and government groups are actively seeking independent timing solutions to ensure uninterrupted operation of the critical infrastructure, including data centers, financial markets, cellular networks, and power distribution. Military users are also interested in robust and reliable geolocation capabilities, even in environments where GPS or GNSS is not available. Commercial and academic entities serving these industries are actively exploring a wide range of technologies to support a comprehensive solution for assured PNT.
Results from three separate user equipment configurations are presented – each of which is intended to increase the real-time performance of the Satellite Time and Location (STL) system, a non-GNSS solution for assured time and location that is highly resilient and physically secure. STL uses the Iridium constellation of 66 communications satellites in Low Earth Orbit (LEO) to transmit specially structured time and location broadcasts. Due to their high RF power and signal coding gain, the STL broadcasts are able to penetrate into difficult attenuation environments, including deep indoors. Like GNSS signals, these broadcasts are specifically designed to allow a receiver to obtain precise time and frequency measurements to derive its position, navigation, and time (PNT). STL is able to augment or serve as a back-up to existing GNSS PNT solutions by providing secure measurements in the presence of high attenuation (deep indoors), active jamming, and/or malicious spoofing.
The first configuration substituted a rubidium clock (a very stable frequency source based on atomic physics rather than quartz crystals) for the Temperature Compensated Crystal Oscillator (TCXO) used as the STL receiver’s timebase during earlier experiments. Using the rubidium clock, the STL receiver demonstrated a Maximum Time Interval Error (MTIE) of 170 nanoseconds over a two-week period at a static location which was precisely known a priori.
The second configuration utilized the same hardware setup, with the receiver configured to a static location that was unknown, and with a rubidium clock that was still warming up 6 hours after a cold start. The demonstrated MTIE for this configuration was 420 nanoseconds over a seven-day period.
The third configuration utilized a nearby STL reference station to transmit localized error measurements to the STL ground segment. The reference receiver used a GPS-derived timebase that was assumed to be perfect. Timing results from 3 receivers, with different internal OCXO oscillators, are presented.

Recent PNT Improvements and Test Results Based on Low Earth Orbit Satellites

/pdf/use-of-traveling-wave-signatures-in-medium-voltage-3d5gt6vdl5.pdf

Use of Traveling Wave Signatures in Medium-Voltage Distribution Systems for Fault Detection and Location

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Use of Traveling Wave Signatures in Medium-Voltage Distribution Systems for Fault Detection and Location