This paper deals with the lacement of a minimal set of phasor measurement units PMU9sT so as to make the system placed at a bus measures the voltage as well as all the current phasors at that bus, requiring the extension of the topological observability theory. This concerns the extension of the concept of spanning tree'to that of spanning measurement subgraph with an actual or a pseudo-measurement assi ned to each of its branches. The minimal PMU set is found t%rough a dual search algorithm which uses both a modified bisecting search and a simulated annealing-based method. The former fixes the number of PMU's while the latter looks for a placement set that leads to an observable network for a fixed number of PMU's. In order to accelerate the procedure, an initial PMU placement is provided by a graph-theoretic procedure which builds a spanning measurement subgraph according to a depth-first search. From computer simulation results performed on various test systems it appears that only one fourth to one third of the system buses need to be provided with PMU% in order to make the system observable. measurement model observabe, \ and thereby linear. A PMU -INTRODUCTIONPresently, the supervision of a power system is performed through an open-loop type centralized control. The control actions are taken by the operators with the help of computer-aided software programs that implement steadystate security functions 1). This is due to the fact that the to capture only quasi-steady state operating conditions, preventin the monitoring of transient phenomena. With the advent o f real-time Phasor Measurement Units PMU's), fast transients can be tracked at high sampling rates 121. Hence, it becomes possible to close the loop, that is, to perform an automatic monitoring and control of the system. This is a faster-than-real-time control that aims at steering the system away from transient or voltage instability through corrective actions initiated during an emergency state. A prerequisite to system monitoring and control is the development of an adequate meter placement scheme. Various lacement methodologies have been proposed in the literature P 2-61. Most of them advocate the use of pilot points located at the center of the coherent regions of a system. These re 'om either contain load buses with similar voltage trends for vogage stability analysis or encompass groups of stiffly interconnected machines with common slow modes of oscillations for transient stability analysis. The only control that has so far been implemented is the secondary voltage control scheme which has been applied to the French [5] and Italian systems [ 6 . Two measurements collected t 6, ough a SCADA system are designed major drawbacks of the coherency approach may be fl oreseen. SM 583-5 PWRS A paper recommended and approved by the IEEE Power System Engineering committee of the IEEE Power Engineering Society for presentation at the IEEE/First, the system may not be decomposable into meaningful clusters, signifying the necessity to monitor all load buses...

Expressions are developed for the components of the linear Lagrangian, linear Eulerian, finite Lagrangian (Green's), and finite Eulerian (Almansi's) strain tensors in terms of a crystal's lattice parameters before and after a deformation. The development has been undertaken with the concepts and notations of linear algebra.

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Several hypothetical silica structures have been generated using a simulated-annealing strategy with an ab initio based covalent-bonding potential. First-principles total-energy pseudopotential methods have been used to examine several promising hypothetical structures and to compare their structural parameters, cohesive energies, and bulk moduli with those of low quartz, low cristobalite, silica sodalite, and stishovite. The cohesive energies of these hypothetical structure types are found to be equivalent to those of low quartz, low cristobalite, and silica sodalite, and significantly lower than that of stishovite.

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