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The spatial correlation properties in the geometrical focal region of a converging, partially coherent vortex wave field are analyzed. Expressions are derived for a pair of points on the axis of symmetry and for a pair of points in the focal plane. It is found that the longitudinal and transverse coherence lengths in the focal region change with the variation of the topological charge and the normalized coherence length of the vortex field. In addition, the degree of coherence is shown to possess phase singularities.

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This paper presents a loss allocation method based on the circuit theories and the concept of orthogonal projection for pool-based electricity markets. The method first defines the current projection component to determine the share of a generator or load on the current through a branch. Then the power flow through each branch is decomposed and the branch loss allocation is obtained. One feature of the method is that it gives intuitively clear explanation of the obtained loss allocation. The two equivalence modes of bus power injections are also discussed. The analysis shows that the equivalence that loads (generators) are converted into equivalent admittances when generators (loads) are converted into current injections reflects the practical situation and thus it is more reasonable and preferable. Comparisons with other methods and validation tests of the proposed method are provided by using several numerical examples.