Step 1: Short-listing Potential Meter Position

A singular value of a matrix is an indication of its scaling property. Performing Singular Value Decomposition (SVD) of a matrix , the following equation is obtained,

and are orthogonal or unitary matrices. The singular value matrix is expressed by , which is a diagonal matrix consisted of non-negative real numbers (where, ). The diagonal elements of are called singular values [ conventionally arranged in descending order. Therefore the first element is the largest singular value of matrix . The largest singular value is a measure of energy preserved by the matrix; it also represents the spectral norm of , The role of singular value is quite significant when a matrix represents a transformation to another vector space for an under-determined or over-determined problem. The linear mapping imposed by the matrix on a vector to vector where can be decomposed as in Fig. 6. 1(a). and represent the gain at input and output directions respectively. The singular value matrix scales the magnitude. Further illustration of a geometrical interpretation of the scaling property of singular value is given in Fig. 6. 1(b). Here = 2 for matrix , that transforms a unit sphere to an ellipsoid. Fig. 6. 1(b) shows that the axes and are

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rotated by , then stretched by forming an ellipse and finally rotates the ellipse to its final direction. These considerations imply that the maximum singular value of a matrix is a measure of its ability to expand or contract a vector it is mapping. The larger/smaller is the value of maximum singular value, the higher is the scale of expansion/contraction.

(a) (b) Fig. 6. 1: Mapping effect and geometrical interpretation of SVD

The scaling impact of the maximum singular value of a matrix has been exploited in the SE problem to search for potential locations to deploy additional sensors. In the SE optimization, the Gauss-Newton linearization problem enables the SE to reduce the sum of measurement residuals in every iteration of (3.11). At the end of recursion process, , as well as , should be as small as possible. A multiplying factor, is introduced here, where _{. Replacing in (6.2), the following}

(6.5) is obtained.

Applying SVD (6.4) to in (6.5) for a network that has M measurements and K states and M ≥ K, the following expressions in (6.6) and (6.7) can be written.

x VT_x Ax ƩVT_x VT Ʃ U

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Here, . The term consists of various types of

measurements having different scales of magnitudes. The voltage measurements are in kV and close to unity in the per unit system; whereas the power measurements are in MW/MVAr and they may vary down to small fractional numbers in per unit values. As is not normalized but rather calculates algebraic differences between the measurements and estimated values, it does not provide complete information about the degree of reduction of measurement residuals corresponding to each measurement. When alternative measurement configurations are to be considered and it has been checked which configuration is most effective to reduce the measurement residuals; instead of examining directly, it would be more useful to observe the transformation effect of on for various measurement configurations. Since the Gauss-Newton WLS method approaches convergences by reducing measurement residuals, the mapping effect on is expected to be contracting in the last iteration. The maximum singular value of can be considered as a measure of the mapping effect on as explained in Fig. 6. 1. If has a small value, the contraction effect on residual vector will be greater.

In addition to the scaling property, the is also useful as an indication of the sensitivity

of a measurement configuration. In (6.6) and (6.7), the perturbation of is mostly affected by the largest component, from the SVD analysis. The greatest changes in will occur to the direction of and in proportion to . As both and set

the direction of changes, is the magnitude of sensitivity of estimation to . The estimated values become less sensitive to the residuals by achieving smaller values of

. The sensitivity of a measurement configuration is important for the distribution

system SE as there will be only few real measurements and placing a sensor on a highly sensitive node may generate erroneous estimation when the sensor provides inaccurate information. Although selection of a more sensitive node to achieve greater effect from the more accurate data expected from real measurements apparently seems rational, that may be true only when there exists an adequate amount of real measurement data. Having an erroneous input can be compensated by many other real measurement values even though

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the flawed data is introduced at a sensitive node. This would not be the case with regard to distribution system scenarios. In the presence of fewer real measurements compared to the amount of pseudo-measurement data, the DSSE optimizer will be greatly influenced by those greater weighted real time data. The probability of having a strong negative effect from erroneous real measurement values is much higher when that is generated at a sensitive node. Therefore, selecting a node having less sensitivity to the SE outcomes is more reasonable for measurement placement when real time measurements data are limited. In essence, selecting a measurement configuration that gives the minimum value of the maximum singular value assists the estimator to achieve reduced estimation errors in two ways: (a) by reducing measurement residuals to achieve good convergence and (b) by selecting less sensitive measurement setups to minimize the adverse effect of erroneous data. The summary of the procedure of short-listing potential meter positions is as follows.

Suppose, there are C available candidate meter positions (MC) to introduce sensors. The

algorithm adds sensors in one place at a time and then performs SE; followed by recording in the last Gauss-Newton iteration step. Addition of sensors at each candidate position thus corresponds with a singular value and a vector of with i = 1 to C is generated after performing SE for all MC. After surveying the magnitude of each element of ,

the algorithm detects those having smaller values. P number of locations are suggested out of MC as potential meter positions (MP), since the measurement setup including real time

data from those positions, attributes smaller values of . Background surveys imply that consideration of the 20-30% of all MC that generate comparatively smaller values of

maximum singular value, is sufficient for the most beneficial meter location (MB)to occur

in the short-listed MP vector.