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Inequalities are frequently used in various fields of mathematics to prove theorems. The existence of inequalities contributes significantly to the foundations of such branches. In this paper, we study the properties of order relations in the system of dual numbers, which is inspired by order relations defined on real numbers. Besides, some special inequalities that are used in various fields of mathematics, such as Cauchy-Schwarz, Minkowski, and Chebyshev are studied in this framework. An example is also provided to validate our research findings. [ABSTRACT FROM AUTHOR]

Modern power systems are highly complicated and nonlinear in nature. Accurate estimation of the power system states (voltage-magnitude and phase-angle) is required for the secure operation of the power system. The presence of bad-data measurements in meters has made this estimation process challenging. An efficient estimator should detect and eliminate the effect of bad data during the estimation process. Least measurement rejected (LMR) is a robust estimator that has been found successful in dealing with various categories of bad data. The performance of LMR depends upon the proper selection of a tolerance for each measurement. This paper presents a novel approach for tolerance value selection to improve the capability of handling different single and multiple bad-data scenarios successfully. The performance of this updated LMR (ULMR) is compared with weighted least squares, weighted least absolute value, and two versions of LMR from the literature. IEEE 30-bus and 118-bus systems are used to demonstrate the robustness of the proposed estimator under different bad-measurement (single and multiple) scenarios. [ABSTRACT FROM AUTHOR]