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In the paper a new grid (potentially linear, nonlinear and even semi-Markovian jump system) was presented. All transition and measurement functions were proposed. Moreover, the transition functions of two types were considered -dependent on one and many different state variables. Also 10 types of measurements were proposed for both nodal and branch cases. Based on the obtained results one can see, which measurement functions are "easy", and which are "hard" for state estimation task.

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In this paper, a new object in the form of a theoretical network is presented, which is useful as a benchmark for particle filtering algorithms designed for multivariable nonlinear systems (potentially linear, nonlinear, and even semi-Markovian jump system). The main goal of the paper is to propose an object that potentially can have similar to the power system grid properties, but with the number of state variables reduced twice (only one state variable for each node, while there are two in the case of power systems). Transition and measurement functions are proposed in the paper, and two types of transition functions are considered: dependent on one or many state variables. In addition, 10 types of measurements are proposed both for branch and nodal cases. The experiments are performed for 14 different, four-dimensional systems. Plants are both linear and highly nonlinear. The results include information about the state estimation quality (based on the mean squared error indicator) and the values of the effective sample size. It is observed how the higher effective sample size resulted in the better estimation quality in subsequent cases. It is also concluded that the very low number of significant particles is the main problem in particle filtering of multivariable systems, and this should be countered. A few potential solutions for the latter are also presented.

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This paper presents a new particle filter algorithm (MultiPDF) for state estimation of nonlinear systems. The proposed method is a modification of the standard particle filter approach. Due to the strong need for the acceleration of calculations and an improvement in the estimation quality of state estimation, the authors propose a method which enables one to divide the main particle filter into smaller sub-filters with an accordingly smaller number of particles for each one of them. The algorithm has been implemented for various numbers of particles and subordinate parallel filters. Estimation quality has been checked for nine nonlinear objects (both one- and multidimensional) and evaluated through the quality index, average root-mean-squared error. The computation time of the particle filter algorithm for several hardware configurations has been compared. Based on the obtained results, it can be concluded that, besides the computation acceleration, the parallelization of the particle filter’s operation also improves the estimation quality.