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Publication# Probabilistic assessment of the process-noise covariance matrix of discrete Kalman filter state estimation of active distribution networks

Abstract

The accuracy of state estimators using the Kalman Filter (KF) is largely influenced by the measurement and the process noise covariance matrices. The former can be directly inferred from the available measurement devices whilst the latter needs to be assessed, as a function of the process model, in order to maximize the KF performances. In this paper we present different approaches that allow assessing the optimal values of the elements composing the process noise covariance matrix within the context of the State Estimation (SE) of Active Distribution Networks (ADNs). In particular, the paper considers a linear SE process based on the availability of synchrophasors measurements. The assessment of the process noise covariance matrix, related to a process model represented by the ARIMA [0,1,0] one, is based either on the knowledge of the probabilistic behavior of nodal network injections/absorptions or on the a-posteriori knowledge of the estimated states and their accuracies. Numerical simulations demonstrating the improvements of the KF-SE accuracy achieved by using the calculated matrix Q are included in the paper. A comparison with the Weighted Least Squares (WLS) method is also given for validation purposes.

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Related concepts (4)

Related publications (1)

Kalman filter

For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

Phasor measurement unit

A phasor measurement unit (PMU) is a device used to estimate the magnitude and phase angle of an electrical phasor quantity (such as voltage or current) in the electricity grid using a common time source for synchronization. Time synchronization is usually provided by GPS or IEEE 1588 Precision Time Protocol, which allows synchronized real-time measurements of multiple remote points on the grid. PMUs are capable of capturing samples from a waveform in quick succession and reconstructing the phasor quantity, made up of an angle measurement and a magnitude measurement.

Electric power distribution

Electric power distribution is the final stage in the delivery of electricity. Electricity is carried from the transmission system to individual consumers. Distribution substations connect to the transmission system and lower the transmission voltage to medium voltage ranging between 2kV and 33kV with the use of transformers. Primary distribution lines carry this medium voltage power to distribution transformers located near the customer's premises.

The evolution from passive to Active Distribution Networks (ADNs) is producing large changes in the operation of these electrical systems. In particular, violations of grid operational constraints, higher dynamics and limited amount of controllable resources represent main limiting factors in the optimal operation of ADNs in presence of massive stochastic distributed generation. In order to deal with these issues, the emergence of ADNs requires the definition of suitable Energy Management Systems to achieve specific operation objectives (i.e., optimal voltage/congestion controls, updated protection schemes etc.). These functions are significantly improved if the system state is known with high accuracy, high refresh rates and low time latencies. Unfortunately, typical refresh rates of traditional State Estimation (SE) processes designed for transmission networks are in the order of few minutes, whereas the time frames of the above functionalities are between few milliseconds to few seconds. Hence, it becomes necessary to define, develop and validate the three-phase Real-Time State Estimation (RTSE) processes characterised by high refresh rates (i.e., in the range of several tens of estimation per second), small latencies (i.e., in the range of few tens of ms) and high accuracy. In this direction, the ADNs SE is facilitated by the emerging technology of Phasor Measurement Units (PMUs) which allow acquiring accurate, time-aligned phasors with typical streaming rates in the order of some tens of f.p.s. Additionally, PMUs measurements of synchrophasors allow formulating the SE problem in a linear way. PMU measurements can be acquired and stored in a real-time (RT) database, provided by Phasor Data Concentrators (PDCs) suitably coupled with the RTSE. This enables, in theory, the performance assessment of the whole RTSE chain. However, the assessment of the RTSE accuracy with real PMUs in a real grid is impossible since the true state is hidden. It is, yet, possible to overcome this limitation by using a Real Time Simulator (RTS) and design a RT setup that allows knowing the true state. This RT setup should be GPS-synchronized, to enable the RTSE accuracy and time latencies assessment. Within the above context, this thesis focuses on the definition of SE methods together with their formal and numerical performance assessment. In particular, the first part of the thesis discusses the formulation of static (e.g., weighted least squares - WLS) and recursive (e.g., Kalman Filter - KF) algorithms fed by synchronized phasors and/or other traditional measurements provided by remote terminal units. Then, the thesis discusses the formal comparison of the accuracy of KF vs. WLS and proves that the former behaves better if its process model is correct. For the case of linear SE, the thesis discusses the method for the verification of the exactness of the so-called measurement noise covariance matrix. The subsequent part of the thesis provides the numerical validation and performance assessment of the RTSE process via offline simulations. This analysis is conducted by using IEEE benchmark distribution and transmission networks as well as real distribution feeders. The last part of the thesis focuses on the experimental validation of the RTSE chain via an experimental RT setup. In this last part, the thesis describes the structure and the individual components simulated in the RT experimental setup as well as the whole validation procedure.