Rotation Representations and Their Conversions

A rigid body motion, which can be decomposed into rotation and translation, is essential for engineers and scientists who deal with moving systems in a space. While translation is as simple as vector addition, rotation is hard to understand because rotations are non-Euclidean, and there are many ways to represent them. Additionally, each representation comes with complex operations, and the conversions between different representations are not unique. Therefore, in this tutorial we review rotation representations which are widely used in industry and academia such as rotation matrices, Euler angles, rotation axis-angles, unit complex numbers, and unit quaternions. In particular, for better understanding we begin with rotations in a two dimensional space and extend them to a three dimensional space. In that context, we learn how to represent rotations in a two dimensional space with rotation angles and unit complex numbers, and extend them respectively to Euler angles and unit quaternions for rotations in a three dimensional space. The definitions and properties of mathematical entities used for representing rotations as well as the conversions between various rotation representations are summarized in tables for the reader’s later convenience.

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On Online Adaptive Direct Data Driven Control

Based on our recent contributions on direct data driven control scheme, this paper continues to do some new research on direct data driven control, paving another way for latter future work on advanced control theory. Firstly, adaptive idea is combined with direct data driven control, one parameter adjustment mechanism is constructed to design the parameterized controller online. Secondly, to show the input-output property for the considered closed loop system, passive analysis is studied to be similar with stability. Thirdly, to validate whether the designed controller is better or not, another safety controller modular is added to achieve the designed or expected control input with the essence of model predictive control. Finally, one simulation example confirms our proposed theories. More generally, this paper studies not only the controller design and passive analysis, but also some online algorithm, such as recursive parameter identification and online subgradient descent algorithm. Furthermore, safety controller modular is firstly introduced in direct data driven control scheme.

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Machine Learning Based Transient Stability Emulation and Dynamic System Equivalencing of Large-Scale AC-DC Grids for Faster-Than-Real-Time Digital Twin

Modern power systems have been expanding significantly including the integration of high voltage direct current (HVDC) systems, bringing a tremendous computational challenge to transient stability simulation for dynamic security assessment (DSA). In this work, a practical method for energy control center with the machine learning (ML) based synchronous generator model (SGM) and dynamic equivalent model (DEM) is proposed to reduce the computational burden of the traditional transient stability (TS) simulation. The proposed ML-based models are deployed on the field programmable gate arrays (FPGAs) for faster-than-real-time (FTRT) digital twin hardware emulation of the real power system. The Gated Recurrent Unit (GRU) algorithm is adopted to train the SGM and DEM, where the training and testing datasets are obtained from the off-line simulation tool DSAToolsTM/TSAT®. A test system containing 15 ACTIVSg 500-bus systems interconnected by a 15-terminal DC grid is established for validating the accuracy of the proposed FTRT digital twin emulation platform. Due to the complexity of emulating large-scale AC-DC grid, multiple FPGA boards are applied, and a proper interface strategy is also proposed for data synchronization. As a result, the efficacy of the hardware emulation is demonstrated by two case studies, where an FTRT ratio of more than 684 is achieved by applying the GRU-SGM, while it reaches over 208 times for hybrid computational-ML based digital twin of AC-DC grid.

*Published in the IEEE Power & Energy Society Section within IEEE Access.

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