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Balanced data is required for deep neural networks (DNNs) when learning to perform power system stability assessment. However, power system measurement data contains relatively few events from where power system dynamics can be learnt. To mitigate this imbalance, we propose a novel data augmentation strategy preserving the dynamic characteristics to be learnt. The augmentation is performed using Variational Mode Decomposition. The detrended and the augmented data are tested for distributions similarity using Kernel Maximum Mean Discrepancy test. In addition, the effectiveness of the augmentation methodology is validated via training an Encoder DNN utilizing original data, testing using the augmented data, and evaluating the Encoder's performance employing several metrics., Comment: To be presented in SYSID 2024

The letter proposes an adaptive model reduction approach based on tensor decomposition to speed up time-domain power system simulation. Taylor series expansion of a power system dynamic model is calculated around multiple equilibria corresponding to different load levels. The terms of Taylor expansion are converted to the tensor format and reduced into smaller-size matrices with the help of tensor decomposition. The approach adaptively changes the complexity of a power system model based on the size of a disturbance to maintain the compromise between high simulation speed and high accuracy of the reduced model. The proposed approach is compared with a traditional linear model reduction approach on the 140-bus 48-machine Northeast Power Coordinating Council system., Comment: 3 pages

The paper proposes a new adaptive approach to power system model reduction for fast and accurate time-domain simulation. This new approach is a compromise between linear model reduction for faster simulation and nonlinear model reduction for better accuracy. During the simulation period, the approach adaptively switches among detailed and linearly or nonlinearly reduced models based on variations of the system state: it employs unreduced models for the fault-on period, uses weighted column norms of the admittance matrix to decide which functions to be linearized in power system differential-algebraic equations for large changes of the state, and adopts a linearly reduced model for small changes of the state. Two versions of the adaptive model reduction approach are introduced. The first version uses traditional power system partitioning where the model reduction is applied to a defined large external area in a power system and the other area defined as the study area keeps full detailed models. The second version applies the adaptive model reduction to the whole system. The paper also conducts comprehensive case studies comparing simulation results using the proposed adaptively reduced models with the linearly reduced model on the Northeast Power Coordinating Council 140-bus 48-machine system., Comment: Submitted to IEEE Transactions on Power Systems

We prove that the higher-dimensional Contou-Carr\`ere symbol is invariant under continuous automorphisms of algebras of iterated Laurent series over a ring. Applying this property, we obtain a new explicit formula for the higher-dimensional Contou-Carr\`ere symbol. Unlike previously known formulas, this formula is given over an arbitrary ring, not necessarily a $\mathbb Q$-algebra, and does not involve algebraic $K$-theory., Comment: 18 pages; minor changes

We study continuous homomorphisms between algebras of iterated Laurent series over a commutative ring. We give a full description of such homomorphisms in terms of a discrete data determined by the images of parameters. In similar terms, we give a criterion of invertibility of an endomorphism and provide an explicit formula for the inverse endomorphism. We study the behavior of the higher-dimensional residue under continuous homomorphisms., Comment: 26 pages; minor changes

We construct a higher-dimensional Contou-Carr\`ere symbol and we study its various fundamental properties. The higher-dimensional Contou-Carr\`ere symbol is defined by means of the boundary map for $K$-groups. We prove its universal property. We provide an explicit formula for the higher-dimensional Contou-Carr\`ere symbol over $\mathbb Q$ and we prove integrality of this formula. A relation with the higher-dimensional Witt pairing is also studied., Comment: 75 pages; minor changes

We define a two-dimensional Contou-Carr\`{e}re symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carr\`{e}re symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We relate also the two-dimensional Contou-Carr\`{e}re symbol with the two-dimensional class field theory., Comment: 58 pages; revised version: added K-theory approach; to appear in Journal of Algebraic Geometry

Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of partial differential operators, and we investigate the properties of these geometric data. This construction is similar to the construction of a formal module of Baker-Akhieser functions. On the other hand, there is a recent generalization of Sato's theory which belongs to the third author of this paper. We compare both approaches to the commutative rings of partial differential operators in two variables., Comment: 25 p V2: minor change V3: revised version, to appear in Selecta Math V4: an inaccuracy in Th.2.1 is fixed

We give a construction of the two-dimensional tame symbol as the commutator of a group-like monoidal groupoid which is obtained from some group of k-linear operators acting in a two-dimensional local field and corresponds to some third cohomology class of this group. We give also the hypothetical method for the proof of the two-dimensional Parshin reciprocity laws. This text was written in 2003 as preprint 03-13 of the Humboldt University of Berlin and was available at http://edoc.hu-berlin.de/docviews/abstract.php?id=26204 (only evident misprints are corrected now). Later E. Frenkel and X. Zhu obtained in arXiv:0810.1487 [math.RT] more general results concerning the third cohomology classes of groups acting on two-dimensional local fields, and the author and X. Zhu obtained in arXiv:1002.4848 [math.AG] the proof of the Parshin reciprocity laws on an algebraic surface similar to the Tate proof of the residue formula on an algebraic curve., Comment: 27 pages, 1 figure

We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using it in the context of two-dimensional local fields and two-dimensional adelic theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface., Comment: 41 pages; final version, to appear in Algebra & Number Theory

We investigate Picard functor of a formal punctured ribbon. We prove that under some conditions this functor is representable by a formal group scheme. Formal punctured ribbons were introduced in arXiv:0708.0985., Comment: 42 pages, minor changes

We investigate formal ribbons on curves. Roughly speaking, formal ribbon is a family of locally linearly compact vector spaces on a curve. We establish a one-to-one correspondence between formal ribbons on curves plus some geometric data and some subspaces of two-dimensional local field., Comment: 38 pages, minor changes

This work introduces adelic constructions of direct images of differentials and symbols in the two-dimensional case in the relative situation. In particular, reciprocity laws for relative residues of differentials and symbols are stated and applied to a construction of the Gysin map for Chow groups., Comment: For introduction and notation, see math.NT/0012131 . Published by Geometry and Topology Monographs at http://www.maths.warwick.ac.uk/gt/GTMon3/m3-II-2.abs.html