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This paper proposes a novel method rooted in differential geometry to approximate the voltage stability boundary of power systems under high variability of renewable generation. We extract intrinsic geometric information of the power flow solution manifold at a given operating point. Specifically, coefficients of the Levi-Civita connection are constructed to approximate the geodesics of the manifold starting at an operating point along any interested directions that represent possible fluctuations in generation and load. Then, based on the geodesic approximation, we further predict the voltage collapse point by solving a few univariate quadratic equations. Conventional methods mostly rely on either expensive numerical continuation at specified directions or numerical optimization. Instead, the proposed approach constructs the Christoffel symbols of the second kind from the Riemannian metric tensors to characterize the complete local geometry which is then extended to the proximity of the stability boundary with efficient computations. As a result, this approach is suitable to handle high-dimensional variability in operating points due to the large-scale integration of renewable resources. Using various case studies, we demonstrate the advantages of the proposed method and provide additional insights and discussions on voltage stability in renewable-rich power systems., Comment: 8 pages, 9 figures

Voltage instability is one of the main causes of power system blackouts. Emerging technologies such as renewable energy integration, distributed energy resources and demand responses may introduce significant uncertainties in analyzing of system-wide voltage stability. This paper starts with summarizing different known voltage instability mechanisms, and then focuses on a class of voltage instability which is induced by the singular surface of the algebraic manifold. We argue and demonstrate that this class can include both dynamic and static voltage instabilities. To determine the minimum distance to the point of voltage collapse, a new formulation is proposed on the algebraic manifold. This formulation is further converted into an optimal control framework for identifying the path with minimum distance on the manifold. Comprehensive numerical studies are conducted on some manifolds of different power system test cases and demonstrate that the proposed method yields candidates for the local shortest paths to the singular surface on the manifold for both the dynamic model and the static model. Simulations show that the proposed method can identify shorter paths on the manifold than the paths associated with the minimum Euclidean distances. Furthermore, the proposed method always locates the right path ending at the correct singular surface which is responsible for the voltage instability; while the Euclidean distance formulation can mistakenly find solutions on the wrong singular surface. A broad range of potential applications using the proposed method are also discussed., Comment: 11 pages, 9 figures

This paper aims to systematically and comprehensively initiate a foundation for using concepts from computational differential geometry as instruments for power flow computing and research. At this point we focus our discussion on the static case, with power flow equations given by quadratic functions defined on voltage space with values in power space; both spaces have real Euclidean coordinates. The central issue is a differential geometric analysis of the power flow solution space boundary (SSB, also in a simplifying way, called saddle node bifurcation set, SNB) both in voltage and in power space. We present different methods for computing tangent vectors, tangent planes and normals of the SSB and the normals' derivatives. Using the latter we compute normal and principal curvatures. All this is needed for tracing the orthogonal projection of points on curves in voltage or power space onto the SSB on points closest to the given points on the curve, thus obtaining estimates for their distance to the SSB. As another example how these concepts can be useful, we present a new high precision continuation method for power flow solutions close to and on the SSB called local inversion of the power flow map from voltage to power space, assuming the dimension of power flow's Jacobean zero space, called KERNEL, is one. For inversion, we present two different geometry-based splitting techniques with one of them using the aforementioned orthogonal tracing method., Comment: 10 pages, 7 figures. arXiv admin note: substantial text overlap with arXiv:1903.11131

This paper aims to systematically and comprehensively initiate a foundation for using concepts from computational differential geometry as instruments for power flow computing and research. At this point we focus our discussion on the static case, with power flow equations given by quadratic functions defined on voltage space with values in power space; both spaces have real Euclidean coordinates. Central issue is a differential geometric analysis of the power flow solution space boundary (SSB) both in voltage and in power space. We present different methods for computing tangent vectors, tangent planes and normals of the SSB and the normals' derivatives. Using the latter we compute normal and principal curvatures. All this is needed for tracing the orthogonal projection of curves in voltage and power space onto the SSB for points on the SSB cosest to given points on the curves, thus obtaining estimates for the distance to the SSB. Furthermore, we present a new high precision continuation method for power flow solutions. We also compute geodesics on the SSB or an implicitly defined submanfold thereof and, used to define geodesic coordinates together with their Jacobians on the manifolds. These computations might be the most innovative and most significant contribution of this paper, because this concept provides a comprehensive coordinate system for sub many folds defined by implicit equations. Therefore while moving on geodesics described by the geodesic coordinates of the sub manifold at hand we get, via systematic navigation guided by geodesic coordinates, access to all feasible operation points of the system. We propose some applications and show some properties of the Jacobian of the power flow map.

Tactile displays have a wide potential field of applications, ranging from enhancing Virtual-Reality scenarios up to aiding telesurgery as well as in fundamental psychological and neurophysiological research. In this paper, we describe an open source hardware and software architecture that is designed to drive a variety of different tactile displays. For demonstration purposes, a tactile computer mouse featuring a simple tactile display, based on lateral piezoelectric (PZT) actuators, is presented. Even though we will focus on driving mechanical actuators in this paper, the system can be extended to different working principles. The suggested architecture is supplied with a custom, easy to use, software stack allowing a simple definition of tactile scenarios as well as user studies while being especially tailored to non-computer scientists. By releasing the OpenTactile system under MIT license we hope to ease the burden of controlling tactile displays as well as designing and reproducing the related experiments., Comment: 17 pages, 10 figures. This paper has not been submitted to any journal yet. It is intended to supplement the information given at the opentactile.org project pages