Indiscernible topological variations in DAE networks

A problem of characterizing conditions under which a topological change in a network of differential–algebraic equations (DAEs) can go undetected is considered. It is shown that initial conditions for which topological changes are indiscernible belong to a generalized eigenspace shared by the nominal system and the system resulting from a topological change. A condition in terms of eigenvectors of the nominal system is derived to check for existence of possibly indiscernible topological changes. For homogeneous networks this condition simplifies to the existence of an eigenvector of the Laplacian of network having equal components. Lastly, a rank condition is derived which can be used to check if a topological change preserves regularity of the nominal network

Load and Renewable-Following Control of Linearization-Free Differential Algebraic Equation Power System Models

Electromechanical transients in power networks are mostly caused by mismatch between power consumption and production, causing generators to deviate from the nominal frequency. To that end, feedback control algorithms have been designed to perform frequency and load/renewables-following control. In particular, the literature addressed a plethora of grid- and frequency-control challenges with a focus on linearized, differential equation models whereby algebraic constraints (i.e., power flows) are eliminated. This is in contrast with the more realistic nonlinear differential algebraic equation (NDAE) models. Yet, as grids are increasingly pushed to their limits via intermittent renewables and varying loads, their physical states risk escaping operating regions due to either a poor prediction or sudden changes in renewables or demands -- deeming a feedback controller based on a linearization point virtually unusable. In lieu of linearized differential equation models, the objective of this paper is to design a simple, purely decentralized, linearization-free, feedback control law for NDAE models of power networks. The objective of such controller is to primarily stabilize frequency oscillations after a large, unknown disturbance in renewables or loads. Although the controller design involves advanced NDAE system theory, the controller itself is as simple as a decentralized proportional or linear quadratic regulator in its implementation. Case studies demonstrate that the proposed controller is able to stabilize dynamic and algebraic states under significant disturbances.Comment: 13 pages, 6 figures, 2 table

Structure Identifiability of an NDS with LFT Parametrized Subsystems

Requirements on subsystems have been made clear in this paper for a linear time invariant (LTI) networked dynamic system (NDS), under which subsystem interconnections can be estimated from external output measurements. In this NDS, subsystems may have distinctive dynamics, and subsystem interconnections are arbitrary. It is assumed that system matrices of each subsystem depend on its (pseudo) first principle parameters (FPPs) through a linear fractional transformation (LFT). It has been proven that if in each subsystem, the transfer function matrix (TFM) from its internal inputs to its external outputs is of full normal column rank (FNCR), while the TFM from its external inputs to its internal outputs is of full normal row rank (FNRR), then the NDS is structurally identifiable. Moreover, under some particular situations like there are no direct information transmission from an internal input to an internal output in each subsystem, a necessary and sufficient condition is established for NDS structure identifiability. A matrix valued polynomial (MVP) rank based equivalent condition is further derived, which depends affinely on subsystem (pseudo) FPPs and can be independently verified for each subsystem. From this condition, some necessary conditions are obtained for both subsystem dynamics and its (pseudo) FPPs, using the Kronecker canonical form (KCF) of a matrix pencil.Comment: 16 page

Global Structure Identifiability and Reconstructibility of an NDS with Descriptor Subsystems

This paper investigates requirements on a networked dynamic system (NDS) such that its subsystem interactions can be solely determined from experiment data or reconstructed from its overall model. The NDS is constituted from several subsystems whose dynamics are described through a descriptor form. Except regularity on each subsystem and the whole NDS, no other restrictions are put on either subsystem dynamics or subsystem interactions. A matrix rank based necessary and sufficient condition is derived for the global identifiability of subsystem interactions, which leads to several conclusions about NDS structure identifiability when there is some a priori information. This matrix also gives an explicit description for the set of subsystem interactions that can not be distinguished from experiment data only. In addition, under a well-posedness assumption, a necessary and sufficient condition is obtained for the reconstructibility of subsystem interactions from an NDS descriptor form model. This condition can be verified with each subsystem separately and is therefore attractive in the analysis and synthesis of a large-scale NDS. Simulation results show that rather than increases monotonically with the distance of subsystem interactions to the undifferentiable set, the magnitude of the external output differences between two NDSs with distinct subsystem interactions increases much more rapidly when one of them is close to be unstable. In addition, directions of probing signals are also very important in distinguishing external outputs of distinctive NDSs.These findings are expected to be helpful in identification experiment designs, etc.Comment: 15 pages, 4 figure

Effects of Aerial LiDAR Data Density on the Accuracy of Building Reconstruction

Previous work has identified a positive relationship between the density of aerial LiDAR input for building reconstruction and the accuracy of the resulting reconstructed models. We hypothesize a point of diminished returns at which higher data density no longer contributes meaningfully to higher accuracy in the end product. We investigate this relationship by subsampling a high-density dataset from the City of Surrey, BC to different densities and inputting each subsampled dataset to reconstruction using two different reconstruction methods. We then determine the accuracy of reconstruction based on manually created reference data, in terms of both 2D footprint accuracy and 3D model accuracy. We find that there is no quantitative evidence for meaningfully improved output accuracy from densities higher than 4 p/m2 for either method, although aesthetic improvements at higher point cloud densities are noted for one method

Neural function approximation on graphs: shape modelling, graph discrimination & compression

Graphs serve as a versatile mathematical abstraction of real-world phenomena in numerous scientific disciplines. This thesis is part of the Geometric Deep Learning subject area, a family of learning paradigms, that capitalise on the increasing volume of non-Euclidean data so as to solve real-world tasks in a data-driven manner. In particular, we focus on the topic of graph function approximation using neural networks, which lies at the heart of many relevant methods. In the first part of the thesis, we contribute to the understanding and design of Graph Neural Networks (GNNs). Initially, we investigate the problem of learning on signals supported on a fixed graph. We show that treating graph signals as general graph spaces is restrictive and conventional GNNs have limited expressivity. Instead, we expose a more enlightening perspective by drawing parallels between graph signals and signals on Euclidean grids, such as images and audio. Accordingly, we propose a permutation-sensitive GNN based on an operator analogous to shifts in grids and instantiate it on 3D meshes for shape modelling (Spiral Convolutions). Following, we focus on learning on general graph spaces and in particular on functions that are invariant to graph isomorphism. We identify a fundamental trade-off between invariance, expressivity and computational complexity, which we address with a symmetry-breaking mechanism based on substructure encodings (Graph Substructure Networks). Substructures are shown to be a powerful tool that provably improves expressivity while controlling computational complexity, and a useful inductive bias in network science and chemistry. In the second part of the thesis, we discuss the problem of graph compression, where we analyse the information-theoretic principles and the connections with graph generative models. We show that another inevitable trade-off surfaces, now between computational complexity and compression quality, due to graph isomorphism. We propose a substructure-based dictionary coder - Partition and Code (PnC) - with theoretical guarantees that can be adapted to different graph distributions by estimating its parameters from observations. Additionally, contrary to the majority of neural compressors, PnC is parameter and sample efficient and is therefore of wide practical relevance. Finally, within this framework, substructures are further illustrated as a decisive archetype for learning problems on graph spaces.Open Acces

Computational Methods for Bayesian Inference in Complex Systems

Bayesian methods are critical for the complete understanding of complex systems. In this approach, we capture all of our uncertainty about a system’s properties using a probability distribution and update this understanding as new information becomes available. By taking the Bayesian perspective, we are able to effectively incorporate our prior knowledge about a model and to rigorously assess the plausibility of candidate models based upon observed data from the system. We can then make probabilistic predictions that incorporate uncertainties, which allows for better decision making and design. However, while these Bayesian methods are critical, they are often computationally intensive, thus necessitating the development of new approaches and algorithms. In this work, we discuss two approaches to Markov Chain Monte Carlo (MCMC). For many statistical inference and system identification problems, the development of MCMC made the Bayesian approach possible. However, as the size and complexity of inference problems has dramatically increased, improved MCMC methods are required. First, we present Second-Order Langevin MCMC (SOL-MC), a stochastic dynamical system-based MCMC algorithm that uses the damped second-order Langevin stochastic differential equation (SDE) to sample a desired posterior distribution. Since this method is based on an underlying dynamical system, we can utilize existing work in the theory for dynamical systems to develop, implement, and optimize the sampler's performance. Second, we present advances and theoretical results for Sequential Tempered MCMC (ST-MCMC) algorithms. Sequential Tempered MCMC is a family of parallelizable algorithms, based upon Transitional MCMC and Sequential Monte Carlo, that gradually transform a population of samples from the prior to the posterior through a series of intermediate distributions. Since the method is population-based, it can easily be parallelized. In this work, we derive theoretical results to help tune parameters within the algorithm. We also introduce a new sampling algorithm for ST-MCMC called the Rank-One Modified Metropolis Algorithm (ROMMA). This algorithm improves sampling efficiency for inference problems where the prior distribution constrains the posterior. In particular, this is shown to be relevant for problems in geophysics. We also discuss the application of Bayesian methods to state estimation, disturbance detection, and system identification problems in complex systems. We introduce a Bayesian perspective on learning models and properties of physical systems based upon a layered architecture that can learn quickly and flexibly. We then apply this architecture to detecting and characterizing changes in physical systems with applications to power systems and biology. In power systems, we develop a new formulation of the Extended Kalman Filter for estimating dynamic states described by differential algebraic equations. This filter is then used as the basis for sub-second fault detection and classification. In synthetic biology, we use a Bayesian approach to detect and identify unknown chemical inputs in a biosensor system implemented in a cell population. This approach uses the tools of Bayesian model selection.</p