arXiv subjects

Saber Jafarpour

Publications and source records attributed to Saber Jafarpour.

45 records · Page 3Linked to original sources

arXiv2020

Transient Stability of Droop-Controlled Inverter Networks with Operating Constraints

Due to the rise of distributed energy resources, the control of networks of grid-forming inverters is now a pressing issue for power system operation. Droop control is a popular control strategy in the literature for frequency control of these inverters. In this paper, we analyze transient stability in droop-controlled inverter networks that are subject to multiple operating constraints. Using a physically-meaningful Lyapunov-like function, we provide two sets of criteria (one mathematical and one computational) to certify that a post-fault trajectory achieves frequency synchronization while respecting operating constraints. We show how to obtain less-conservative transient stability conditions by incorporating information from loop flows, i.e., net flows of active power around cycles in the network. Finally, we use these conditions to quantify the scale of parameter disturbances to which the network is robust. We illustrate our results with numerical case studies of the IEEE 24-bus system.

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arXiv2020

Graph-Theoretic Stability Conditions for Metzler Matrices and Monotone Systems

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we derive two sets of stability conditions based on two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems; based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability.

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arXiv2020

Singular Perturbation and Small-signal Stability for Inverter Networks

This paper examines small-signal stability of electrical networks composed dominantly of three-phase grid-following inverters. We show that the mere existence of a high-voltage power flow solution does not necessarily imply small-signal stability; this motivates us to develop a framework for stability analysis that systematically acknowledges inverter dynamics. We identify a suitable time-scale decomposition for the inverter dynamics, and using singular perturbation theory, obtain an analytic sufficient condition to verify small-signal stability. Compared to the alternative of performing an eigenvalue analysis of the full-order network dynamics, our analytic sufficient condition reduces computational complexity and yields insights on the role of network topology and constitution as well as inverter-filter and control parameters on small-signal stability. Numerical simulations for a radial network validate the approach and illustrate the efficiency of our analytic conditions for designing and monitoring grid-tied inverter networks.

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arXiv2019

On small-time local controllability

In this paper, we study small-time local controllability of real analytic control-affine systems under small perturbations of their vector fields. Consider a real analytic control system $\mathcal{X}$ which is small-time locally controllable and whose reachable sets shrink with the polynomial rate of order $N$ with respect to time. We will prove a general theorem which states that any real analytic control-affine system whose vector fields are perturbations of the vector fields of $\mathcal{X}$ with polynomials of order higher than $N$ is again small-time locally controllable. In particular, we show that this result connects two long-standing open conjectures about small-time local controllability of systems.

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arXiv2018

Synchronization of Kuramoto Oscillators: Inverse Taylor Expansions

Synchronization in networks of coupled oscillators is a widely studied topic with extensive scientific and engineering applications. In this paper, we study the frequency synchronization problem for networks of Kuramoto oscillators with arbitrary topology and heterogeneous edge weights. We propose a novel equivalent transcription for the equilibrium synchronization equation. Using this transcription, we develop a power series expansion to compute the synchronized solution of the Kuramoto model as well as a sufficient condition for the strong convergence of this series expansion. Truncating the power series provides (i) an efficient approximation scheme for computing the synchronized solution, and (ii) a simple-to-check, statistically-correct hierarchy of increasingly accurate synchronization tests. This hierarchy of tests provides a theoretical foundation for and generalizes the best-known approximate synchronization test in the literature. Our numerical experiments illustrate the accuracy and the computational efficiency of the truncated series approximation compared to existing iterative methods and existing synchronization tests.

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arXiv2018

Synchronization of Coupled Oscillators: The Taylor Expansion of the Inverse Kuramoto Map

Synchronization in the networks of coupled oscillators is a widely studied topic in different areas. It is well-known that synchronization occurs if the connectivity of the network dominates heterogeneity of the oscillators. Despite extensive study on this topic, the quest for sharp closed-form synchronization tests is still in vain. In this paper, we present an algorithm for finding the Taylor expansion of the inverse Kuramoto map. We show that this Taylor series can be used to obtain a hierarchy of increasingly accurate approximate tests with low computational complexity. These approximate tests are then used to estimate the threshold of synchronization as well as the position of the synchronization manifold of the network.

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arXiv2018

Synchronization of Kuramoto Oscillators via Cutset Projections

Synchronization in coupled oscillators networks is a remarkable phenomenon of relevance in numerous fields. For Kuramoto oscillators the loss of synchronization is determined by a trade-off between coupling strength and oscillator heterogeneity. Despite extensive prior work, the existing sufficient conditions for synchronization are either very conservative or heuristic and approximate. Using a novel cutset projection operator, we propose a new family of sufficient synchronization conditions; these conditions rigorously identify the correct functional form of the trade-off between coupling strength and oscillator heterogeneity. To overcome the need to solve a nonconvex optimization problem, we then provide two explicit bounding methods, thereby obtaining (i) the best-known sufficient condition for unweighted graphs based on the 2-norm, and (ii) the first-known generally-applicable sufficient condition based on the $\infty$-norm. We conclude with a comparative study of our novel $\infty$-norm condition for specific topologies and IEEE test cases; for most IEEE test cases our new sufficient condition is one to two orders of magnitude more accurate than previous rigorous tests.

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arXiv2016

Local and global holomorphic extensions of time-varying real analytic vector fields

In this paper, we consider time-varying real analytic vector fields as curves on the space of real analytic vector fields. Using a suitable topology on the space of real analytic vector fields, we study and characterize different properties of time-varying real analytic vector fields. We study holomorphic extensions of time-varying real analytic vector fields and show that under suitable integrability conditions, a time-varying real analytic vector field on a manifold can be extended to a time-varying holomorphic vector field on a neighbourhood of that manifold. Moreover, we develop an operator setting, where the nonlinear differential equation governing the flow of a time-varying real analytic vector field can be considered as a linear differential equation on an infinite dimensional locally convex vector space. Using the holomorphic extension results, we show that the integrability of the time-varying vector field ensures the convergence of the sequence of Picard iterations for this linear differential equation. This gives us a series representation for the flow of an integrable time-varying real analytic vector field. We also define the exponential map between integrable time-varying real analytic vector fields and their flows. Using the holomorphic extensions of time-varying real analytic vector fields, we show that the exponential map is sequentially continuous.

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arXiv2014

Mathematical models for geometric control theory

Just as an explicit parameterisation of system dynamics by state, i.e., a choice of coordinates, can impede the identification of general structure, so it is too with an explicit parameterisation of system dynamics by control. However, such explicit and fixed parameterisation by control is commonplace in control theory, leading to definitions, methodologies, and results that depend in unexpected ways on control parameterisation. In this paper a framework is presented for modelling systems in geometric control theory in a manner that does not make any choice of parameterisation by control; the systems are called "tautological control systems." For the framework to be coherent, it relies in a fundamental way on topologies for spaces of vector fields. As such, classes of systems are considered possessing a variety of degrees of regularity: finitely differentiable; Lipschitz; smooth; real analytic. In each case, explicit geometric seminorms are provided for the topologies of spaces of vector fields that enable straightforward descriptions of time-varying vector fields and control systems. As part of the development, theorems are proved for regular (including real analytic) dependence on initial conditions of flows of vector fields depending measurably on time. Classes of "ordinary" control systems are characterised that interact with the regularity under consideration in a comprehensive way. In this framework, for example, the statement that "a smooth or real analytic control-affine system is a smooth or real analytic control system" becomes a theorem. Correspondences between ordinary control systems and tautological control systems are carefully examined, and trajectory correspondence between the two classes is proved for control-affine systems and for systems with general control dependence when the control set is compact.

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