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The small-signal stability is an integral part of the power system security analysis. The introduction of renewable source related uncertainties is making the stability assessment difficult as the equilibrium point is varying rapidly. This paper focuses on the Differential Algebraic Equation (DAE) formulation of power systems and bridges the gap between the conventional reduced system and the original one using logarithmic norm. We propose a sufficient condition for stability using Bilinear Matrix Inequality and its inner approximation as Linear Matrix Inequality. Another contribution is the construction of robust stability regions in state-space in contrast to most existing approaches trying same in the parameter space. Performance evaluation of the sufficiency condition and its inner approximation has been given with the necessary and sufficient condition for a small-scale test case. The paper provides a necessary base to develop tractable construction techniques for the robust stability region of power systems.

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain $L^1$ convergence in the white-noise limit.

The Primal-Dual (PD) algorithm is widely used in convex optimization to determine saddle points. While the stability of the PD algorithm can be easily guaranteed, strict contraction is nontrivial to establish in most cases. This work focuses on continuous, possibly non-autonomous PD dynamics arising in a network context, in distributed optimization, or in systems with multiple time-scales. We show that the PD algorithm is indeed strictly contracting in specific metrics and analyze its robustness establishing stability and performance guarantees for different approximate PD systems. We derive estimates for the performance of multiple time-scale multi-layer optimization systems, and illustrate our results on a primal-dual representation of the Automatic Generation Control of power systems., Comment: 6 pages, 1 figures, published on LCSS and CDC 2018

The convex restriction of the power flow feasible sets identifies the convex subset of power injections where the solution for power flow is guaranteed to exist and satisfy the operational constraints. In contrast to convex relaxations, the convex restriction provides a sufficient condition for power flow feasibility and is particularly useful for problems involving uncertainty in the power generation and demand. In this paper, we present a general framework of constructing convex restriction of an algebraic set defined by equality and inequality constraints and apply the framework to power flow feasibility problem. The procedure results in convex quadratic constraints that provide a sufficiently large region for practical operation of the grid.

The Generalized Langevin Equation (GLE) is a Stochastic Integro-Differential Equation that is commonly used to describe the velocity of microparticles that move randomly in viscoelastic fluids. Such particles commonly exhibit what is known as anomalous subdiffusion, which is to say that their position Mean-Squared Displacement (MSD) scales sublinearly with time. While it is common in the literature to observe that there is a relationship between the MSD and the memory structure of the GLE, and there exist special cases where explicit solutions exist, this connection has never been fully characterized. Here, we establish a class of memory kernels for which the GLE is well-defined; we investigate the associated regularity properties of solutions; and we prove that large-time asymptotic behavior of the particle MSD is entirely determined by the tail behavior of the GLE's memory kernel.

We propose a scalable optimization framework for estimating convex inner approximations of the steady-state security sets. The framework is based on Brouwer fixed point theorem applied to a fixed-point form of the power flow equations. It establishes a certificate for the self-mapping of a polytope region constructed around a given feasible operating point. This certificate is based on the explicit bounds on the nonlinear terms that hold within the self-mapped polytope. The shape of the polytope is adapted to find the largest approximation of the steady-state security region. While the corresponding optimization problem is nonlinear and non-convex, every feasible solution found by local search defines a valid inner approximation. The number of variables scales linearly with the system size, and the general framework can naturally be applied to other nonlinear equations with affine dependence on inputs. Test cases, with the system sizes up to $1354$ buses, are used to illustrate the scalability of the approach. The results show that the approximated regions are not unreasonably conservative and that they cover substantial fractions of the true steady-state security regions for most medium-sized test cases., Comment: 11 pages, 5 figures

Power injection uncertainties in distribution power grids, which are mostly induced by aggressive introduction of intermittent renewable sources, may drive the system away from normal operating regimes and potentially lead to the loss of long-term voltage stability (LTVS). Naturally, there is an ever increasing need for a tool for assessing the LTVS of a distribution system. This paper presents a fast and reliable tool for constructing \emph{inner approximations} of LTVS regions in multidimensional injection space such that every point in our constructed region is guaranteed to be solvable. Numerical simulations demonstrate that our approach outperforms all existing inner approximation methods in most cases. Furthermore, the constructed regions are shown to cover substantial fractions of the true voltage stability region. The paper will later discuss a number of important applications of the proposed technique, including fast screening for viable injection changes, constructing an effective solvability index and rigorously certified loadability limits., Comment: 11 pages, 9 figs, published on Transactions on Smart Grid

This paper studies the contraction properties of nonlinear differential-algebraic equation (DAE) systems. Specifically we develop scalable techniques for constructing the attraction regions associated with a particular stable equilibrium, by establishing the relation between the contraction rates of the original systems and the corresponding virtual extended systems. We show that for a contracting DAE system, the reduced system always contracts faster than the extended ones; furthermore, there always exists an extension with contraction rate arbitrarily close to that of the original system. The proposed construction technique is illustrated with a power system example in the context of transient stability assessment., Comment: 9 pages, 3 figures, submitted to TAC

Power flow solvable boundary plays an important role in contingency analysis, security assessment, and planning processes. However, to construct the real solvable boundary in multidimensional parameter space is burdensome and time consuming. In this paper, we develop a new technique to approximate the solvable boundary of distribution systems based on Banach fixed point theorem. Not only the new technique is fast and non-iterative, but also the approximated boundary is more valuable to system operators in the sense that it is closer to the feasible region. Moreover, a simple solvable criterion is also introduced that can serve as a security constraint in various planning and operational problems., Comment: 5 pages, 6 figures, accepted to PESGM 2015

Dynamic response of loads has a significant effect on system stability and directly determines the stability margin of the operating point. Inherent uncertainty and natural variability of load models make the stability assessment especially difficult and may compromise the security of the system. We propose a novel mathematical "robust stability" criterion for the assessment of small-signal stability of operating points. Whenever the criterion is satisfied for a given operating point, it provides mathematical guarantees that the operating point will be stable with respect to small disturbances for any dynamic response of the loads. The criterion can be naturally used for identification of operating regions secure from the occurrence of Hopf bifurcation. Several possible applications of the criterion are discussed, most importantly the concept of Robust Stability Assessment (RSA) that could be integrated in dynamic security assessment packages and used in contingency screening and other planning and operational studies., Comment: 16 pages, 20 figures, IEEE Transactions on Power Systems 2015, in press

High levels of penetration of distributed generation and aggressive reactive power compensation may result in the reversal of power flows in future distribution grids. The voltage stability of these operating conditions may be very different from the more traditional power consumption regime. This paper focused on demonstration of multistability phenomenon in radial distribution systems with reversed power flow, where multiple stable equilibria co-exist at the given set of parameters. The system may experience transitions between different equilibria after being subjected to disturbances such as short-term losses of distributed generation or transient faults. Convergence to an undesirable equilibrium places the system in an emergency or \textit{in extremis} state. Traditional emergency control schemes are not capable of restoring the system if it gets entrapped in one of the low voltage equilibria. Moreover, undervoltage load shedding may have a reverse action on the system and can induce voltage collapse. We propose a novel pulse emergency control strategy that restores the system to the normal state without any interruption of power delivery. The results are validated with dynamic simulations of IEEE $13$-bus feeder performed with SystemModeler software. The dynamic models can be also used for characterization of the solution branches via a novel approach so-called the admittance homotopy power flow method., Comment: 13 pages, 22 figures. IEEE Transactions on Smart Grid 2015, in press

In complex power systems, nonlinear load flow equations have multiple solutions. Under typical load conditions only one solution is stable and corresponds to a normal operating point, whereas the second solution is not stable and is never realized in practice. However, in future distribution grids with high penetration of distributed generators more stable solutions may appear because of active or reactive power reversal. The systems can operate at different states, and additional control measures may be required to ensure that it remains at the appropriate point. This paper focuses on the analysis of several cases where multiple solution phenomena is observed. A non-iterative approach for solving load flow equations based on the Gr\"{o}bner basis is introduced to overcome the convergence and computational efficiency associated with standard iterative approaches. All the solutions of load flow problems with their existence boundaries are analyzed for a simple 3-bus model. Furthermore, the stability of the solutions is analyzed using a derived aggregated load dynamics model, and suggestions for preventive control are proposed and discussed. The failure of na\"{i}ve voltage stability criteria is demonstrated and new voltage stability criteria is proposed. Some of the new solutions of load flow equations are proved to be stable and/or acceptable to the EN 50610 voltage fluctuation standard., Comment: 6 pages, 5 figures, IEEE PES General Meeting 2014

High levels of penetration of distributed generation and aggressive reactive power compensation with modern power electronics may result in the reversal of active and reactive power flows in future distribution grids. The voltage stability of these operating conditions may be very different from the more traditional power consumption regime. We study the stability characteristics of distribution networks with reversed power flow. After introducing a universal algebraic approach to characterize all the solutions of the power flow equations, we show that new solutions appear in the reversed power flow regime even in the simplest three bus systems. We show that the some of these solutions are stable and the system may exhibit a phenomenon of multistability, where multiple stable equilibria co-exist at the given set of parameters, and the system may converge to an undesirable equilibrium after a disturbance. These predictions are validated with dynamic simulations of two different systems. Under certain conditions the new states are viable and may be characterized by relatively high voltages. Possible approaches towards reactive power/voltage regulation as well as permissible distributed generation capacity in future power systems are proposed and discussed in the end of the paper., Comment: This paper has been withdrawn by the authors. The revised of this paper is now available as arXiv:1407.1355