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.

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.

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The wide deployment of renewable generation and the gradual decrease in the overall system inertia make modern power grids more vulnerable to transient instabilities and unacceptable frequency fluctuations. Time-domain simulationbased assessment of the system robustness against uncertain and stochastic disturbances is extremely time-consuming. In this paper, we develop an alternative approach, which has its roots in the input-output stability analysis for Lur'e systems. Our approach consists of a mathematically rigorous characterization of the external disturbances that the power system is transiently stable and the frequency constraints are not violated. The derived certificate is efficiently constructed via convex optimization and is shown to be non-conservative for different IEEE test cases.Index Terms-Input-output stability, small-gain analysis, constrained input constrained output stability, sector-bound nonlinearity, transient stability, frequency constraints.

The aggressive integration of distributed renewable sources is changing the dynamics of the electric power grid in an unexpected manner. As a result, maintaining conventional performance specifications, such as transient stability, may not be sufficient to ensure its reliable operation in stressed conditions. In this paper, we introduce a novel criteria in transient stability with consideration of operational constraints over frequency deviation and angular separation. In addition, we provide a robustness measure of the region of attraction, which can quantify the ability of the post-fault system to remain synchronized even under disturbances. To assess this new stability specification, we adopt the notion of Input-to-State Stability (ISS) to the context of power systems and introduce a new class of convex Lyapunov functions, which will result in tractable convex-optimization-based stability certificates. As a result, we are able to quantify the level of disturbance a power system can withstand while maintaining its safe operation. We illustrate the introduced stability specification and certificate on the IEEE 9 bus system.