The paper proposes a framework for modeling and analysis of the dynamics of supply, demand, and clearing prices in power system with real-time retail pricing and information asymmetry. Real-time retail pricing is characterized by passing on the real-time wholesale electricity prices to the end consumers, and is shown to create a closed-loop feedback system between the physical layer and the market layer of the power system. In the absence of a carefully designed control law, such direct feedback between the two layers could increase volatility and lower the system's robustness to uncertainty in demand and generation. A new notion of generalized price-elasticity is introduced, and it is shown that price volatility can be characterized in terms of the system's maximal relative price elasticity, defined as the maximal ratio of the generalized price-elasticity of consumers to that of the producers. As this ratio increases, the system becomes more volatile, and eventually, unstable. As new demand response technologies and distributed storage increase the price-elasticity of demand, the architecture under examination is likely to lead to increased volatility and possibly instability. This highlights the need for assessing architecture systematically and in advance, in order to optimally strike the trade-offs between volatility, economic efficiency, and system reliability.

Abstract. We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, path-dependent quadratic, and maximum/minimumof-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We derive approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs. This provides worst-case perfomance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

The paper proposes a mechanism for real-time retail pricing of electricity in smart power grids, with price stability as the primary concern. In previous articles, the authors argued that relaying the real-time wholesale market prices to the end consumers creates a closed loop feedback system which could be unstable or lack robustness, leading to extreme price volatility. In this paper, a mathematical model is developed for characterization of the dynamic evolution of supply, (elastic) demand, and market clearing (locational marginal) prices under real-time pricing. It is assumed that the real-time prices for retail consumers are derived from the Locational Marginal Prices (LMPs) of the wholesale balancing markets. The main contribution of the paper is in presenting a stabilizing pricing algorithm and characterization of its effects on system efficiency. Numerical simulations conform with our analysis and show the stabilizing effect of the mechanism and its robustness to disturbances.

The paper proposes a mathematical model for the dynamic evolution of supply, demand, and clearing prices under a class of real-time pricing mechanisms characterized by passing on the real-time wholesale prices to the end consumers. The effects that such mechanisms could pose on the stability and efficiency of the entire system is investigated and several stability criteria are presented. It is shown that relaying the realtime wholesale electricity prices to the end consumers creates a closed loop feedback system which could be unstable or lack robustness, leading to extreme price volatility. Finally, a result is presented which characterizes the efficiency losses incurred when, in order to achieve stability, the wholesale prices are adjusted by a static pricing function before they are passed on to the retail consumers.