Mardavij Roozbehani

Volatility of Power Grids Under Real-Time Pricing

The paper proposes a framework for modeling and analysis of the dynamics of supply, demand, and clearing prices in power systems with real-time retail pricing and information asymmetry. Characterized by passing on the real-time wholesale electricity prices to the end consumers, real-time pricing creates a closed-loop feedback system between the physical layer and the market layer of the system. In the absence of a carefully designed control law, such direct feedback can increase sensitivity and lower the systems robustness to uncertainty in demand and generation. It is shown that price volatility can be characterized in terms of the systems 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 may become more volatile. Since new demand response technologies increase the price-elasticity of demand, and since increased penetration of distributed generation can also increase the uncertainty in price-based demand response, the theoretical findings suggest that the architecture under examination can potentially lead to increased volatility. This study highlights the need for assessing architecture systematically and in advance, in order to optimally strike the trade-offs between volatility/robustness and performance metrics such as economic efficiency and environmental efficiency.

On the stability of wholesale electricity markets under real-time pricing

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 real-time 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.

Dynamic Pricing and Stabilization of Supply and Demand in Modern Electric Power Grids

The paper proposes a mechanism for real-time pricing of electricity in smart power grids, with price stability as the primary concern. In previous publications 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 an effective 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.

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

We introduce the framework of path-complete graph Lyapunov functions for ap- proximation 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/minimum- of-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 performance bounds for path-dependent quadratic Lyapunov functions and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

Analysis of the joint spectral radius via lyapunov functions on path-complete graphs

We study the problem of approximating the joint spectral radius (JSR) of a finite set of matrices. Our approach is based on the analysis of the underlying switched linear system via inequalities imposed between 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, maximum/minimum-of-quadratics Lyapunov functions. We characterize all path-complete graphs consisting of two nodes on an alphabet of two matrices and compare their performance. For the general case of any set of n x n matrices we propose semidefinite programs of modest size that approximate the JSR within a multiplicative factor of 1/4√n of the true value. We establish a notion of duality among path-complete graphs and a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.

Optimal utilization of storage and the induced price elasticity of demand in the presence of ramp constraints

This paper is concerned with optimal utilization of storage, characterization of the economic value of storage in the presence of ramp-rate constraints and stochastically-varying electricity prices, and characterization of the price elasticity of demand induced by optimal utilization of storage. The ramp constraints limit the charging and discharging rate of storage, and can be due to the physical limitations of the storage device or the power lines. Such constraints make analytical characterization of optimal policies particularly difficult. In this paper, the optimal utilization problem is addressed in a finite-horizon stochastic dynamic programming framework, and an analytical characterization of the value function along with recursive formulas for computation of the associated optimal policy are derived. It is shown that the value function associated with the dynamic programming problem is a piecewise linear convex function of the storage state, i.e., the amount of stored energy. Furthermore, while the economic value of storage capacity is a non-decreasing function of price volatility, it is shown that due to finite ramping rates, the value of storage saturates quickly as the capacity increases, regardless of price volatility. Finally, it is shown that optimal utilization of storage by consumers could induce a considerable amount of price elasticity, particularly near the average price.

Stability of linear systems with interval time delays excluding zero

The stability of linear systems with multiple, time-invariant, independent and uncertain delays is investigated. Each delay is assumed to reside within a known interval excluding zero. A delay-free sufficient comparison system is formed by replacing the delay elements with parameter-dependent filters, satisfying certain properties. It is shown that robust stability of this finite dimensional parameter-dependent comparison system, guarantees stability of the original time-delay system. This result is novel in the sense that it does not require any a priori knowledge regarding stability of the time-delay system for some fixed delay. When the parameter-dependent filters are formed in a particular manner using Pade approximations, an upper bound on the degree-of-conservatism of the comparison system may be obtained, which is independent of the time-delay system considered. With this, it is shown that the conservatism of this comparison system may be made arbitrarily small. A linear matrix ineqaulity (LMI) formulation is presented for analysis of the stability of the parameter-dependent comparison system. In the single-delay case, an eigenvalue criterion is also available for stability analysis which incurs no additional conservatism

The intertemporal utility of demand and price elasticity of consumption in power grids with shiftable loads

This paper presents a mathematical model of consumer behavior in response to stochastically-varying electricity prices, and a characterization of price elasticity of consumption induced by optimally shifting flexible demands within a fixed time window. The approach is based on deriving the optimal load-shifting policy through a finite horizon stochastic dynamic program, and the analysis is performed under both perfect and partial information about price distribution. An aggregate demand model is constructed from individual demands with random arrivals and random deadlines. Under this model, the aggregate demand becomes a function of price only, and thus allows for quantitative characterization of the utility of demand and price elasticity. While the demand for electricity is often deemed to be highly inelastic, it is shown in this paper that optimal load-shifting can create a considerable amount of price elasticity, even when the aggregate consumption over a long period remains constant.

Modeling, optimization and computation for software verification

Modeling and analysis techniques are presented for real-time, safety-critical software. Software analysis is the task of verifying whether the computer code will execute safely, free of run-time errors. The critical properties that prove safe execution include bounded-ness of variables and termination of the program in finite time. In this paper, dynamical system representations of computer programs along with specific models that are pertinent to analysis via an optimization-based search for system invariants are developed. It is shown that the automatic search for system invariants that establish the desired properties of computer code, can be formulated as a convex optimization problem, such as linear programming, semidefinite programming, and/or sum of squares programming.

Efficiency-Risk Tradeoffs in Electricity Markets with Dynamic Demand Response

In order to study the impact of dynamic demand response in the future smart grid, we examine in an abstract framework how a tradeoff between efficiency and risk arises under different market architectures. We first examine the system performance under noncooperative and cooperative market architectures. The statistics of the stationary aggregate demand processes show that, although the noncooperative load scheduling scheme leads to an efficiency loss, the stationary distribution of the corresponding aggregate demand process has a smaller tail, resulting in less frequent aggregate demand spikes. Cooperative dynamic demand response, on the other hand, makes the market place more efficient at the cost of increased risk of aggregate demand spikes. The market architecture determines the locus of the system performance with respect to the tradeoff curve. We also investigate how a properly designed real-time electricity pricing mechanism can help the system operator achieve a target tradeoff between efficiency and risk in a noncooperative market. We further provide a convex characterization of the Pareto front of system performance measures, which serves as a benchmark of the tradeoffs for the system operator to evaluate the pricing rules.