Eoin Devane

Primary Frequency Regulation With Load-Side Participation—Part I: Stability and Optimality

We present a method to design distributed generation and demand control schemes for primary frequency regulation in power networks that guarantee asymptotic stability and ensure fairness of allocation. We impose a passivity condition on net power supply variables and provide explicit steady-state conditions on a general class of generation and demand control dynamics that ensure convergence of solutions to equilibria that solve an appropriately constructed network optimization problem. We also show that the inclusion of controllable demand results in a drop in steady-state frequency deviations. We discuss how various classes of dynamics used in recent studies fit within our framework and show that this allows for less conservative stability and optimality conditions. We illustrate our results with simulations on the IEEE 68-bus transmission system and the IEEE 37-bus distribution system with static and dynamic demand response schemes.

Stability and convergence of distributed algorithms for the OPF problem

Many modern power networks are partitioned in nature, with disjoint components of the overall network controlled by competing operators. The problem of solving the Optimal Power Flow (OPF) problem in a distributed manner is therefore of significant interest. For networks in which the high-level structure has tree topology, we analyze a dual decomposition approach to solving a recent convex relaxation of the OPF problem for the overall network in a distributed manner. Incorporating higher-order dynamics in terms of local auxiliary variables, we prove a result of guaranteed convergence to the solution set for sufficiently small values of the step size.

Stability of a general class of distributed algorithms for power control in time-dependent wireless networks

In order for a wireless network to function effectively, the signal power of each users transmitter must be sufficiently large to ensure a reliable uplink connection to the receiver, but not so large as to cause interference with neighboring users. We consider a general class of distributed algorithms for the control of transmitter power allocations in wireless networks with a general form of interference nonlinearity. In particular, we allow this interference to have explicit time-dependence, allowing our analysis to remain valid for network configurations that vary with time. We employ appropriately constructed Lyapunov functions to show that any bounded power distribution obtained from these algorithms is uniformly asymptotically stable. Further, we use Lyapunov-Razumikhin functions to show that, even when the system incorporates heterogeneous, time-varying delays, any solution along which the generalized system nonlinearity is bounded must also be uniformly asymptotically stable. Moreover, in both of these cases this stability is shown to be global, meaning that every power distribution has the same asymptotic behavior. These results are also used in the paper to derive time-invariant asymptotic bounds for the trajectories when the system nonlinearities are appropriately bounded.

Primary Frequency Regulation With Load-Side Participation—Part II: Beyond Passivity Approaches

We consider the problem of distributed generation and demand control for primary frequency regulation in power networks, such that stability and optimality of the power allocation can be guaranteed. It was shown in Part I of this work, that by imposing an input strict passivity condition on the net supply dynamics at each bus, combined with a decentralized condition on their steady-state behavior, convergence to optimality can be guaranteed for broad classes of generation and demand control dynamics in a general network. In this paper, we show that by taking into account additional local information, the input strict passivity condition can be relaxed to less restrictive decentralized conditions. These conditions extend the classes of generation and load dynamics for which convergence to optimality can be guaranteed beyond the class of passive systems, thus, allowing to reduce the conservatism in the analysis and feedback design.

Stability and optimality of distributed schemes for secondary frequency regulation in power networks

We present a method for designing distributed generation and demand control schemes for secondary frequency regulation in power networks such that asymptotic stability and an economically optimal power allocation can be guaranteed. A dissipativity condition is imposed on net power supply variables to provide stability guarantees. Furthermore, economic optimality is achieved by explicit decentralized steady state conditions on the generation and controllable demand. We discuss how various classes of dynamics used in recent studies fit within our framework and give examples of higher order generation and controllable demand dynamics that can be included within our analysis. We also discuss how the dissipativity condition imposed can be easily verified for linear systems by solving an appropriate LMI. Our results are illustrated with simulations on the IEEE 68 bus system which demonstrate that the inclusion of controllable loads offer improved transient behavior and that an optimal power allocation among controllable loads is achieved.

Delay-Independent Asymptotic Stability in Monotone Systems

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimonotonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied primarily on the conditions of homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems by deriving convergence results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results shows that if the undelayed system has a convergent trajectory that is unbounded in all components as t → -∞, then the system is globally asymptotically stable for arbitrary bounded time-varying delays. This follows from a more general result derived in the paper that allows to quantify delay-independent regions of attraction, which can be used to prove global asymptotic stability for various classes of systems. These also recover various known delay-independent stability results that are discussed within the paper.

Delay-independent asymptotic stability in monotone systems

Monotone systems comprise an important class of dynamical systems that are of interest both for their wide applicability and because of their interesting mathematical properties. It is known that under the property of quasimono-tonicity time-delayed systems become monotone, and some remarkable properties have been reported for such systems. These include, for example, the fact that for linear systems global asymptotic stability of the undelayed system implies global asymptotic stability for the delayed system under arbitrary bounded delays. Nevertheless, extensions to nonlinear systems have thus far relied on various restrictive conditions, such as homogeneity and subhomogeneity, and it has been conjectured that these can be relaxed. Our aim in this paper is to show that this is feasible for a general class of nonlinear monotone systems, by deriving asymptotic stability results in which simple properties of the undelayed system lead to delay-independent stability. In particular, one of our results is to show that if the undelayed system has a convergent trajectory that is unbounded in all components as t → -∞ then the system is globally asymptotically stable for arbitrary time-varying delays. This follows from a more general result derived in the paper where delay-independent regions of attraction are quantified from the asymptotic behavior of individual trajectories of the undelayed system. This result recovers various known delay-independent stability results, and several examples are included in the paper to illustrate the significance of the proposed stability conditions.

On the stability and optimality of primary frequency regulation with load-side participation

We present a method to design distributed generation and demand control schemes for primary frequency regulation in power networks that guarantee asymptotic stability and ensure fairness of allocation. We impose a passivity condition on net power supply variables and provide explicit steady state conditions on a general class of generation and demand control dynamics that ensure convergence of solutions to equilibria that solve an appropriately constructed network optimization problem. We discuss how various classes of dynamics used in recent studies fit within our framework and show that, in some cases, this allows for less conservative stability and optimality conditions. We illustrate our results with simulations on the IEEE 68 bus system and observe that both static and dynamic demand response schemes that fit within our framework offer improved transient and steady state behavior compared with control of generation alone.

Stability and optimality of distributed secondary frequency control schemes in power networks

We present a systematic method for designing distributed generation and demand control schemes for secondary frequency regulation in power networks such that stability and an economically optimal power allocation can be guaranteed. We consider frequency dynamics given by swing equation along with generation, controllable demand, and a secondary control scheme that makes use of local frequency measurements and a locally exchanged signal. A decentralized dissipativity condition is imposed on net power supply variables to provide stability guarantees. Furthermore, economic optimality is achieved by explicit steady state conditions on the generation and controllable demand. A distinctive feature of the proposed stability analysis is the fact that it can cope with generation and demand dynamics that are of general higher order. Moreover, we discuss how the proposed framework captures various classes of power supply dynamics used in recent studies. In case of linear dynamics, the proposed dissipativity condition can be efficiently verified using an appropriate linear matrix inequality. Moreover, it is shown how the addition of a suitable observer layer can relax the requirement for demand measurements in the secondary controller. The efficiency and practicality of the proposed results are demonstrated with simulations on the Northeast Power Coordinating Council (NPCC) 140-bus and a 9-bus system.

Delay-independent incremental stability in time-varying monotone systems satisfying a generalized condition of two-sided scalability

Monotone systems generated by delay differential equations with explicit time-variation are of importance in the modeling of a number of significant practical problems, including the analysis of communications systems, population dynamics, and consensus protocols. In such problems, it is often of importance to be able to guarantee delay-independent incremental asymptotic stability, whereby all solutions converge toward each other asymptotically, thus allowing the asymptotic properties of all trajectories of the system to be determined by simply studying those of some particular convenient solution. It is known that the classical notion of quasimonotonicity renders time-delayed systems monotone. However, this is not sufficient alone to obtain such guarantees. In this work we show that by combining quasimonotonicity with a condition of scalability motivated by wireless networks, it is possible to guarantee incremental asymptotic stability for a general class of systems that includes a variety of interesting examples. Furthermore, we obtain as a corollary a result of guaranteed convergence of all solutions to a quantifiable invariant set, enabling time-invariant asymptotic bounds to be obtained for the trajectories even if the precise values of time-varying parameters are unknown.