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.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/ACC.2015.717206

A System Reference Frame Approach for Stability Analysis and Control of Power Grids

During the last decades, significant advances have been made in the area of power system stability and control. Nevertheless, when this analysis is carried out by means of decentralized conditions in a general network, it has been based on conservative assumptions such as the adoption of lossless networks. In the current paper, we present a novel approach for decentralized stability analysis and control of power grids through the transformation of both the network and the bus dynamics into the system reference frame. In particular, the aforementioned transformation allows us to formulate the network model as an input-output system that is shown to be passive even if the network's lossy nature is taken into account. We then introduce a broad class of bus dynamics that are viewed as multivariable input/output systems compatible with the network formulation, and appropriate passivity conditions are imposed on those that guarantee stability of the power network. We discuss the opportunities and advantages offered by this approach while explaining how this allows the inclusion of advanced models for both generation and power flows. Our analysis is verified through applications to the Two Area Kundur and the IEEE 68-bus test systems with both primary frequency and voltage regulation mechanisms included

Frequency regulation with thermostatic load participation in power networks

We consider the problem of controlling thermostatic loads such that ancillary services are provided to the power network within the secondary frequency control timeframe. This problem has been widely studied in the literature, where stochastic control schemes have been proposed to avoid the possibility of load synchronization, which induces persistent frequency oscillations. However, stochastic schemes introduce delays in the response of thermostatic loads that may limit their ability to provide support at urgencies. In this paper, we present a deterministic control mechanism for thermostatic loads such that those switch when prescribed frequency thresholds are exceeded in order to provide ancillary services to the power network. For the considered scheme, we propose appropriate conditions for the design of the frequency thresholds that bound the coupling between frequency and thermostatic load dynamics, so as to avoid synchronization phenomena. In particular, we show that as the number of loads tends to infinity, there exist arbitrarily long time intervals where the frequency deviations are arbitrarily small.ERC starting grant 67977

Frequency and Voltage Regulation in Hybrid AC/DC Networks

Hybrid AC/DC networks are a key technology for sustainable electrical power systems, due to the increasing number of converter-based distributed energy resources such as solar or wind. In this paper, we consider the design of control schemes for hybrid AC/DC networks, focusing especially on the control of the interlinking converters (ILC(s)). We present two control schemes: firstly for decentralized primary control, and secondly, a distributed controller to achieve secondary control objectives as well. In the primary case, the stability of the controlled system is proven in a general hybrid AC/DC network which may include asynchronous AC subsystems. Furthermore, it is demonstrated that power-sharing across the AC/DC network is significantly improved compared to previously proposed dual droop control. The proposed scheme for secondary control guarantees the convergence of the AC system frequencies and the average DC voltage of each DC subsystem to their nominal values respectively. An optimal power allocation is also achieved at steady-state. The applicability and effectiveness of the proposed algorithms are verified by advanced simulations on a test hybrid AC/DC network in Simscape Power Systems.ERC starting grant 67977

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.Engineering and Physical Sciences Research Counci

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

In order for a wireless network to function effectively, the signal power of each user's 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

Variance bounds for a class of biochemical reactions with bursts using a discrete expansion

We consider the problem of quantifying the variance in the number of molecules of a species, in biochemical reactions with nonlinear reaction rates. We address this problem for a particular configuration where a species is formed with bursts, with a nonlinear rate that depends on another spontaneously formed species. By making use of an appropriately formulated expansion based on the Newton series, in conjunction with spectral properties of the master equation, we derive an analytical expression that provides a hard bound for the variance. We also show that this bound is exact when the propensities are linear. Furthermore, numerical simulations demonstrate that this is very close to the actual variance.ERC starting grant 67977

Control of Interlinking Converters in Hybrid AC/DC Grids: Network Stability and Scalability

Hybrid AC/DC networks are an effective solution for future power systems, due to their ability to combine advantages of both AC and DC networks. However, they bring new technological challenges, one key area being the control of such a network. The network, and especially the interlinking converter (ILC), must be controlled to ensure that the DC and AC subsystems coordinate to stabilize the network and allocate power appropriately. This is an area which has attracted considerable recent interest due to the non-triviality of the control design. One promising tool is passivity theory which allows the derivation of decentralized conditions through which the stability of the network can be guaranteed. This paper investigates the application of a passivity framework to AC/DC grids, using a typical lossless line assumption. By ensuring that an appropriately formulated passivity condition is satisfied by the AC and DC buses, and the interlinking converter, the stability of the interconnection can be guaranteed. We also discuss how the ILC controller may be designed to achieve an appropriate power allocation between AC and DC sources. Simulation results demonstrate that the proposed ILC control design regulates the frequency and voltages of the hybrid AC/DC network with a stable operation maintained.ERC starting grant 67977

Optimized Dispatch of Energy Storage Systems in Unbalanced Distribution Networks

© 2010-2012 IEEE. This paper presents a method to achieve optimal active and reactive power contributions from each energy storage system in an unbalanced distribution network to minimize power loss, while ensuring network current and voltage constraints are satisfied. By modeling loads as either constant current or constant impedance, the ac optimal power flow is transformed into a noniterative convex optimization problem. The application of capacity constraints, voltage constraints, and energy storage constraints in an unbalanced three-phase four-wire system is considered, addressing specific issues pertaining to unbalanced networks such as voltage unbalance and neutral voltage displacement. The proposed method is then used to demonstrate optimized dispatch of energy storage systems in a suitable four-wire unbalanced distribution test network. The contribution of losses in the neutral wire to the total losses is also determined for a test system under a range of operating conditions and various neutral earthing systems, highlighting the importance of considering this in a typical unbalanced distribution network

An Improved Control Scheme for Grid-forming Inverters

In order to reduce the reliance of power grids on conventional (and often non-renewable) generation, reliable and dispatchable converter-interfaced distributed generators (DGs) are required. Instead of relying on large rotating machines for frequency and voltage regulation, it becomes crucial to develop improved control schemes for grid-forming inverters. In this paper we propose a simple and effective frequency and voltage control scheme that offers desirable dynamic response and power sharing. The proposed controller is compared to the conventional hierarchical control scheme via a stability analysis of the overall system dynamics and time-domain simulation results. It is shown that the transient performance and the stability properties are significantly improved.ERC starting grant 67977