Optimal mean-square performance for networked control systems with unreliable communication

In this work we study mean-square (MS) stability and mean-square (MS) performance for discrete-time, finite-dimensional linear time-varying systems with dynamics subject to i.i.d. random variation. We do so primarily in the context of networked control systems (NCS) where the network communication channels are unreliable and modeled as multiplicative stochastic uncertainties, e.g. wireless links subject to packet dropouts and modeled as Bernoulli processes. We first focus on the analysis problem in general. We derive a convex feasibility problem and associated convex optimization problem which can be used to determine the MS stability and MS performance respectively of a given system. Since this analysis theory is derived in terms of the feasibility of and optimization of a linear cost subject to linear matrix inequalities (LMIs), it serves as the foundation from which a solution methodology for numerous controller synthesis problems can be derived. Next we formulate the main synthesis problem we consider in this work: a networked control system where both the sensor measurements for the plant(s) and the commands from the controller are transmitted via unreliable communication channels. We treat the unreliable communication links as i.i.d. random processes. We assume that the plant(s) and links are subject to additive exogenous noise, and that we have access to a reliable but delayed acknowledgment of whether or not the controller commands were received by the plant(s) on the previous time step. Finally we restrict the controller to be finite-dimensional, linear, have no structural dependence on the particular path history of the random processes, and scale in size and complexity linearly with the number of random channels. We then show that this synthesis problem has a MS stabilizing solution if and only if two simpler convex problems have MS stabilizing solutions, and moreover that the optimal MS performance solution to this synthesis problem if it exists can be obtained by solving a sequence of these simpler convex problems. Additionally, we show that the overall optimal MS performance cost is the sum of two components which can be determined from the solutions to the special problems. That is, we derive a separation principle for our problem analogous to the classical H2 synthesis

Stability analysis and robust control of power networks in stochastic environment

The modern power grid is moving towards a cleaner form of energy, renewable energy to meet the ever-increasing demand and new technologies are being installed in the power network to monitor and maintain a stable operation. Further, the interactions in the network are not anymore localized but take place over a system, and the control centers are located remotely, thus involving control of network components over communication channels. Further, given the rapid integration of wind energy, it is essential to study the impact of wind variability on the system stability and frequency regulation. Hence, we model the unreliable and intermittent nature of wind energy with stochastic uncertainty. Moreover, the phasor measurement unit (PMU) data from the power network is transmitted to the control center over communication channels, and it is susceptible to inherent communication channel uncertainties, cyber attacks, and hence, the data at the receiving end cannot be accurate. In this work, we model these communication channels with stochastic uncertainties to study the impact of stochastic uncertainty on the stability and wide area control of power network. The challenging aspect of the stability analysis of stochastic power network is that the stochastic uncertainty appears multiplicative as well as additive in the system dynamics. The notion of mean square exponential stability is considered to study the properties of stochastic power network expressed as a networked control system (NCS) with stochastic uncertainty. We develop, necessary and sufficient conditions for mean square exponential stability which are shown in terms of the input-output property of deterministic or nominal system dynamics captured by the mean square system norm and variance of the channel uncertainty. For a particular case of single input channel uncertainty, we also prove a fundamental limitation result that arises in the mean square exponential stabilization of the continuous-time linear system. Overall, the theoretical contributions in this work generalize the existing results on stability analysis from discrete-time linear systems to continuous-time linear systems with multiplicative uncertainty. The stability results can also be interpreted as a small gain theorem for continuous-time stochastic systems. Linear Matrix Inequalities (LMI)-based optimization formulation is provided for the computation of mean square system norm for stability analysis and controller synthesis. An IEEE 68 bus system is considered, and the fragility of the decentralized load-side primary frequency controller with uncertain wind is shown. The critical variance value is shown to decrease with the increase in the cost of the controllable loads and with the rise in penetration of wind farms. Next, we model the power network with detailed higher order differential equations for synchronous generator (SG), wind turbine generator (WTG). The network power flow equations are expressed as algebraic equations. The resultant system is described by a detailed higher order nonlinear differential-algebraic model. It is shown that the uncertainty in the wind speed appears multiplicative in the system dynamics. Stochastic stability of such systems is characterized based on the developed results on mean square exponential stability. In particular, we study the stochastic small signal stability of the resultant system and characterize the critical variance in wind speeds, beyond which the grid dynamics becomes mean square unstable. The power fluctuations in the demand side and intermittent generation (from renewables) cause frequency excursions from the nominal value. In this context, we consider the controllable loads which can vary their power to achieve frequency regulation based on the frequency feedback from the network. Two different load-side frequency controller strategies, decentralized and distributed frequency controllers are studied in the presence of stochastic wind. Finally, the time-domain simulations on an IEEE 39 bus system (by replacing some of the traditional SGs with WTG) are shown using the wind speeds modeled as stochastic as well as actual wind speeds obtained from the wind farm located near Ames, Iowa. It can be seen that, with an increase in the penetration of wind generation in the network, the network turns mean square unstable. Furthermore, we capture the mean square unstable behavior of the power network with increased penetration of renewables using the statistics of actual wind analytically and complement them through linear and nonlinear time domain simulations. Finally, we analyze the vulnerability of communication channel to stochastic uncertainty on an IEEE 39 bus system and design a wide area controller that is robust to various sources of uncertainties that arise in the communication channels. Further, the PMU measurements and wide area control inputs are rank ordered based on their criticality