Madhu N. Belur

A Generalized Computational Method to Determine Stability of a Multi-inverter Microgrid

Microgrid-containing parallel-connected inverters, where each inverter is controlled by decentralized active power/voltage frequency and reactive power/voltage magnitude droop control laws results in flexible and expandable systems. These systems have been known to have stability problems for large values of active power/voltage frequency droop control gain. However, so far the stability analysis of multi-inverter systems has always been performed in a computationally intensive manner by considering the entire microgrid. In a practical microgrid, where the number of inverters may be large or the capacity of the units may differ, it becomes essential to develop a method by which stability can be examined without much computational burden. The system of differential algebraic equations has been simplified using justifiable assumptions to result in a final expression that allows the stability of the microgrid to be examined separately with respect to the droop control laws of each inverter transformed into an equivalent network. Moreover, the procedure allows taking into consideration the R/X ratio of the interconnecting cables. Analysis of final expressions validate the stability results reported in literature. Experimental results on hardware show the stable operation of the microgrid.

Stabilization, pole placement, and regular implementability

In this paper, we study control by interconnection of linear differential systems. We give necessary and sufficient conditions for. regular implementability of a given linear differential system. We formulate the problems of stabilization and pole placement as problems of finding a suitable, regularly implementable sub-behavior of the manifest plant behavior. The problem formulations and their resolutions are completely representation free, and specified only in terms of the system dynamics. Control is viewed as regular interconnection. A controller is a system that constrains the plant behavior through a distinguished set of variables, namely, the control variables. The issue of implementation of a controller in the feedback configuration and its relation to regularity of interconnection is addressed. Freedom of disturbances in a plant and regular interconnection with a controller also turn out to be inter-related.

The canonical controllers and regular interconnection

We study the control problem from the point of view of the behavioral systems theory. Two controller constructions, called canonical controllers, are introduced. We prove that for linear time-invariant behaviors, the canonical controllers implement the desired behavior if and only if there exists a controller that implements it. We also investigate the regularity of the canonical controllers, and establish the fact that they are maximally irregular. This means a canonical controller is regular if and only if every other controller that implements the desired behavior is regular.

Robust Active Queue Management for Wireless Networks

Active Queue Management (AQM) algorithms have been extensively studied in the literature in the context of wired networks. In this paper, we study AQM for wireless networks. Unlike a wired link, which is assumed to have a fixed capacity, a wireless link has a capacity that is time-varying due to fading. Thus, the controller is required to meet performance objectives in the presence of these capacity variations. We propose a robust controller design that maintains the queue length close to an operating point. We treat capacity variations as an external disturbance and design a robust controller using H∞ control techniques. We also consider the effect of round-trip time in our model. Our method of incorporating the delay into the discretized model simplifies controller design by allowing direct use of systematic controller design methods and/or design packages. We demonstrate the robustness of the controller to changes in the load condition and in the round-trip time through ns-2 simulations.

Analysis and Mitigation of Voltage Offsets in Multi-inverter Microgrids

This paper studies microgrids where loads are supplied by parallel-connected inverters controlled by decentralized active power/voltage frequency and reactive power-/voltage-magnitude droop control laws. A paralleled ac system, such as a multiinverter microgrid, is susceptible to circulating currents due to differences in voltage magnitude, frequency, phase angle, or dc offset. Circulating currents due to differences in voltage magnitude and dc offset have been known issues reported in literature. However, an in-depth analysis of the problem is required to ascertain the deviation of the system-operating condition from the desired condition. This paper provides a mathematical model that predicts the effect of voltage-magnitude offsets on reactive power sharing between inverters. Simulation and experimental results verify the accuracy of the analytical results obtained from the mathematical model. We examine the effect of dc-circulating currents and propose a simple capacitor emulation control law implemented in software to eliminate dc-circulating currents. This solution is a possible alternative for hardware implementation to eliminate dc-circulating currents. The effectiveness of the capacitor emulation control law has been verified through experimental results.

The strict dissipativity synthesis problem and the rank of the coupling QDF

The problem of existence of a controlled behavior that is strictly dissipative with respect to a quadratic supply rate is studied. The relation between strictness and the rank of a suitable coupling condition that combines the dissipativity properties of the hidden behavior and the orthogonal complement of the plant behavior is analyzed.

Dissipativity of Uncontrollable Systems, Storage Functions, and Lyapunov Functions

Dissipative systems have played an important role in the analysis and synthesis of dynamical systems. The commonly used definition of dissipativity often requires an assumption on the controllability of the system. In this paper we use a definition of dissipativity that is slightly different (and less often used in the literature) to study a linear, time-invariant, possibly uncontrollable dynamical system. We provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not “mixed”; i.e., no pair of uncontrollable poles is symmetric about the imaginary axis. This condition is known to be related to the solvability of a Lyapunov equation; we link Lyapunov functions for autonomous systems to storage functions of an uncontrollable system. The set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature. We show that for an uncontrollable system the set of storage functions is unbounded, and that the unboundedness arises precisely due to the set of Lyapunov functions for an autonomous linear system being unbounded. Further, we show that stabilizability of a system results in this unbounded set becoming bounded from below. Positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive. In this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions. In most of the results in this paper, we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis; we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity.

On the dissipativity of uncontrollable systems

This paper deals with dissipativity of uncontrollable linear time-invariant systems with quadratic supply rates and storage functions. A definition of dissipativity appropriate for this class of systems is given. We present a necessary and sufficient condition for dissipativeness in the single input/single output case.

Algorithmic issues in the synthesis of dissipative systems

In this paper we discuss algorithmic issues that arise in the problem of synthesis of dissipative systems. We deal with linear differential systems that can be controlled only through a restricted set of variables called the control variables. The main feature of this paper is that we assume the system dynamics to be specified in the most general form: a latent variable representation. Starting from such a representation, we provide concrete algorithms that finally fetch a controller to implement the desired behavior. Many other peripheral algorithmic issues that crop up are also studied.

Singular LQ Control, Optimal PD Controller and Inadmissible Initial Conditions

We consider a classical control problem: the infinite horizon singular LQ problem, i.e., some inputs are unpenalized in the quadratic performance index. In this case, it is known that the slow dynamics is constrained to be in a proper subspace of the state-space, with the optimal input for the slow dynamics implementable by feedback. In this technical note we show that both the fast dynamics and the slow dynamics can be implemented by a feedback controller. Moreover, we show that the feedback controller cannot be a static feedback controller but can be PD, i.e., proportional+differentiate exactly once, in the state. We show that the closed loop system is a singular descriptor state space system and we also characterize the conditions on the system/performance index for existence of inadmissible initial conditions, i.e., initial conditions that cause impulsive solutions. There are no inadmissible initial conditions in the controlled system if and only if in the strictly proper transfer matrix from the unpenalized inputs to the penalized states, there exists at least one maximal minor of relative degree equal to the number of unpenalized inputs. In addition to the above, we prove solvability of the infinite horizon singular LQ problem under milder assumptions than in the literature.