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With the increasing use of renewable generation in power systems, responsive resources will be necessary to support primary frequency control in future low-inertia/under-damped power systems. Flexible loads can provide fast-frequency response services if coordinated effectively. However, practical implementations of such synthetic damping services require both effective local sensing and control at the device level and an ability to accurately estimate online and predict the available synthetic damping from a fleet. In addition, the inherent trade-off between a fleet being available for fast frequency response while providing other ancillary services needs to be characterized. In this context, the manuscript presents a novel, fully decentralized, packet-based controller for diverse flexible loads that dynamically prioritizes and interrupts loads to engender synthetic damping suitable for primary frequency control. Moreover, the packet-based control methodology is shown to accurately characterize the available synthetic damping in real-time, which is useful to aggregators and system operators. Furthermore, spectral analysis of historical frequency regulation data is used to produce a probabilistic bound on the expected available synthetic damping for primary frequency control from a fleet and the trade-off from concurrently providing secondary frequency control services. Finally, numerical simulation on IEEE test networks demonstrates the effectiveness of the proposed methodology., Comment: Submitted to IEEE Transactions on Power systems

The safety region of operation of a system is the subset of allowed outputs for which no undesirable outcome would occur. Knowing if a system would ever leave its safety regions of operation is important information for the planning and control of systems. The computation of the set of outputs also known as the reachable set of the system is, in many cases, intensive and a simple overestimation of it is preferred, instead. There are several perspectives to address this problem including set-based approaches, Mixed-Monotonicity, Hamilton-Jacobi reachability, neural networks, and recently reachability via Chen-Fliess series. In the present work, a Chen-Fliess series representing the input-output behavior of a dynamical system along with interval arithmetic is used to overestimate the reachable set of a system. The advantage of this combination of techniques is that it provides a closed-form of the overestimating set. Examples are presented to illustrate the results.

Distributed energy resources (DERs) such as responsive loads and energy storage systems are valuable resources available to grid operators for balancing supply-demand mismatches via load coordination. However, consumer acceptance of load coordination schemes depends on ensuring quality of service (QoS), which embodies device-level constraints. Since each device has its own internal energy state, the effect of QoS on the fleet can be cast as fleet-wide energy limits within which the aggregate "state of charge" (SoC) must be actively maintained. This requires coordination of DERs that is cognizant of the SoC, responsive to grid conditions, and depends on fast communication networks. To that effect, this paper presents a novel real-time grid-and-DER co-simulation platform for validating advanced DER coordination schemes and characterizing the capability of such a DER fleet. In particular, we present how the co-simulation platform is suitable for: i) testing real-time performance of a large fleet of DERs in delivering advanced grid services, including frequency regulation; ii) online state estimation to characterize the corresponding SoC of a large fleet of DERs; and iii) incorporating practical limitations of DERs and communications and analyzing the effects on fleet-wide performance. To illustrate these benefits of the presented grid-DER co-simulation platform, we employ the advanced DER coordination scheme called packetized energy management (PEM), which is a novel device-driven, asynchronous, and randomizing control paradigm for DERs. A fleet of thousands of PEM-enabled DERs are then added to a realistic and dynamical model of the Vermont transmission system to complete validation of the co-simulation platform.

Given an additive network of input-output systems where each node of the network is modeled by a locally convergent Chen-Fliess series, two basic properties of the network are established. First, it is shown that every input-output map between a given pair of nodes has a locally convergent Chen-Fliess series representation. Second, sufficient conditions are given under which the input-output map between a pair of nodes has a well defined relative degree as defined by its generating series. This analysis leads to the conclusion that this relative degree property is generic in a certain sense.

This paper addresses the question: how many distributed energy resources (DERs) are needed to provide $\pm1$MW of flexibility over a number of hours? For this purpose, a metric based on an ISO's own performance score is proposed. Then, a systematic procedure is presented and validated that makes use of either a simulator or the solution to an optimization problem based on a nominal analytical formulation to get flexibility in terms of kW-per-device. Furthermore, simulation-based analysis indicates that flexibility from different DER fleets adds linearly, that is, the total flexibility provided by a mixture of different DER types can be obtained as a convex combination of their individual kW-per-device flexibility. The proposed methodology is validated on ($i$) a centralized coordinator and ($ii$) a device driven DER coordination scheme called packetized energy management (PEM). Furthermore, the effect of heterogeneity as well as PEM specific parameters such as packet length and mean time-to-request on flexibility is also quantified.

This paper presents a novel frequency-responsive control scheme for demand-side resources, such as electric water heaters. A frequency-dependent control law is designed to provide damping from distributed energy resources (DERs) in a fully decentralized fashion. This local control policy represents a frequency-dependent threshold for each DER that ensures that the aggregate response provides damping during frequency deviations. The proposed decentralized policy is based on an adaptation of a packet-based DER coordination scheme where each device send requests for energy access (also called an "energy packet") to an aggregator. The number of previously accepted active packets can then be used a-priori to form an online estimate of the aggregate damping capability of the DER fleet in a dynamic power system. A simple two-area power system is used to illustrate and validate performance of the decentralized control policy and the accuracy of the online damping estimating for a fleet of 400,000 DERs., Comment: 7 pages, 6 figures, 3 tables, submitted to IEEE smart grid conference 2020

A learning control system is presented suitable for control affine nonlinear plants based on discrete-time Chen-Fliess series and capable of incorporating knowledge of a given physical model. The underlying noncommutative algebraic and combinatorial structures needed to realize the multivariable case are also described. The method is demonstrated using a two-input, two-output Lotka-Volterra system., Comment: Fixed typos, updated references

A detailed expose of the Hopf algebra approach to interconnected input-output systems in nonlinear control theory is presented. The focus is on input-output systems that can been represented in terms of Chen-Fliess functional expansions or Fliess operators. This provides a starting point for a discrete-time version of this theory. In particular, the notion of a discrete-time Fliess operator is given and a class of parallel interconnections is described., Comment: 43 pages, review article

A case study is presented of a collision avoidance system that directly integrates the kinematics of a bi-steerable car with a suitable path planning algorithm. The first step is to identify a path using the method of rapidly exploring random trees, and then a spline approximation is computed. The second step is to solve the output tracking problem by explicitly computing the left inverse of the kinematics of the system to render the Taylor series of the desired input for each polynomial section of the spline approximation. The method is demonstrated by numerical simulation.

A convenient way to represent a nonlinear input-output system in control theory is via a Chen-Fliess functional expansion or Fliess operator. The general goal of this paper is to describe how to approximate Fliess operators with iterated sums and to provide accurate error estimates for two different scenarios, one where the series coefficients are growing at a local convergence rate, and the other where they are growing at a global convergence rate. In each case, it is shown that the error estimates are achievable in the sense that worst case inputs can be identified which hit the error bound. The paper then focuses on the special case where the operators are rational, i.e., they have rational generating series. It is shown in this situation that the iterated sum approximation can be realized by a discrete-time state space model which is a rational function of the input and state affine. In addition, this model comes from a specific discretization of the bilinear realization of the rational Fliess operator.

This paper provides a dendriform-tree setting for Fliess operators with matrix-valued inputs. This class of analytic nonlinear input-output systems is convenient, for example, in quantum control. In particular, a description of such Fliess operators is provided using planar binary trees. Sufficient conditions for convergence of the defining series are also given.

Given two nonlinear input-output systems written in terms of Chen-Fliess functional expansions, it is known that the feedback interconnected system is always well defined and in the same class. An explicit formula for the generating series of a single-input, single-output closed-loop system was provided by the first two authors in earlier work via Hopf algebra methods. This paper is a sequel. It has four main innovations. First, the full multivariable extension of the theory is presented. Next, a major simplification of the basic set up is introduced using a new type of grading that has recently appeared in the literature. This grading also facilitates a fully recursive algorithm to compute the antipode of the Hopf algebra of the output feedback group, and thus, the corresponding feedback product can be computed much more efficiently. The final innovation is an improved convergence analysis of the antipode operation, namely, the radius of convergence of the antipode is computed., Comment: background material expanded and reorganized, a few typos fixed

In this work a combinatorial description is provided of a Faa di Bruno type Hopf algebra which naturally appears in the context of Fliess operators in nonlinear feedback control theory. It is a connected graded commutative and non-cocommutative Hopf algebra defined on rooted circle trees. A cancellation free forest formula for its antipode is given., Comment: revised and updated

This paper aims to provide conditions under which a quantum stochastic differential equation can serve as a model for interconnection of a bilinear system evolving on an operator group SU(2) and a linear quantum system representing a quantum harmonic oscillator. To answer this question we derive algebraic conditions for the preservation of canonical commutation relations (CCRs) of quantum stochastic differential equations (QSDE) having a subset of system variables satisfying the harmonic oscillator CCRs, and the remaining variables obeying the CCRs of SU(2). Then, it is shown that from the physical realizability point of view such QSDEs correspond to bilinear-linear quantum cascades.

The goal of this paper is to provide conditions under which a quantum stochastic differential equation (QSDE) preserves the commutation and anticommutation relations of the SU(n) algebra, and thus describes the evolution of an open n-level quantum system. One of the challenges in the approach lies in the handling of the so-called anomaly coefficients of SU(n). Then, it is shown that the physical realizability conditions recently developed by the authors for open n-level quantum systems also imply preservation of commutation and anticommutation relations.

This paper considers the physical realizability condition for multi-level quantum systems having polynomial Hamiltonian and multiplicative coupling with respect to several interacting boson fields. Specifically, it generalizes a recent result the authors developed for two-level quantum systems. For this purpose, the algebra of SU(n) was incorporated. As a consequence, the obtained condition is given in terms of the structure constants of SU(n)., Comment: arXiv admin note: text overlap with arXiv:1208.3256

Coherent feedback control considers purely quantum controllers in order to overcome disadvantages such as the acquisition of suitable quantum information, quantum error correction, etc. These approaches lack a systematic characterization of quantum realizability. Recently, a condition characterizing when a system described as a linear stochastic differential equation is quantum was developed. Such condition was named physical realizability, and it was developed for linear quantum systems satisfying the quantum harmonic oscillator canonical commutation relations. In this context, open two-level quantum systems escape the realm of the current known condition. When compared to linear quantum system, the challenges in obtaining such condition for such systems radicate in that the evolution equation is now a bilinear quantum stochastic differential equation and that the commutation relations for such systems are dependent on the system variables. The goal of this paper is to provide a necessary and sufficient condition for the preservation of the Pauli commutation relations, as well as to make explicit the relationship between this condition and physical realizability.