arXiv subjects

Saverio Bolognani

Publications and source records attributed to Saverio Bolognani.

70 records · Page 4Linked to original sources

arXiv2020

Wasserstein Distributionally Robust Look-Ahead Economic Dispatch

We consider the problem of look-ahead economic dispatch (LAED) with uncertain renewable energy generation. The goal of this problem is to minimize the cost of conventional energy generation subject to uncertain operational constraints. The risk of violating these constraints must be below a given threshold for a family of probability distributions with characteristics similar to observed past data or predictions. We present two data-driven approaches based on two novel mathematical reformulations of this distributionally robust decision problem. The first one is a tractable convex program in which the uncertain constraints are defined via the distributionally robust conditional-value-at-risk. The second one is a scalable robust optimization program that yields an approximate distributionally robust chance-constrained LAED. Numerical experiments on the IEEE 39-bus system with real solar production data and forecasts illustrate the effectiveness of these approaches. We discuss how system operators should tune these techniques in order to seek the desired robustness-performance trade-off and we compare their computational scalability.

math.OC↗

arXiv2020

Limit Behavior and the Role of Augmentation in Projected Saddle Flows for Convex Optimization

In this paper, we study the stability and convergence of continuous-time Lagrangian saddle flows to solutions of a convex constrained optimization problem. Convergence of these flows is well-known when the underlying saddle function is either strictly convex in the primal or strictly concave in the dual variables. In this paper, we show convergence under non-strict convexity when a simple, unilateral augmentation term is added. For this purpose, we establish a novel, non-trivial characterization of the limit set of saddle-flow trajectories that allows us to preclude limit cycles. With our presentation we try to unify several existing problem formulations as a projected dynamical system that allows projection of both the primal and dual variables, thus complementing results available in the recent literature.

math.OC↗

arXiv2020

Timescale Separation in Autonomous Optimization

Autonomous optimization refers to the design of feedback controllers that steer a physical system to a steady state that solves a predefined, possibly constrained, optimization problem. As such, no exogenous control inputs such as setpoints or trajectories are required. Instead, these controllers are modeled after optimization algorithms that take the form of dynamical systems. The interconnection of this type of optimization dynamics with a physical system is however not guaranteed to be stable unless both dynamics act on sufficiently different timescales. In this paper, we quantify the required timescale separation and give prescriptions that can be directly used in the design of this type of feedback controllers. Using ideas from singular perturbation analysis, we derive stability bounds for different feedback laws that are based on common continuous-time optimization schemes. In particular, we consider gradient descent and its variations, including projected gradient, and Newton gradient. We further give stability bounds for momentum methods and saddle-point flows. Finally, we discuss how optimization algorithms like subgradient and accelerated gradient descent, while well-behaved in offline settings, are unsuitable for autonomous optimization due to their general lack of robustness.

math.OC↗

arXiv2020

Projected Dynamical Systems on Irregular, Non-Euclidean Domains for Nonlinear Optimization

Continuous-time projected dynamical systems are an elementary class of discontinuous dynamical systems with trajectories that remain in a feasible domain by means of projecting outward-pointing vector fields. They are essential when modeling physical saturation in control systems, constraints of motion, as well as studying projection-based numerical optimization algorithms. Motivated by the emerging application of feedback-based continuous-time optimization schemes that rely on the physical system to enforce nonlinear hard constraints, we study the fundamental properties of these dynamics on general locally-Euclidean sets. Among others, we propose the use of Krasovskii solutions, show their existence on nonconvex, irregular subsets of low-regularity Riemannian manifolds, and investigate how they relate to conventional Carathéodory solutions. Furthermore, we establish conditions for uniqueness, thereby introducing a generalized definition of prox-regularity which is suitable for non-flat domains. Finally, we use these results to study the stability and convergence of projected gradient flows as an illustrative application of our framework. We provide simple counter-examples for our main results to illustrate the necessity of our already weak assumptions.

math.OC↗

arXiv2020

Non-convex Feedback Optimization with Input and Output Constraints

In this paper, we present a novel control scheme for feedback optimization. That is, we propose a discrete-time controller that can steer the steady state of a physical plant to the solution of a constrained optimization problem without numerically solving the problem. Our controller can be interpreted as a discretization of a continuous-time projected gradient flow. Compared to other schemes used for feedback optimization, such as saddle-point flows or inexact penalty methods, our algorithm combines several desirable properties: It asymptotically enforces constraints on the plant steady-state outputs, and temporary constraint violations can be easily quantified. Our algorithm requires only reduced model information in the form of steady-state input-output sensitivities of the plant. Further, as we prove in this paper, global convergence is guaranteed even for non-convex problems. Finally, our algorithm is straightforward to tune, since the step-size is the only tuning parameter.

eess.SY↗

arXiv2019

A Market Mechanism for Virtual Inertia

One of the recognized principal issues brought along by the steadfast migration towards power electronic interfaced energy sources is the loss of rotational inertia. In conventional power systems, the inertia of the synchronous machines plays a crucial role in safeguarding against any drastic variations in frequency by acting as a buffer in the event of large and sudden power generation-demand imbalances. In future power electronic-based power systems, the same role can be played by strategically located virtual inertia devices. However, the question looms large as to how the system operators would procure and pay for these devices. In this article, we propose a market mechanism inspired by the ancillary service markets in power supply. We consider a linear network reduced power system model along with a robust $\mathcal{H}_2$ performance metric penalizing the worst-case primary control effort. With a social welfare maximization problem for the system operator as a benchmark, we construct a market mechanism in which bids are invited from agents providing virtual inertia, who in turn are compensated via a Vickrey-Clarke-Groves payment rule. The resulting mechanism ensures truthful bidding to be the dominant bidding strategy and guarantees non-negative payoffs for the agents. A three-region case study is considered in simulations, and a comparison with a regulatory approach to the same problem is presented.

math.OC↗

arXiv2019

Today Me, Tomorrow Thee: Efficient Resource Allocation in Competitive Settings using Karma Games

We present a new type of coordination mechanism among multiple agents for the allocation of a finite resource, such as the allocation of time slots for passing an intersection. We consider the setting where we associate one counter to each agent, which we call karma value, and where there is an established mechanism to decide resource allocation based on agents exchanging karma. The idea is that agents might be inclined to pass on using resources today, in exchange for karma, which will make it easier for them to claim the resource use in the future. To understand whether such a system might work robustly, we only design the protocol and not the agents' policies. We take a game-theoretic perspective and compute policies corresponding to Nash equilibria for the game. We find, surprisingly, that the Nash equilibria for a society of self-interested agents are very close in social welfare to a centralized cooperative solution. These results suggest that many resource allocation problems can have a simple, elegant, and robust solution, assuming the availability of a karma accounting mechanism.

cs.MA↗

arXiv2019

On the existence and linear approximation of the power flow solution in power distribution networks

We consider the problem of deriving an explicit approximate solution of the nonlinear power equations that describe a balanced power distribution network. We give sufficient conditions for the existence of a practical solution to the power flow equations, and we propose an approximation that is linear in the active and reactive power demands of the PQ buses. For this approximation, which is valid for generic power line impedances and grid topology, we derive a bound on the approximation error as a function of the grid parameters. We illustrate the quality of the approximation via simulations, we show how it can also model the presence of voltage controlled (PV) buses, and we discuss how it generalizes the DC power flow model to lossy networks.

math.OC↗

arXiv2019

Hierarchical and Distributed Monitoring of Voltage Stability in Distribution Networks

We consider the problem of quantifying and assessing the steady-state voltage stability in radial distribution networks. Our approach to the voltage stability problem is based on a local, approximate, and yet highly accurate characterization of the determinant of the Jacobian of the power flow equations parameterized according to the branch-flow model. The proposed determinant approximation allows us to construct a voltage stability index that can be computed in a fully distributed or in a hierarchical fashion, resulting in a scalable approach to the assessment of steady-state voltage stability. Finally, we provide upper bounds for the approximation error and we numerically validate the quality and the robustness of the proposed approximation with the IEEE 123-bus test feeder.

math.OC↗

arXiv2018

Time-varying Projected Dynamical Systems with Applications to Feedback Optimization of Power Systems

This paper is concerned with the study of continuous-time, non-smooth dynamical systems which arise in the context of time-varying non-convex optimization problems, as for example the feedback-based optimization of power systems. We generalize the notion of projected dynamical systems to time-varying, possibly non-regular, domains and derive conditions for the existence of so-called Krasovskii solutions. The key insight is that for trajectories to exist, informally, the time-varying domain can only contract at a bounded rate whereas it may expand discontinuously. This condition is met, in particular, by feasible sets delimited via piecewise differentiable functions under appropriate constraint qualifications. To illustrate the necessity and usefulness of such a general framework, we consider a simple yet insightful power system example, and we discuss the implications of the proposed conditions for the design of feedback optimization schemes.

math.OC↗

arXiv2018

Stability of Dynamic Feedback Optimization with Applications to Power Systems

We consider the problem of optimizing the steady state of a dynamical system in closed loop. Conventionally, the design of feedback optimization control laws assumes that the system is stationary. However, in reality, the dynamics of the (slow) iterative optimization routines can interfere with the (fast) system dynamics. We provide a study of the stability and convergence of these feedback optimization setups in closed loop with the underlying plant, via a custom-tailored singular perturbation analysis result. Our study is particularly geared towards applications in power systems and the question whether recently developed online optimization schemes can be deployed without jeopardizing dynamic system stability.

math.OC↗

arXiv2018

Generic Existence of Unique Lagrange Multipliers in AC Optimal Power Flow

Solutions to nonlinear, nonconvex optimization problems can fail to satisfy the KKT optimality conditions even when they are optimal. This is due to the fact that unless constraint qualifications (CQ) are satisfied, Lagrange multipliers may fail to exist. Even if the KKT conditions are applicable, the multipliers may not be unique. These possibilities also affect AC optimal power flow (OPF) problems which are routinely solved in power systems planning, scheduling and operations. The complex structure -- in particular the presence of the nonlinear power flow equations which naturally exhibit a structural degeneracy -- make any attempt to establish CQs for the entire class of problems very challenging. In this paper, we resort to tools from differential topology to show that for AC OPF problems in various contexts the linear independence constraint qualification is satisfied almost certainly, thus effectively obviating the usual assumption on CQs. Consequently, for any local optimizer there generically exists a unique set of multipliers that satisfy the KKT conditions.

math.OC↗

arXiv2017

A distributed voltage stability margin for power distribution networks

We consider the problem of characterizing and assessing the voltage stability in power distribution networks. Different from previous formulations, we consider the branch-flow parametrization of the power system state, which is particularly effective for radial networks. Our approach to the voltage stability problem is based on a local, approximate, yet highly accurate characterization of the determinant of the power flow Jacobian. Our determinant approximation allows us to construct a voltage stability index that can be computed in a fully scalable and distributed fashion. We provide an upper bound on the approximation error, and we show how the proposed index outperforms other voltage indices that have been recently proposed in the literature.

math.OC↗

arXiv2014

Distributed reactive power feedback control for voltage regulation and loss minimization

We consider the problem of exploiting the microgenerators dispersed in the power distribution network in order to provide distributed reactive power compensation for power losses minimization and voltage regulation. In the proposed strategy, microgenerators are smart agents that can measure their phasorial voltage, share these data with the other agents on a cyber layer, and adjust the amount of reactive power injected into the grid, according to a feedback control law that descends from duality-based methods applied to the optimal reactive power flow problem. Convergence to the configuration of minimum losses and feasible voltages is proved analytically for both a synchronous and an asynchronous version of the algorithm, where agents update their state independently one from the other. Simulations are provided in order to illustrate the performance and the robustness of the algorithm, and the innovative feedback nature of such strategy is discussed.

math.OC↗

arXiv2012

A distributed control strategy for reactive power compensation in smart microgrids

We consider the problem of optimal reactive power compensation for the minimization of power distribution losses in a smart microgrid. We first propose an approximate model for the power distribution network, which allows us to cast the problem into the class of convex quadratic, linearly constrained, optimization problems. We then consider the specific problem of commanding the microgenerators connected to the microgrid, in order to achieve the optimal injection of reactive power. For this task, we design a randomized, gossip-like optimization algorithm. We show how a distributed approach is possible, where microgenerators need to have only a partial knowledge of the problem parameters and of the state, and can perform only local measurements. For the proposed algorithm, we provide conditions for convergence together with an analytic characterization of the convergence speed. The analysis shows that, in radial networks, the best performance can be achieved when we command cooperation among units that are neighbors in the electric topology. Numerical simulations are included to validate the proposed model and to confirm the analytic results about the performance of the proposed algorithm.

math.OC↗

arXiv2009

Engineering Stable Discrete-Time Quantum Dynamics via a Canonical QR Decomposition

We analyze the asymptotic behavior of discrete-time, Markovian quantum systems with respect to a subspace of interest. Global asymptotic stability of subspaces is relevant to quantum information processing, in particular for initializing the system in pure states or subspace codes. We provide a linear-algebraic characterization of the dynamical properties leading to invariance and attractivity of a given quantum subspace. We then construct a design algorithm for discrete-time feedback control that allows to stabilize a target subspace, proving that if the control problem is feasible, then the algorithm returns an effective control choice. In order to prove this result, a canonical QR matrix decomposition is derived, and also used to establish the control scheme potential for the simulation of open-system dynamics.

quant-ph↗