Mathematics

Forecasting and decision-making are generally modeled as two sequential steps with no feedback, following an open-loop approach. In power systems, operators first forecast loads trying to minimize errors with respect to historical data. They also size reserve requirements based on error estimates. Next, energy and reserves are scheduled and the system is operated following the dispatch schedule, deploying reserves as needed to accommodate forecast errors. However, co-optimizing these processes may lead to better decisions and result in lower operating costs than when they are considered sequentially. In this paper, we present a new closed-loop learning framework in which the processes of forecasting and decision-making are merged and co-optimized through a bilevel optimization problem. We prove asymptotic convergence of the method and propose two solution approaches: an exact method based on the KKT conditions of the second level problem, and a scalable heuristic approach suitable for decomposition methods. We benchmark our methodology with the standard sequential least-squares forecast and dispatch planning process. We apply the proposed methodology to an illustrative system and to a wide range of systems ranging from dozens to thousands of buses. Our results show that the proposed approach yields consistently better performance than the standard open-loop approach.

We show that sparsity constrained optimization problems over low dimensional spaces tend to have a small duality gap. We use the Shapley-Folkman theorem to derive both data-driven bounds on the duality gap, and an efficient primalization procedure to recover feasible points satisfying these bounds. These error bounds are proportional to the rate of growth of the objective with the target cardinality, which means in particular that the relaxation is nearly tight as soon as the target cardinality is large enough so that only uninformative features are added.

In these notes we collect some results from several of the authors' works in order to make available a single source and show how the approximate geometric methods for regulation have been developed, and how the control design strategy has evolved from the theoretical methods, involving the regulator equations, to what we now call the regularized controller. In between these two extremes we developed, in a series of works, a fairly rigorous analysis of the regularization scheme leading to the regularized dynamic regulator equations and an iterative scheme that produces very accurate tracking and disturbance rejection control laws. In our most recent work we have extended dynamic regulator equations to what we now refer to as the regularized controller. This new formulation has only recently being applied to examples including linear and nonlinear delay equations.

This paper presents a methodology for freight traffic assignment in a large-scale road-rail intermodal network under uncertainty. Network uncertainties caused by natural disasters have dramatically increased in recent years. Several of these disasters (e.g., Hurricane Sandy, Mississippi River Flooding, Hurricane Harvey) severely disrupted the U.S. freight transport network, and consequently, the supply chain. To account for these network uncertainties, a stochastic freight traffic assignment model is formulated. An algorithmic framework, involving the sample average approximation and gradient projection algorithm, is proposed to solve this challenging problem. The developed methodology is tested on the U.S. intermodal network with freight flow data from the Freight Analysis Framework. The experiments consider four types of natural disasters that have different risks and impacts on the transportation network: earthquake, hurricane, tornado, and flood. The results demonstrate the feasibility of the model and algorithmic framework to obtain freight flows for a realistic-sized network in reasonable time (between 417 and 716 minutes). It is found that for all disaster scenarios the freight ton-miles are higher compared to the base case without uncertainty. The increase in freight ton-miles is the highest under the flooding scenario; this is due to the fact that there are more states in the flood-risk areas and they are scattered throughout the U.S.

We study the problem of minimizing the sum of potentially non-differentiable convex cost functions with partially overlapping dependences in an asynchronous manner, where communication in the network is not coordinated. We study the behavior of an asynchronous algorithm based on dual decomposition and block coordinate subgradient methods under assumptions weaker than those used in the literature. At the same time, we allow different agents to use local stepsizes with no global coordination. Sufficient conditions are provided for almost sure convergence to the solution of the optimization problem. Under additional assumptions, we establish a sublinear convergence rate that in turn can be strengthened to linear convergence rate if the problem is strongly convex and has Lipschitz gradients. We also extend available results in the literature by allowing multiple and potentially overlapping blocks to be updated at the same time with non-uniform and potentially time varying probabilities assigned to different blocks. A numerical example is provided to illustrate the effectiveness of the algorithm.

We propose a fully asynchronous networked aggregative game (Asy-NAG) where each player minimizes a cost function that depends on its local action and the aggregate of all players' actions. In sharp contrast to the existing NAGs, each player in our Asy-NAG can compute an estimate of the aggregate action at any wall-clock time by only using (possibly stale) information from nearby players of a directed network. Such an asynchronous update does not require any coordination among players. Moreover, we design a novel distributed algorithm with an aggressive mechanism for each player to adaptively adjust the optimization stepsize per update. Particularly, the slow players in terms of updating their estimates smartly increase their stepsizes to catch up with the fast ones. Then, we develop an augmented system approach to address the asynchronicity and the information delays between players, and rigorously show the convergence to a Nash equilibrium of the Asy-NAG via a perturbed coordinate algorithm which is also of independent interest. Finally, we evaluate the performance of the distributed algorithm through numerical simulations.

Communication delays and synchronization are major bottlenecks for parallel computing, and tolerating asynchrony is therefore crucial for accelerating parallel computation. Motivated by optimization problems that do not satisfy convexity assumptions, we present an asynchronous block coordinate descent algorithm for nonconvex optimization problems whose objective functions satisfy the Polyak-Lojasiewicz condition. This condition is a generalization of strong convexity to nonconvex problems and requires neither convexity nor uniqueness of minimizers. Under only assumptions of mild smoothness of objective functions and bounded delays, we prove that a linear convergence rate is obtained. Numerical experiments for logistic regression problems are presented to illustrate the impact of asynchrony upon convergence.

Path-tracking control of wheeled mobile robot (WMR) has gained a lot of research attention, primarily because of its wide applicability -- for example intelligent wheelchairs, exploration-assistant remote WMR. Recent increase in remote and autonomous operations\requirements for WMR has led to more and more use of IoT devices within the control loop. Consequently, providing interfaces for malicious interactions through false data injection attacks (FDIA). Moreover, optimization-based FDIAs have been shown to cause catastrophic consequences in feedback control systems while by-passing any residual-based monitoring system. Since these attacks target system measurement process, this paper focuses on the problem of improving the resiliency of dynamical observers against FDIA. Specifically, we propose an attack-resilient pruning algorithm which attempts to exclude compromised channels from being processed by the observer. The proposed pruning algorithm improves attack-localization precision to 100% with high probability, which correspondingly improves the resiliency of the underlying UKF to FDIA. The improvements due to the developed resilient pruning-based observer is validated through a numerical simulation of a two-layer path-tracking control platform of differential-driven wheeled mobile robot (DDWMR) under FDIA.

Using the Attouch-Théra duality, we study the cycles, gap vectors and fixed point sets of compositions of proximal mappings. Sufficient conditions are given for the existence of cycles and gap vectors. A primal-dual framework provides an exact relationship between the cycles and gap vectors. We also introduce the generalized cycle and gap vectors to tackle the case when the classical ones do not exist. Examples are given to illustrate our results.

With the considerable increase of Distributed Energy Resources (DER), the reliable and cost-effective operation of the electricity distribution grid becomes challenging. This relies on computationally dependable and tractable optimisation solvers, which may handle: 1) non-linear AC power flow constraints, and 2) the multiple numbers of input variables and objectives of DER, over the operational horizon. In this paper, we introduce an application of a high-performance MultiPeriod AC Optimal Power Flow (MPOPF) solver, called "BATTPOWER", to simulate active distribution grids for a near-future scenario. A large-scale Norwegian distribution grid along with a large population of Electric Vehicles (EV) are here taken as the case-study. We suggest and analyse three operational strategies for the Distribution System Operator (DSO): (a) uncoordinated (dumb) charging, (b) coordinated charging with the objective of energy cost-minimisation, and (c) coordinated charging with the objective of energy cost-minimisation along with operational constraints of the grid. The results demonstrate that the uncoordinated charging would lead to: 1) overloading of lines and transformers when the share of EVs are above 20\%, and 2) higher operational costs than the proposed control strategies of (b) and (c). Although the implementation of strategy (b) is not difficult to apply, nonetheless operational line/transformer limits are violated when the populations of EVs are growing above 36\%. This implies that current market design must be altered to allow active control of a large proportion of DERs within grid operational limits to achieve cost minimization at system level.