We extend the regret analysis of the online distributed weighted dual averaging (DWDA) algorithm [1] to the dynamic setting and provide the tightest dynamic regret bound known to date with respect to the time horizon for a distributed online convex optimization (OCO) algorithm. Our bound is linear in the cumulative difference between consecutive optima and does not depend explicitly on the time horizon. We use dynamic-online DWDA (D-ODWDA) and formulate a performance-guaranteed distributed online demand response approach for heating, ventilation, and air-conditioning (HVAC) systems of commercial buildings. We show the performance of our approach for fast timescale demand response in numerical simulations and obtain demand response decisions that closely reproduce the centralized optimal ones.
University of California, Berkeley（加州大学伯克利分校）；University of Melbourne（墨尔本大学）；University of Toronto（多伦多大学）
We incorporate future information in the form of the estimated value of future gradients in online convex optimization. This is motivated by demand response in power systems, where forecasts about the current round, e.g., the weather or the loads' behavior, can be used to improve on predictions made with only past observations. Specifically, we introduce an additional predictive step that follows the standard online convex optimization step when certain conditions on the estimated gradient and descent direction are met. We show that under these conditions and without any assumptions on the pre…
Polytechnique Montréal（蒙特利尔理工学院）；University of Toronto（多伦多大学）；University of California, Berkeley（加州大学伯克利分校）
We consider online optimization with binary decision variables and convex loss functions. We design a new algorithm, binary online gradient descent (bOGD) and bound its expected dynamic regret. We provide a regret bound that holds for any time horizon and a specialized bound for finite time horizons. First, we present the regret as the sum of the relaxed, continuous round optimum tracking error and the rounding error of our update in which the former asymptomatically decreases with time under certain conditions. Then, we derive a finite-time bound that is sublinear in time and linear in the cu…
Experian Data Lab（益博睿数据实验室）；University of California San Diego（加州大学圣地亚哥分校）
We apply column generation to approximating complex structured objects via a set of primitive structured objects under either the cross entropy or L2 loss. We use L1 regularization to encourage the use of few structured primitive objects. We attack approximation using convex optimization over an infinite number of variables each corresponding to a primitive structured object that are generated on demand by easy inference in the Lagrangian dual. We apply our approach to producing low rank approximations to large 3-way tensors.
This paper presents an asynchronous incremental aggregated gradient algorithm and its implementation in a parameter server framework for solving regularized optimization problems. The algorithm can handle both general convex (possibly non-smooth) regularizers and general convex constraints. When the empirical data loss is strongly convex, we establish linear convergence rate, give explicit expressions for step-size choices that guarantee convergence to the optimum, and bound the associated convergence factors. The expressions have an explicit dependence on the degree of asynchrony and recover …
Demand response has been implemented by distribution system operators to reduce peak demand and mitigate contingency issues on distribution lines and substations. Specifically, the campus based commercial buildings make the major contributions to peak demand in a distribution system. Note that prior works neglect the consumers comfort level in performing demand response, which limits their applications as the incentives are not worth as compared to the loss in comfort levels for most time. Thus, a framework to comprehensively consider both operating costs and comfort levels is necessary. Moreo…
Scalable demand response of residential electric loads has been a timely research topic in recent years. The commercial coming of age or residential demand response requires a scalable control architecture that is both efficient and practical to use. This work presents such a strategy for domestic hot water heaters and present a commercial proof-of-concept deployment. The strategy combines state of the art in aggregate-and-dispatch with a novel dispatch strategy leveraging recent developments in reinforcement learning and is tested in a hardware-in-the-loop simulation environment. The results …
As microgrids have advanced from early prototypes to relatively mature technologies, converting data center integrated commercial buildings to microgrids provides economic, reliability and resiliency enhancements for the building owners. Thus, microgrid design and economically sizing distributed energy resources (DER) are becoming more demanding to gain widespread microgrids commercial viability. In this paper, an optimal DER sizing formulation for a hybrid AC/DC microgrid configuration has been proposed to leverage all benefits that AC or DC microgrid could solely contribute. Energy storage (…
Neglecting consumers' comfort always leads to failure or slow-response to demand response request. In this paper, we propose several comprehensive comfort level models for various appliances in campus-based commercial buildings (CBs). The objective of the proposed system is to minimize O\&M costs of campus-based CBs and maximize various comfort levels simultaneously under the worst-case scenarios. Adaptive robust optimization (ARO) is leveraged to handle various uncertainties within the proposed system: (i) demand response signals sending from the distribution system operator (DSO); (ii) arriv…
This paper investigates the water network's potential ability to provide demand response services to the power grid under the framework of a distribution-level water-energy nexus (micro-WEN). In particular, the hidden controllability of water loads, such as irrigation systems, was closely studied to improve the flexibility of electrical grids. A optimization model is developed for the demand-side management (DSM) of micro-WEN, and the simulation results assert that grid flexibility indeed benefits from controllable water loads. Although the proposed optimal DSM model is an intractable mixed-in…
The 2028 LA Olympics is approaching, and the mass transportation network requires to be expanded. The green electric bus transportation system is gaining increasing attention as an essential step to mitigate emission concerns. However, the introduction of a large electric bus fleet will increase the peak load demand, which will adversely affect the grid. This research addresses the issue of an increase in the peak load demand due to the introduction of the electric bus fleet. Therefore, the potential implementation of stationary battery energy storage systems is investigated to shave the peak …
This paper presents a method to obtain a convex inner approximation that aims to improve the feasibility of optimal power flow (OPF) models in distribution feeders. For a resistive distribution network, both real and reactive power effect the node voltages and this makes it necessary to consider both when formulating the OPF problem. Inaccuracy in linearized OPF models may lead to under and over voltages when dispatching flexible demand, at scale, in response to whole-sale market or grid conditions. In order to guarantee feasibility, this paper obtains an inner convex set in which the dispatch…
We consider a Cournot oligopoly model where multiple suppliers (oligopolists) compete by choosing quantities. We compare the social welfare achieved at a Cournot equilibrium to the maximum possible, for the case where the inverse market demand function is convex. We establish a lower bound on the efficiency of Cournot equilibria in terms of a scalar parameter derived from the inverse demand function, namely, the ratio of the slope of the inverse demand function at the Cournot equilibrium to the average slope of the inverse demand function between the Cournot equilibrium and a social optimum. A…
This paper investigates the impact of the changes in the demand of power systems on the quality of the solution procured by the convex relaxation methods for the AC optimal power flow (ACOPF) problem. This investigation needs various measures to evaluate the tightness of the solution procured by the convex relaxation approaches. Therefore, three tightness measures are leveraged to illustrate the performance of convex relaxation methods under different demand scenarios. The main issue of convex relaxation methods is recovering an optimal solution which is not necessarily feasible for the origin…
This paper introduces a computationally efficient comparative approach to classical pricing rules for day-ahead electricity markets, namely Convex Hull Pricing, IP Pricing and European-like market rules, in a Power Exchange setting with non-convex demand bids. These demand bids can, for example, be useful to large industrial consumers, and extend demand block orders in use by European Power Exchanges. For this purpose, we show that Convex Hull Prices can be efficiently computed using continuous relaxations for bidding products involving start-up costs, minimum power output levels and ramp cons…
The AC Optimal power flow (OPF) problem is one of the most fundamental problems in power systems engineering. For the past decades, researchers have been relying on unproven heuristics to tackle OPF. The hardness of OPF stems from two issues: (1) non-convexity and (2) combinatoric constraints (e.g., discrete power extraction constraints). The recent advances in providing sufficient conditions on the exactness of convex relaxation of OPF can address the issue of non-convexity. To complete the understanding of OPF, this paper presents a polynomial-time approximation algorithm to solve the convex…
In this paper, a green transportation location problem is considered with uncertain demand parameter. Increasing robustness influences the number of trucks for sending goods and products, and consequently, makes the air pollution enhance. In this paper, two green approaches are introduced which demand is the main uncertain parameter in both. These approaches are addressed to provide a trade-off between using available trucks and buying new hybrid trucks for evaluating total costs besides air pollution. Due to growing complexity, a Lagrangian decomposition algorithm is applied to find a tight l…
We develop a novel optimization model to maximize the profit of a Demand-Side Platform (DSP) while ensuring that the budget utilization preferences of the DSP's advertiser clients are adequately met. Our model is highly flexible and can be applied in a Real-Time Bidding environment (RTB) with arbitrary auction types, e.g., both first and second price auctions. Our proposed formulation leads to a non-convex optimization problem due to the joint optimization over both impression allocation and bid price decisions. Using Fenchel duality theory, we construct a dual problem that is convex and can b…
Online Resource Allocation with Convex-set Machine-Learned Advice
Sloan School of Management, Massachusetts Institute of Technology（麻省理工学院斯隆管理学院）；Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology（麻省理工学院电气工程与计算机科学系）；Department of Industrial Engineering and Decision Analytics, Hong Kong University of Science and Technology（香港科学与技术大学工业工程与决策分析系）
We consider convex programming problems with integrality constraints that are invariant under a linear symmetry group. To decompose such problems we introduce the new concept of core points, i.e., integral points whose orbit polytopes are lattice-free. For symmetric integer linear programs we describe two algorithms based on this decomposition. Using a characterization of core points for direct products of symmetric groups, we show that prototype implementations can compete with state-of-the-art commercial solvers, and solve an open MIPLIB problem.
In the freeway network control (FNC) problem, the operation of a traffic network is optimized using only flow control. For special cases of the FNC problem, in particular the case when all merging flows are controlled, there exist tight convex relaxations of the corresponding optimization problem. However, model uncertainty, in particular regarding the fundamental diagram and predictions of future traffic demand, can be a problem in practice. This uncertainty poses a challenge to control approaches that pursue a model- and optimization-based strategy. In this work, we propose a robust counterp…
Recently, there is growing interest and need for dynamic pricing algorithms, especially, in the field of online marketplaces by offering smart pricing options for big online stores. We present an approach to adjust prices based on the observed online market data. The key idea is to characterize optimal prices as minimizers of a total expected revenue function, which turns out to be convex. We assume that consumers face information processing costs, hence, follow a discrete choice demand model, and suppliers are equipped with quantity adjustment costs. We prove the strong smoothness of the tota…
Inspired by the decomposition in the hybrid quantum-classical optimization algorithm we introduced in arXiv:1902.04215, we propose here a new (fully classical) approach to solving certain non-convex integer programs using Graver bases. This method is well suited when (a) the constraint matrix $A$ has a special structure so that its Graver basis can be computed systematically, (b) several feasible solutions can also be constructed easily and (c) the objective function can be viewed as many convex functions quilted together. Classes of problems that satisfy these conditions include Cardinality B…
This paper considers a class of generalized convex games where each player is associated with a convex objective function, a convex inequality constraint and a convex constraint set. The players aim to compute a Nash equilibrium through communicating with neighboring players. The particular challenge we consider is that the component functions are unknown a priori to associated players. We study two distributed computation algorithms and analyze their convergence properties in the presence of data transmission delays and dynamic changes of network topologies. The algorithm performance is verif…
Motivated by potential applications in power systems, we study a problem of optimizing a sum of $n$ convex functions on dynamic networks of $n$ nodes when each function is known to only a single node. The nodes' variables, while satisfy their local constraints, are coupled through a linear constraint. Our main contribution is to design a fully distributed primal-dual method for this problem. Under some fairly standard assumptions on objective functions, strong convexity and smoothness, we provide an explicit analysis for the convergence rate of our method on different networks. In particular, …
We consider a robust optimization problem in an electric power system under uncertain demand and availability of renewable energy resources. Solving the deterministic alternating current optimal power flow (ACOPF) problem has been considered challenging since the 1960s due to its nonconvexity. Linear approximation of the AC power flow system sees pervasive use, but does not guarantee a physically feasible system configuration. In recent years, various convex relaxation schemes for the ACOPF problem have been investigated, and under some assumptions, a physically feasible solution can be recove…
This material provides thorough tutorials on some optimization techniques frequently used in various engineering disciplines, including convex optimization, linearization techniques and mixed-integer linear programming, robust optimization, and equilibrium/game problems. It discusses how to reformulate a difficult (non-convex, multi-agent, min-max) problem to a solver-compatible form (semidefinite program, mixed-integer linear program) via convexification, linearization, and decomposition, so the original problem can be reliably solved by commercial/open-source software. Fundamental algorithms…
We consider the problem of smart and flexible loads providing contingency reserves to the electric grid and provide a Distributed Gradient Projection (DGP) algorithm to minimize loads' disutility while providing contingency services. Each load uses locally obtained grid-frequency measurements and inter-load communication to coordinate their actions, and the privacy of each load is preserved: only gradient information is exchanged---not disutility or consumption information. We provide a proof of convergence of the proposed DGP algorithm, and we compare its performance through simulations to th…
We develop an optimization model and corresponding algorithm for the management of a demand-side platform (DSP), whereby the DSP aims to maximize its own profit while acquiring valuable impressions for its advertiser clients. We formulate the problem of profit maximization for a DSP interacting with ad exchanges in a real-time bidding environment in a cost-per-click/cost-per-action pricing model. Our proposed formulation leads to a nonconvex optimization problem due to the joint optimization over both impression allocation and bid price decisions. We use Lagrangian relaxation to develop a trac…
Though distribution system operators have been adding more sensors to their networks, they still often lack an accurate real-time picture of the behavior of distributed energy resources such as demand responsive electric loads and residential solar generation. Such information could improve system reliability, economic efficiency, and environmental impact. Rather than installing additional, costly sensing and communication infrastructure to obtain additional real-time information, it may be possible to use existing sensing capabilities and leverage knowledge about the system to reduce the need…
In this paper we study a continuous time stochastic inventory model for a commodity traded in the spot market and whose supply purchase is affected by price and demand uncertainty. A firm aims at meeting a random demand of the commodity at a random time by maximizing total expected profits. We model the firm's optimal procurement problem as a singular stochastic control problem in which controls are nondecreasing processes and represent the cumulative investment made by the firm in the spot market (a so-called stochastic "monotone follower problem"). We assume a general exponential Lévy proces…
It is well-known that demand response can improve the system efficiency as well as lower consumers' (prosumers') electricity bills. However, it is not clear how we can either qualitatively identify the prosumer with the most impact potential or quantitatively estimate each prosumer's contribution to the total social welfare improvement when additional resource capacity/flexibility is introduced to the system with demand response, such as allowing net-selling behavior. In this work, we build upon existing literature on the electricity market, which consists of price-taking prosumers each with v…
