arXiv subjects

Soummya Kar

Publications and source records attributed to Soummya Kar.

At least 73 records · Page 4Linked to original sources

arXiv2020

Learning to Solve AC Optimal Power Flow by Differentiating through Holomorphic Embeddings

Alternating current optimal power flow (AC-OPF) is one of the fundamental problems in power systems operation. AC-OPF is traditionally cast as a constrained optimization problem that seeks optimal generation set points whilst fulfilling a set of non-linear equality constraints -- the power flow equations. With increasing penetration of renewable generation, grid operators need to solve larger problems at shorter intervals. This motivates the research interest in learning OPF solutions with neural networks, which have fast inference time and is potentially scalable to large networks. The main difficulty in solving the AC-OPF problem lies in dealing with this equality constraint that has spurious roots, i.e. there are assignments of voltages that fulfill the power flow equations that however are not physically realizable. This property renders any method relying on projected-gradients brittle because these non-physical roots can act as attractors. In this paper, we show efficient strategies that circumvent this problem by differentiating through the operations of a power flow solver that embeds the power flow equations into a holomorphic function. The resulting learning-based approach is validated experimentally on a 200-bus system and we show that, after training, the learned agent produces optimized power flow solutions reliably and fast. Specifically, we report a 12x increase in speed and a 40% increase in robustness compared to a traditional solver. To the best of our knowledge, this approach constitutes the first learning-based approach that successfully respects the full non-linear AC-OPF equations.

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arXiv2020

Push-SAGA: A decentralized stochastic algorithm with variance reduction over directed graphs

In this paper, we propose Push-SAGA, a decentralized stochastic first-order method for finite-sum minimization over a directed network of nodes. Push-SAGA combines node-level variance reduction to remove the uncertainty caused by stochastic gradients, network-level gradient tracking to address the distributed nature of the data, and push-sum consensus to tackle the challenge of directed communication links. We show that Push-SAGA achieves linear convergence to the exact solution for smooth and strongly convex problems and is thus the first linearly-convergent stochastic algorithm over arbitrary strongly connected directed graphs. We also characterize the regimes in which Push-SAGA achieves a linear speed-up compared to its centralized counterpart and achieves a network-independent convergence rate. We illustrate the behavior and convergence properties of Push-SAGA with the help of numerical experiments on strongly convex and non-convex problems.

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arXiv2020

Variance-Reduced Decentralized Stochastic Optimization with Accelerated Convergence

This paper describes a novel algorithmic framework to minimize a finite-sum of functions available over a network of nodes. The proposed framework, that we call~\GTVR, is stochastic and decentralized, and thus is particularly suitable for problems where large-scale, potentially private data, cannot be collected or processed at a centralized server. The \GTVR~framework leads to a family of algorithms with two key ingredients: (i) \textit{local variance reduction}, that enables estimating the local batch gradients from arbitrarily drawn samples of local data; and, (ii) \textit{global gradient tracking}, which fuses the gradient information across the nodes. Naturally, combining different variance reduction and gradient tracking techniques leads to different algorithms of interest with valuable practical tradeoffs and design considerations. Our focus in this paper is on two instantiations of the~$\GTVR$ framework, namely~\textbf{\texttt{GT-SAGA}} and~\textbf{\texttt{GT-SVRG}}, that, similar to their centralized counterparts (\SAGA~and~\SVRG), exhibit a compromise between space and time. We show that both~\textbf{\texttt{GT-SAGA}} and~\textbf{\texttt{GT-SVRG}} achieve accelerated linear convergence for smooth and strongly convex problems and further describe the regimes in which they achieve non-asymptotic, network-independent linear convergence rates that are faster with respect to the existing decentralized first-order schemes. Moreover, we show that both algorithms achieve a linear speedup in such regimes, in that, the total number of gradient computations required at each node is reduced by a factor of $1/n$, where $n$ is the number of nodes, compared to their centralized counterparts that process all data at a single node. Extensive simulations illustrate the convergence behavior of the corresponding algorithms.

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arXiv2020

COHORT: Coordination of Heterogeneous Thermostatically Controlled Loads for Demand Flexibility

Demand flexibility is increasingly important for power grids. Careful coordination of thermostatically controlled loads (TCLs) can modulate energy demand, decrease operating costs, and increase grid resiliency. We propose a novel distributed control framework for the Coordination Of HeterOgeneous Residential Thermostatically controlled loads (COHORT). COHORT is a practical, scalable, and versatile solution that coordinates a population of TCLs to jointly optimize a grid-level objective, while satisfying each TCL's end-use requirements and operational constraints. To achieve that, we decompose the grid-scale problem into subproblems and coordinate their solutions to find the global optimum using the alternating direction method of multipliers (ADMM). The TCLs' local problems are distributed to and computed in parallel at each TCL, making COHORT highly scalable and privacy-preserving. While each TCL poses combinatorial and non-convex constraints, we characterize these constraints as a convex set through relaxation, thereby making COHORT computationally viable over long planning horizons. After coordination, each TCL is responsible for its own control and tracks the agreed-upon power trajectory with its preferred strategy. In this work, we translate continuous power back to discrete on/off actuation, using pulse width modulation. COHORT is generalizable to a wide range of grid objectives, which we demonstrate through three distinct use cases: generation following, minimizing ramping, and peak load curtailment. In a notable experiment, we validated our approach through a hardware-in-the-loop simulation, including a real-world air conditioner (AC) controlled via a smart thermostat, and simulated instances of ACs modeled after real-world data traces. During the 15-day experimental period, COHORT reduced daily peak loads by an average of 12.5% and maintained comfortable temperatures.

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arXiv2020

Distributed Gradient Methods for Nonconvex Optimization: Local and Global Convergence Guarantees

The article discusses distributed gradient-descent algorithms for computing local and global minima in nonconvex optimization. For local optimization, we focus on distributed stochastic gradient descent (D-SGD)--a simple network-based variant of classical SGD. We discuss local minima convergence guarantees and explore the simple but critical role of the stable-manifold theorem in analyzing saddle-point avoidance. For global optimization, we discuss annealing-based methods in which slowly decaying noise is added to D-SGD. Conditions are discussed under which convergence to global minima is guaranteed. Numerical examples illustrate the key concepts in the paper.

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arXiv2020

A Circuit-Theoretic Approach to State Estimation

Traditional state estimation (SE) methods that are based on nonlinear minimization of the sum of localized measurement error functionals are known to suffer from non-convergence and large residual errors. In this paper we propose an equivalent circuit formulation (ECF)-based SE approach that inherently considers the complete network topology and associated physical constraints. We analyze the mathematical differences between the two approaches and show that our approach produces a linear state-estimator that is mathematically a quadratic programming (QP) problem with closed-form solution. Furthermore, this formulation imposes additional topology-based constraints that provably shrink the feasible region and promote convergence to a more physically meaningful solution. From a probabilistic viewpoint, we show that our method applies prior knowledge into the estimate, thus converging to a more physics-based estimate than the traditional observation-driven maximum likelihood estimator (MLE). Importantly, incorporation of the entire system topology and underlying physics, while being linear, makes ECF-based SE advantageous for large-scale systems.

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arXiv2020

The economics of utility-scale portable energy storage systems in a high-renewable grid

Battery storage is expected to play a crucial role in the low-carbon transformation of energy systems. The deployment of battery storage in the power gird, however, is currently severely limited by its low economic viability, which results from not only high capital costs but also the lack of flexible and efficient utilization schemes and business models. Making utility-scale battery storage portable through trucking unlocks its capability to provide various on-demand services. We introduce the potential applications of utility-scale portable energy storage and investigate its economics in California using a spatiotemporal decision model that determines the optimal operation and transportation schedules of portable storage. We show that mobilizing energy storage can increase its life-cycle revenues by 70% in some areas and improve renewable energy integration by relieving local transmission congestion. The life-cycle revenue of spatiotemporal arbitrage can fully compensate for the costs of portable energy storage system in several regions in California, including San Diego and the San Francisco Bay Area.

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arXiv2020

Distributed Gradient Flow: Nonsmoothness, Nonconvexity, and Saddle Point Evasion

The paper considers distributed gradient flow (DGF) for multi-agent nonconvex optimization. DGF is a continuous-time approximation of distributed gradient descent that is often easier to study than its discrete-time counterpart. The paper has two main contributions. First, the paper considers optimization of nonsmooth, nonconvex objective functions. It is shown that DGF converges to critical points in this setting. The paper then considers the problem of avoiding saddle points. It is shown that if agents' objective functions are assumed to be smooth and nonconvex, then DGF can only converge to a saddle point from a zero-measure set of initial conditions. To establish this result, the paper proves a stable manifold theorem for DGF, which is a fundamental contribution of independent interest. In a companion paper, analogous results are derived for discrete-time algorithms.

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arXiv2020

S-ADDOPT: Decentralized stochastic first-order optimization over directed graphs

In this report, we study decentralized stochastic optimization to minimize a sum of smooth and strongly convex cost functions when the functions are distributed over a directed network of nodes. In contrast to the existing work, we use gradient tracking to improve certain aspects of the resulting algorithm. In particular, we propose the~\textbf{\texttt{S-ADDOPT}} algorithm that assumes a stochastic first-order oracle at each node and show that for a constant step-size~$α$, each node converges linearly inside an error ball around the optimal solution, the size of which is controlled by~$α$. For decaying step-sizes~$\mathcal{O}(1/k)$, we show that~\textbf{\texttt{S-ADDOPT}} reaches the exact solution sublinearly at~$\mathcal{O}(1/k)$ and its convergence is asymptotically network-independent. Thus the asymptotic behavior of~\textbf{\texttt{S-ADDOPT}} is comparable to the centralized stochastic gradient descent. Numerical experiments over both strongly convex and non-convex problems illustrate the convergence behavior and the performance comparison of the proposed algorithm.

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arXiv2020

Power System Dispatch with Marginal Degradation Cost of Battery Storage

Battery storage is essential for the future smart grid. The inevitable cell degradation renders the battery lifetime volatile and highly dependent on battery dispatch, and thus incurs opportunity cost. This paper rigorously derives the marginal degradation cost of battery for power system dispatch. The derived optimal marginal degradation cost is time-variant to reflect the time value of money and the functionality fade of battery and takes the form of a constant value divided by a discount factor plus a term related to battery state of health. In case studies, we demonstrate the evolution of the optimal marginal costs of degradation that corresponds to the optimal long-term dispatch outcome. We also show that the optimal marginal cost of degradation depends on the marginal cost of generation in the grid.

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arXiv2020

Resilient Distributed Field Estimation

We study resilient distributed field estimation under measurement attacks. A network of agents or devices measures a large, spatially distributed physical field parameter. An adversary arbitrarily manipulates the measurements of some of the agents. Each agent's goal is to process its measurements and information received from its neighbors to estimate only a few specific components of the field. We present $\mathbf{SAFE}$, the Saturating Adaptive Field Estimator, a consensus+innovations distributed field estimator that is resilient to measurement attacks. Under sufficient conditions on the compromised measurement streams, the physical coupling between the field and the agents' measurements, and the connectivity of the cyber communication network, $\mathbf{SAFE}$ guarantees that each agent's estimate converges almost surely to the true value of the components of the parameter in which the agent is interested. Finally, we illustrate the performance of $\mathbf{SAFE}$ through numerical examples.

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arXiv2020

Gradient tracking and variance reduction for decentralized optimization and machine learning

Decentralized methods to solve finite-sum minimization problems are important in many signal processing and machine learning tasks where the data is distributed over a network of nodes and raw data sharing is not permitted due to privacy and/or resource constraints. In this article, we review decentralized stochastic first-order methods and provide a unified algorithmic framework that combines variance-reduction with gradient tracking to achieve both robust performance and fast convergence. We provide explicit theoretical guarantees of the corresponding methods when the objective functions are smooth and strongly-convex, and show their applicability to non-convex problems via numerical experiments. Throughout the article, we provide intuitive illustrations of the main technical ideas by casting appropriate tradeoffs and comparisons among the methods of interest and by highlighting applications to decentralized training of machine learning models.

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arXiv2019

Variance-Reduced Decentralized Stochastic Optimization with Gradient Tracking--Part I: GT-SAGA

In this paper, we study decentralized empirical risk minimization problems, where the goal is to minimize a finite-sum of smooth and strongly-convex functions available over a network of nodes. In this Part I, we propose \textbf{\texttt{GT-SAGA}}, a decentralized stochastic first-order algorithm based on gradient tracking \cite{DSGT_Pu,DSGT_Xin} and a variance-reduction technique called SAGA \cite{SAGA}. We develop the convergence analysis and the iteration complexity of this algorithm. We further demonstrate various trade-offs and discuss scenarios in which \textbf{\texttt{GT-SAGA}} achieves superior performance (in terms of the number of local gradient computations required) with respect to existing decentralized schemes. In Part II \cite{GT_SVRG} of this two-part paper, we develop and analyze \textbf{\texttt{GT-SVRG}}, a decentralized gradient tracking based implementation of SVRG \cite{SVRG}, another well-known variance-reduction technique.

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arXiv2019

Variance-Reduced Decentralized Stochastic Optimization with Gradient Tracking -- Part II: GT-SVRG

Decentralized stochastic optimization has recently benefited from gradient tracking methods \cite{DSGT_Pu,DSGT_Xin} providing efficient solutions for large-scale empirical risk minimization problems. In Part I \cite{GT_SAGA} of this work, we develop \textbf{\texttt{GT-SAGA}} that is based on a decentralized implementation of SAGA \cite{SAGA} using gradient tracking and discuss regimes of practical interest where \textbf{\texttt{GT-SAGA}} outperforms existing decentralized approaches in terms of the total number of local gradient computations. In this paper, we describe \textbf{\texttt{GT-SVRG}} that develops a decentralized gradient tracking based implementation of SVRG \cite{SVRG}, another well-known variance-reduction technique. We show that the convergence rate of \textbf{\texttt{GT-SVRG}} matches that of \textbf{\texttt{GT-SAGA}} for smooth and strongly-convex functions and highlight different trade-offs between the two algorithms in various settings.

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arXiv2019

The Economic End of Life of Electrochemical Energy Storage

The useful life of electrochemical energy storage (EES) is a critical factor to EES planning, operation, and economic assessment. Today, systems commonly assume a physical end-of-life criterion, retiring EES when the remaining capacity reaches a threshold below which the EES is of little use because of functionality degradation. Here, we propose an economic end of life criterion, where EES is retired when it cannot earn positive net economic benefit in its intended application. This criterion depends on the use case and degradation characteristics of the EES, but is independent of initial capital cost. Using an intertemporal operational framework to consider functionality and profitability degradation, our case study shows that the economic end of life could occur significantly faster than the physical end of life. We argue that both criteria should be applied in EES system planning and assessment. We also analyze how R&D efforts should consider cycling capability and calendar degradation rate when considering the economic end-of-life of EES.

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arXiv2019

MATCHA: Speeding Up Decentralized SGD via Matching Decomposition Sampling

This paper studies the problem of error-runtime trade-off, typically encountered in decentralized training based on stochastic gradient descent (SGD) using a given network. While a denser (sparser) network topology results in faster (slower) error convergence in terms of iterations, it incurs more (less) communication time/delay per iteration. In this paper, we propose MATCHA, an algorithm that can achieve a win-win in this error-runtime trade-off for any arbitrary network topology. The main idea of MATCHA is to parallelize inter-node communication by decomposing the topology into matchings. To preserve fast error convergence speed, it identifies and communicates more frequently over critical links, and saves communication time by using other links less frequently. Experiments on a suite of datasets and deep neural networks validate the theoretical analyses and demonstrate that MATCHA takes up to $5\times$ less time than vanilla decentralized SGD to reach the same training loss.

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arXiv2019

An introduction to decentralized stochastic optimization with gradient tracking

Decentralized solutions to finite-sum minimization are of significant importance in many signal processing, control, and machine learning applications. In such settings, the data is distributed over a network of arbitrarily-connected nodes and raw data sharing is prohibitive often due to communication or privacy constraints. In this article, we review decentralized stochastic first-order optimization methods and illustrate some recent improvements based on gradient tracking and variance reduction, focusing particularly on smooth and strongly-convex objective functions. We provide intuitive illustrations of the main technical ideas as well as applications of the algorithms in the context of decentralized training of machine learning models.

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arXiv2019

Distributed Gradient Descent: Nonconvergence to Saddle Points and the Stable-Manifold Theorem

The paper studies a distributed gradient descent (DGD) process and considers the problem of showing that in nonconvex optimization problems, DGD typically converges to local minima rather than saddle points. The paper considers unconstrained minimization of a smooth objective function. In centralized settings, the problem of demonstrating nonconvergence to saddle points of gradient descent (and variants) is typically handled by way of the stable-manifold theorem from classical dynamical systems theory. However, the classical stable-manifold theorem is not applicable in distributed settings. The paper develops an appropriate stable-manifold theorem for DGD showing that convergence to saddle points may only occur from a low-dimensional stable manifold. Under appropriate assumptions (e.g., coercivity), this result implies that DGD typically converges to local minima and not to saddle points.

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