arXiv subjects

Krishnamurthy Dvijotham

Publications and source records attributed to Krishnamurthy Dvijotham.

At least 55 records · Page 3Linked to original sources

arXiv2019

Convex Restriction of Power Flow Feasibility Sets

The convex restriction of the power flow feasible sets identifies the convex subset of power injections where the solution for power flow is guaranteed to exist and satisfy the operational constraints. In contrast to convex relaxations, the convex restriction provides a sufficient condition for power flow feasibility and is particularly useful for problems involving uncertainty in the power generation and demand. In this paper, we present a general framework of constructing convex restriction of an algebraic set defined by equality and inequality constraints and apply the framework to power flow feasibility problem. The procedure results in convex quadratic constraints that provide a sufficiently large region for practical operation of the grid.

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arXiv2018

Verification of deep probabilistic models

Probabilistic models are a critical part of the modern deep learning toolbox - ranging from generative models (VAEs, GANs), sequence to sequence models used in machine translation and speech processing to models over functional spaces (conditional neural processes, neural processes). Given the size and complexity of these models, safely deploying them in applications requires the development of tools to analyze their behavior rigorously and provide some guarantees that these models are consistent with a list of desirable properties or specifications. For example, a machine translation model should produce semantically equivalent outputs for innocuous changes in the input to the model. A functional regression model that is learning a distribution over monotonic functions should predict a larger value at a larger input. Verification of these properties requires a new framework that goes beyond notions of verification studied in deterministic feedforward networks, since requiring worst-case guarantees in probabilistic models is likely to produce conservative or vacuous results. We propose a novel formulation of verification for deep probabilistic models that take in conditioning inputs and sample latent variables in the course of producing an output: We require that the output of the model satisfies a linear constraint with high probability over the sampling of latent variables and for every choice of conditioning input to the model. We show that rigorous lower bounds on the probability that the constraint is satisfied can be obtained efficiently. Experiments with neural processes show that several properties of interest while modeling functional spaces can be modeled within this framework (monotonicity, convexity) and verified efficiently using our algorithms

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arXiv2018

A Framework for Robust Steady-State Voltage Stability of Distribution Systems

Power injection uncertainties in distribution power grids, which are mostly induced by aggressive introduction of intermittent renewable sources, may drive the system away from normal operating regimes and potentially lead to the loss of long-term voltage stability (LTVS). Naturally, there is an ever increasing need for a tool for assessing the LTVS of a distribution system. This paper presents a fast and reliable tool for constructing \emph{inner approximations} of LTVS regions in multidimensional injection space such that every point in our constructed region is guaranteed to be solvable. Numerical simulations demonstrate that our approach outperforms all existing inner approximation methods in most cases. Furthermore, the constructed regions are shown to cover substantial fractions of the true voltage stability region. The paper will later discuss a number of important applications of the proposed technique, including fast screening for viable injection changes, constructing an effective solvability index and rigorously certified loadability limits.

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arXiv2018

Constructing Convex Inner Approximations of Steady-State Security Regions

We propose a scalable optimization framework for estimating convex inner approximations of the steady-state security sets. The framework is based on Brouwer fixed point theorem applied to a fixed-point form of the power flow equations. It establishes a certificate for the self-mapping of a polytope region constructed around a given feasible operating point. This certificate is based on the explicit bounds on the nonlinear terms that hold within the self-mapped polytope. The shape of the polytope is adapted to find the largest approximation of the steady-state security region. While the corresponding optimization problem is nonlinear and non-convex, every feasible solution found by local search defines a valid inner approximation. The number of variables scales linearly with the system size, and the general framework can naturally be applied to other nonlinear equations with affine dependence on inputs. Test cases, with the system sizes up to $1354$ buses, are used to illustrate the scalability of the approach. The results show that the approximated regions are not unreasonably conservative and that they cover substantial fractions of the true steady-state security regions for most medium-sized test cases.

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arXiv2018

A Sufficient Condition for Small-Signal Stability and Construction of Robust Stability Region

The small-signal stability is an integral part of the power system security analysis. The introduction of renewable source related uncertainties is making the stability assessment difficult as the equilibrium point is varying rapidly. This paper focuses on the Differential Algebraic Equation (DAE) formulation of power systems and bridges the gap between the conventional reduced system and the original one using logarithmic norm. We propose a sufficient condition for stability using Bilinear Matrix Inequality and its inner approximation as Linear Matrix Inequality. Another contribution is the construction of robust stability regions in state-space in contrast to most existing approaches trying same in the parameter space. Performance evaluation of the sufficiency condition and its inner approximation has been given with the necessary and sufficient condition for a small-scale test case. The paper provides a necessary base to develop tractable construction techniques for the robust stability region of power systems.

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arXiv2018

Training verified learners with learned verifiers

This paper proposes a new algorithmic framework, predictor-verifier training, to train neural networks that are verifiable, i.e., networks that provably satisfy some desired input-output properties. The key idea is to simultaneously train two networks: a predictor network that performs the task at hand,e.g., predicting labels given inputs, and a verifier network that computes a bound on how well the predictor satisfies the properties being verified. Both networks can be trained simultaneously to optimize a weighted combination of the standard data-fitting loss and a term that bounds the maximum violation of the property. Experiments show that not only is the predictor-verifier architecture able to train networks to achieve state of the art verified robustness to adversarial examples with much shorter training times (outperforming previous algorithms on small datasets like MNIST and SVHN), but it can also be scaled to produce the first known (to the best of our knowledge) verifiably robust networks for CIFAR-10.

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arXiv2018

Safe Exploration in Continuous Action Spaces

We address the problem of deploying a reinforcement learning (RL) agent on a physical system such as a datacenter cooling unit or robot, where critical constraints must never be violated. We show how to exploit the typically smooth dynamics of these systems and enable RL algorithms to never violate constraints during learning. Our technique is to directly add to the policy a safety layer that analytically solves an action correction formulation per each state. The novelty of obtaining an elegant closed-form solution is attained due to a linearized model, learned on past trajectories consisting of arbitrary actions. This is to mimic the real-world circumstances where data logs were generated with a behavior policy that is implausible to describe mathematically; such cases render the known safety-aware off-policy methods inapplicable. We demonstrate the efficacy of our approach on new representative physics-based environments, and prevail where reward shaping fails by maintaining zero constraint violations.

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arXiv2017

High-voltage solution in radial power networks:Existence, properties and equivalent algorithms

The AC power flow equations describe the steady-state behavior of the power grid. While many algorithms have been developed to compute solutions to the power flow equations, few theoretical results are available characterizing when such solutions exist, or when these algorithms can be guaranteed to converge. In this paper, we derive necessary and sufficient conditions for the existence and uniqueness of a power flow solution in balanced radial distribution networks with homogeneous (uniform $R/X$ ratio) distribution lines. We study three distinct solution methods: fixed point iterations}, convex relaxations, and energy functions we show that the three algorithms successfully find a solution if and only if a solution exists. Moreover, all three algorithms always find the unique high-voltage solution to the power flow equations, the existence of which we formally establish.}%, which can be formally defined as the high-voltage solution -- any other solution of the power flow equations has smaller or equal voltage magnitudes at every bus. At this solution, we prove that (i) voltage magnitudes are increasing functions of the reactive power injections, (ii) the solution is a continuous function of the injections, and (iii) the solution is the last one to vanish as the system is loaded past the feasibility boundary.

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arXiv2017

Solvability regions of affinely parameterized quadratic equations

Quadratic systems of equations appear in several applications. The results in this paper are motivated by quadratic systems of equations that describe equilibrium behavior of physical infrastructure networks like the power and gas grids. The quadratic systems in infrastructure networks are parameterized- the parameters can represent uncertainty (estimation error in resistance/inductance of a power transmission line, for example)or controllable decision variables (power outputs of generators,for example). It is then of interest to understand conditions on the parameters under which the quadratic system is guaranteed to have a solution within a specified set (for example, bounds on voltages and flows in a power grid). Given nominal values of the parameters at which the quadratic system has a solution and the Jacobian of the quadratic system at the solution i snon-singular, we develop a general framework to construct convex regions around the nominal value such that the system is guaranteed to have a solution within a given distance of the nominal solution. We show that several results from recen tliterature can be recovered as special cases of our framework,and demonstrate our approach on several benchmark power systems.

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arXiv2016

Opportunities for Price Manipulation by Aggregators in Electricity Markets

Aggregators are playing an increasingly crucial role in the integration of renewable generation in power systems. However, the intermittent nature of renewable generation makes market interactions of aggregators difficult to monitor and regulate, raising concerns about potential market manipulation by aggregators. In this paper, we study this issue by quantifying the profit an aggregator can obtain through strategic curtailment of generation in an electricity market. We show that, while the problem of maximizing the benefit from curtailment is hard in general, efficient algorithms exist when the topology of the network is radial (acyclic). Further, we highlight that significant increases in profit are possible via strategic curtailment in practical settings.

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arXiv2016

Graphical Models for Optimal Power Flow

Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NP-hard in general, and weakly NP-hard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a tree-structured graphical model where the nodal variables are low-dimensional vectors. We adapt the standard dynamic programming algorithm for inference over a tree-structured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for "smart grid" applications like control of distributed energy resources. We evaluate our technique numerically on several benchmark networks and show that practical OPF problems can be solved effectively using this approach.

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arXiv2016

Market Dynamics of Best-Response with Lookahead

One attractive approach to market dynamics is the level $k$ model in which a level $0$ player adopts a very simple response to current conditions, a level $1$ player best-responds to a model in which others take level $0$ actions, and so forth. (This is analogous to $k$-ply exploration of game trees in AI, and to receding-horizon control in control theory.) If players have deterministic mental models with this kind of finite-level response, there is obviously no way their mental models can all be consistent. Nevertheless, there is experimental evidence that people act this way in many situations, motivating the question of what the dynamics of such interactions lead to. We address this question in the setting of Fisher Markets with constant elasticities of substitution (CES) utilities, in the weak gross substitutes (WGS) regime. We show that despite the inconsistency of the mental models, and even if players' models change arbitrarily from round to round, the market converges to its unique equilibrium. (We show this for both synchronous and asynchronous discrete-time updates.) Moreover, the result is computationally feasible in the sense that the convergence rate is linear, i.e., the distance to equilibrium decays exponentially fast. To the best of our knowledge, this is the first result that demonstrates, in Fisher markets, convergence at any rate for dynamics driven by a plausible model of seller incentives. Even for the simple case of (level $0$) best-response dynamics, where we observe that convergence at some rate can be derived from recent results in convex optimization, our result is the first to demonstrate a linear rate of convergence.

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arXiv2015

A differential analysis of the power flow equations

The AC power flow equations are fundamental in all aspects of power systems planning and operations. They are routinely solved using Newton-Raphson like methods. However, there is little theoretical understanding of when these algorithms are guaranteed to find a solution of the power flow equations or how long they may take to converge. Further, it is known that in general these equations have multiple solutions and can exhibit chaotic behavior. In this paper, we show that the power flow equations can be solved efficiently provided that the solution lies in a certain set. We introduce a family of convex domains, characterized by Linear Matrix Inequalities, in the space of voltages such that there is at most one power flow solution in each of these domains. Further, if a solution exists in one of these domains, it can be found efficiently, and if one does not exist, a certificate of non-existence can also be obtained efficiently. The approach is based on the theory of monotone operators and related algorithms for solving variational inequalities involving monotone operators. We validate our approach on IEEE test networks and show that practical power flow solutions lie within an appropriately chosen convex domain.

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arXiv2015

Solving the power flow equations: a monotone operator approach

The AC power flow equations underlie all operational aspects of power systems. They are solved routinely in operational practice using the Newton-Raphson method and its variants. These methods work well given a good initial "guess" for the solution, which is always available in normal system operations. However, with the increase in levels of intermittent generation, the assumption of a good initial guess always being available is no longer valid. In this paper, we solve this problem using the theory of monotone operators. We show that it is possible to compute (using an offline optimization) a "monotonicity domain" in the space of voltage phasors. Given this domain, there is a simple efficient algorithm that will either find a solution in the domain, or provably certify that no solutions exist in it. We validate the approach on several IEEE test cases and demonstrate that the offline optimization can be performed tractably and the computed "monotonicity domain" includes all practically relevant power flow solutions.

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arXiv2015

Construction of power flow feasibility sets

We develop a new approach for construction of convex analytically simple regions where the AC power flow equations are guaranteed to have a feasible solutions. Construction of these regions is based on efficient semidefinite programming techniques accelerated via sparsity exploiting algorithms. Resulting regions have a simple geometric shape in the space of power injections (polytope or ellipsoid) and can be efficiently used for assessment of system security in the presence of uncertainty. Efficiency and tightness of the approach is validated on a number of test networks.

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arXiv2015

Natural Gas Flow Solutions with Guarantees: A Monotone Operator Theory Approach

We consider balanced flows in a natural gas transmission network and discuss computationally hard problems such as establishing if solution of the underlying nonlinear gas flow equations exists, if it is unique, and finding the solution. Particular topologies, e.g. trees, are known to be easy to solve based on a variational description of the gas flow equations, but these approaches do not generalize. In this paper, we show that the gas flow problem can be solved efficiently using the tools of monotone operator theory, provided that we look for solution within certain monotonicity domains. We characterize a family of monotonicity domains, described in terms of Linear Matrix Inequalities (LMI) in the state variables, each containing at most one solution. We also develop an efficient algorithm to choose a particular monotonicity domain, for which the LMI based condition simplifies to a bound on the flows. Performance of the technique is illustrated on exemplary gas networks.

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arXiv2015

Convexity of Energy-Like Functions: Theoretical Results and Applications to Power System Operations

Power systems are undergoing unprecedented transformations with the incorporation of larger amounts of renewable energy sources, distributed generation and demand response. All these changes, while potentially making power grids more responsive, efficient and resilient, also pose significant implementation challenges. In particular, operating the new power grid will require new tools and algorithms capable of predicting if the current state of the system is operationally safe. In this paper we study and generalize the so-called energy function as a tool to design algorithms to test if a high-voltage power transmission system is within the allowed operational limits. In the past the energy function technique was utilized primarily to access the power system transient stability. In this manuscript, we take a new look at energy functions and focus on an aspect that has previously received little attention: Convexity. We characterize the domain of voltage magnitudes and phases within which the energy function is convex. We show that the domain of the energy function convexity is sufficiently large to include most operationally relevant and practically interesting cases. We show how the energy function convexity can be used to analyze power flow equations, e.g. to certify solution uniqueness or non-existence within the domain of convexity. This and other useful features of the generalized energy function are described and illustrated on IEEE 14 and 118 bus models.

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arXiv2014

Universal Convexification via Risk-Aversion

We develop a framework for convexifying a fairly general class of optimization problems. Under additional assumptions, we analyze the suboptimality of the solution to the convexified problem relative to the original nonconvex problem and prove additive approximation guarantees. We then develop algorithms based on stochastic gradient methods to solve the resulting optimization problems and show bounds on convergence rates. %We show a simple application of this framework to supervised learning, where one can perform integration explicitly and can use standard (non-stochastic) optimization algorithms with better convergence guarantees. We then extend this framework to apply to a general class of discrete-time dynamical systems. In this context, our convexification approach falls under the well-studied paradigm of risk-sensitive Markov Decision Processes. We derive the first known model-based and model-free policy gradient optimization algorithms with guaranteed convergence to the optimal solution. Finally, we present numerical results validating our formulation in different applications.

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