Krishnamurthy Dvijotham

A nullspace analysis of the nuclear norm heuristic for rank minimization

The problem of minimizing the rank of a matrix subject to linear equality constraints arises in applications in machine learning, dimensionality reduction, and control theory, and is known to be NP-hard. A popular heuristic minimizes the nuclear norm (sum of the singular values) of the matrix instead of the rank, and was recently shown to give an exact solution in several scenarios. In this paper, we present a new analysis for this heuristic based on a property of the nullspace of the operator defining the constraints, called the spherical section property. We give conditions for the exact recovery of all matrices up to a certain rank, and show that these conditions hold with high probability for operators generated from random Gaussian ensembles. Our analysis provides simpler proofs than existing isometry-based methods, as well as robust recovery results when the matrix is not exactly low-rank.

Storage Sizing and Placement through Operational and Uncertainty-Aware Simulations

As the penetration level of transmission-scale time-intermittent renewable generation resources increases, control of flexible resources will become important to mitigating the fluctuations due to these new renewable resources. Flexible resources may include new or existing synchronous generators as well as new energy storage devices. Optimal placement and sizing of energy storage to minimize costs of integrating renewable resources is a difficult optimization problem. Further, optimal planning procedures typically do not consider the effect of the time dependence of operations and may lead to unsatisfactory results. Here, we use an optimal energy storage control algorithm to develop a heuristic procedure for energy storage placement and sizing. We perform operational simulation under various time profiles of intermittent generation, loads and interchanges (artificially generated or from historical data) and accumulate statistics of the usage of storage at each node under the optimal dispatch. We develop a greedy heuristic based on the accumulated statistics to obtain a minimal set of nodes for storage placement. The quality of the heuristic is explored by comparing our results to the obvious heuristic of placing storage at the renewables for IEEE benchmarks and real-world network topologies.

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.

Convexity of optimal linear controller design

We develop a general class of stochastic optimal control problems for which the problem of designing optimal linear feedback gains is convex. The class of problems includes arbitrary time varying linear systems and costs that are mixtures of exponentiated quadratics. This allows us to model problems with quadratic state costs and linear constraints on states and state transitions. Further, convexity in the feedback gains lets us impose arbitrary convex constraints or penalties on the feedback matrix: Thus we can model problems like distributed control (by imposing a sparsity structure on the feedback matrix) and variable-stiffness control (by applying time-varying penalties to feedback gain matrices). We show that the convex optimization problem can be solved efficiently by using the structure of the matrices involved. Finally, we present an application of these ideas to a practical problem arising in distributed control of power systems.

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.

Convex Structured Controller Design in Finite Horizon

We consider the problem of synthesizing optimal linear feedback policies subject to arbitrary convex constraints on the feedback matrix. This is known to be a hard problem in the usual formulations (H2, H∞, LQR) and previous works have focused on characterizing classes of structural constraints that allow an efficient solution through convex optimization or dynamic programming techniques. In this paper, we propose a new control objective for finite horizon discrete-time problems and show that this formulation makes the problem of computing optimal linear feedback matrices convex under arbitrary convex constraints on the feedback matrix. This allows us to solve problems in decentralized control (sparsity in the feedback matrices), control with delays, and variable impedance control. Although the control objective is nonstandard, we present theoretical and empirical evidence showing that it agrees well with standard notions of control. We show that the theoretical approach carries over to nonlinear systems, although the computational tractability of the extension is not investigated in this paper. We present numerical experiments validating our approach.

Distributed control of generation in a transmission grid with a high penetration of renewables

Deviations of grid frequency from the nominal frequency are an indicator of the global imbalance between generation and load. Two types of control, a distributed proportional control and a centralized integral control, are currently used to keep frequency deviations small. Although generation-load imbalance can be very localized, both controls primarily rely on frequency deviation as their input. The time scales of control require the outputs of the centralized integral control to be communicated to distant generators every few seconds. We reconsider this control/communication architecture and suggest a hybrid approach that utilizes parameterized feedback policies that can be implemented in a fully distributed manner because the inputs to these policies are local observables at each generator. Using an ensemble of forecasts of load and time-intermittent generation representative of possible future scenarios, we perform a centralized off-line stochastic optimization to select the generator-specific feedback parameters. These parameters need only be communicated to generators once per control period (60 minutes in our simulations). We show that inclusion of local power flows as feedback inputs is crucial and reduces frequency deviations by a factor of ten. We demonstrate our control on a detailed transmission model of the Bonneville Power Administration (BPA). Our findings suggest that a smart automatic and distributed control, relying on advanced off-line and system-wide computations communicated to controlled generators infrequently, may be a viable control and communication architecture solution. This architecture is suitable for a future situation when generation-load imbalances are expected to grow because of increased penetration of time-intermittent generation.

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 letter, 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) transmission lines. We study three distinct solution methods: 1) fixed point iterations; 2) convex relaxations; and 3) 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. At this solution, we prove that: 1) voltage magnitudes are increasing functions of the reactive power injections; 2) the solution is a continuous function of the injections; and 3) the solution is the last one to vanish as the system is loaded past the feasibility boundary.

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 tree-structured 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. Numerical evaluations on several benchmark networks show that practical OPF problems can be solved effectively using this approach.

Solvability Regions of Affinely Parameterized Quadratic Equations

Quadratic systems of equations appear in several applications. The results in this letter are motivated by quadratic systems of equations that describe equilibrium behavior of physical infrastructure networks like power and gas grids. The quadratic systems in infrastructure networks are parameterized—the parameters can represent uncertainty or controllable decision variables. 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. Given nominal values of the parameters at which the quadratic system has a solution, we develop a general framework to construct regions around the nominal parameter value such that the system is guaranteed to have a solution within a given distance of the nominal solution. The regions are described by explicit norm-like constraints on the parameters. We compare the results to previous approaches in the context of power systems.