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Accurate dynamic modeling of power systems is essential to assess the stability of electrical power systems when faced with disturbances, which can trigger cascading failures leading to blackouts. A continuum model proves to be effective in capturing Electromechanical Wave (EMW) propagation characteristics, including its velocity, arrival time, and deviations. Analyzing these characteristics enables the assessment of the impacts of EMW on the performance of the protection system. Prior research has often modeled nonlinear EMW propagation through Partial Differential Equations (PDEs) within a homogeneous and uniform frame structure, assuming constant bus voltages across the entire power system. However, this assumption can produce inaccurate results. In this paper, we relax this assumption by introducing a second-order nonlinear hyperbolic EMW propagation equation model that accounts for voltage variations. Additionally, we present numerical solutions for the EMW propagation equation using the Lax-Wendroff integration method. To validate our approach, we conduct simulations on two test systems: a two-bus one-machine system and the New England 39-bus 10-machine system. The simulation results demonstrate the effectiveness of our proposed model and emphasize the importance of including the bus voltage equations in the analysis.

Existing methods to determine the stability of a power system to small perturbations are based on eigenvalue analysis and focus on the asymptotic (long-term) behavior of the power grid. During the preasymptotic (short-term) transient, however, the system can exhibit large growth that is not explained by eigenvalues alone. In this paper we propose a new framework to determine the maximum (optimal) preasymptotic growth using the singular value decomposition. The approach is tailored to the analysis of quantities of interest in power system dynamics, such as the set of rotor speed deviations. Matrix-free techniques are developed to avoid the explicit formation of dense matrices and enable the analysis of large-scale systems without reaching memory bounds. Extensive results carried out from small to very large-scale systems (e.g., 70k-bus system) verify the theoretical aspects of the technique.

We investigate how to port the standard interior-point method to new exascale architectures for block-structured nonlinear programs with state equations. Computationally, we decompose the interior-point algorithm into two successive operations: the evaluation of the derivatives and the solution of the associated Karush-Kuhn-Tucker (KKT) linear system. Our method accelerates both operations using two levels of parallelism. First, we distribute the computations on multiple processes using coarse parallelism. Second, each process uses a SIMD/GPU accelerator locally to accelerate the operations using fine-grained parallelism. The KKT system is reduced by eliminating the inequalities and the state variables from the corresponding equations, to a dense matrix encoding the sensitivities of the problem's degrees of freedom, drastically minimizing the memory exchange. We demonstrate the method's capability on the supercomputer Polaris, a testbed for the future exascale Aurora system. Each node is equipped with four GPUs, a setup amenable to our two-level approach. Our experiments on the stochastic optimal power flow problem show that the method can achieve a 50x speed-up compared to the state-of-the-art method., Comment: 23 pages, 8 figures

A machine learning technique is proposed for quantifying uncertainty in power system dynamics with spatiotemporally correlated stochastic forcing. We learn one-dimensional linear partial differential equations for the probability density functions of real-valued quantities of interest. The method is suitable for high-dimensional systems and helps to alleviate the curse of dimensionality., Comment: 5 pages, 2 figures, Accepted to 2022 IEEE PES GM

Randomized linear solvers randomly compress and solve a linear system with compelling theoretical convergence rates and computational complexities. However, such solvers suffer a substantial disconnect between their theoretical rates and actual efficiency in practice. Fortunately, these solvers are quite flexible and can be adapted to specific problems and computing environments to ensure high efficiency in practice, even at the cost of lower effectiveness (i.e., having a slower theoretical rate of convergence). While highly efficient adapted solvers can be readily designed by application experts, will such solvers still converge and at what rate? To answer this, we distill three general criteria for randomized adaptive solvers, which, as we show, will guarantee a worst-case exponential rate of convergence of the solver applied to consistent and inconsistent linear systems irrespective of whether such systems are over-determined, under-determined or rank-deficient. As a result, we enable application experts to design randomized adaptive solvers that achieve efficiency and can be verified for effectiveness using our theory. We demonstrate our theory on twenty-six solvers, nine of which are novel or novel block extensions of existing methods to the best of our knowledge., Comment: 44 pages, 5 figures, 3 Tables

The interior-point method (IPM) has become the workhorse method for nonlinear programming. The performance of IPM is directly related to the linear solver employed to factorize the Karush--Kuhn--Tucker (KKT) system at each iteration of the algorithm. When solving large-scale nonlinear problems, state-of-the art IPM solvers rely on efficient sparse linear solvers to solve the KKT system. Instead, we propose a novel reduced-space IPM algorithm that condenses the KKT system into a dense matrix whose size is proportional to the number of degrees of freedom in the problem. Depending on where the reduction occurs we derive two variants of the reduced-space method: linearize-then-reduce and reduce-then-linearize. We adapt their workflow so that the vast majority of computations are accelerated on GPUs. We provide extensive numerical results on the optimal power flow problem, comparing our GPU-accelerated reduced space IPM with Knitro and a hybrid full space IPM algorithm. By evaluating the derivatives on the GPU and solving the KKT system on the CPU, the hybrid solution is already significantly faster than the CPU-only solutions. The two reduced-space algorithms go one step further by solving the KKT system entirely on the GPU. As expected, the performance of the two reduction algorithms depends intrinsically on the number of available degrees of freedom: their performance is poor when the problem has many degrees of freedom, but the two algorithms are up to 3 times faster than Knitro as soon as the relative number of degrees of freedom becomes smaller., Comment: 21 pages, 3 figures, 2 tables

This paper presents an efficient method for extracting the second-order sensitivities from a system of implicit nonlinear equations on upcoming graphical processing units (GPU) dominated computer systems. We design a custom automatic differentiation (AutoDiff) backend that targets highly parallel architectures by extracting the second-order information in batch. When the nonlinear equations are associated to a reduced space optimization problem, we leverage the parallel reverse-mode accumulation in a batched adjoint-adjoint algorithm to compute efficiently the reduced Hessian of the problem. We apply the method to extract the reduced Hessian associated to the balance equations of a power network, and show on the largest instances that a parallel GPU implementation is 30 times faster than a sequential CPU reference based on UMFPACK., Comment: SIAM-PP22

We propose a novel feasible-path algorithm to solve the optimal power flow (OPF) problem for real-time use cases. The method augments the seminal work of Dommel and Tinney with second-order derivatives to work directly in the reduced space induced by the power flow equations. In the reduced space, the optimization problem includes only inequality constraints corresponding to the operational constraints. While the reduced formulation directly enforces the physical constraints, the operational constraints are softly enforced through Augmented Lagrangian penalty terms. In contrast to interior-point algorithms (state-of-the art for solving OPF), our algorithm maintains feasibility at each iteration, which makes it suitable for real-time application. By exploiting accelerator hardware (Graphic Processing Units) to compute the reduced Hessian, we show that the second-order method is numerically tractable and is effective to solve both static and real-time OPF problems., Comment: 7 pages, 3 figures, conference proceeding

In this work we present a novel technique, based on a trust-region optimization algorithm and second-order trajectory sensitivities, to compute the extreme trajectories of power system dynamic simulations given a bounded set that represents parametric uncertainty. We show how this method, while remaining computationally efficient compared with sampling-based techniques, overcomes the limitations of previous sensitivity-based techniques to approximate the bounds of the trajectories, when the local approximation loses validity because of the nonlinearity. In addition, we show how this method can be adapted to account for those cases in which the initial conditions depend on the uncertain parameter. To conclude, we present several numerical experiments that showcase the accuracy and scalability of the technique, including a demonstration on the IEEE New England test system.

Deterministic and randomized, row-action and column-action linear solvers have become increasingly popular owing to their simplicity, low computational and memory complexities, and ease of composition with other techniques. Moreover, in order to achieve high-performance, such solvers must often be adapted to the given problem structure and to the hardware platform on which the problem will be solved. Unfortunately, determining whether such adapted solvers will converge to a solution has required equally unique analyses. As a result, adapted, reliable solvers are slow to be developed and deployed. In this work, we provide a general set of assumptions under which such adapted solvers are guaranteed to converge with probability one, and provide worst case rates of convergence. As a result, we can provide practitioners with guidance on how to design highly adapted, randomized or deterministic, row-action or column-action linear solvers that are also guaranteed to converge.

In this paper we focus on the parameter estimation of dynamic load models with stochastic terms, in particular, load models where protection settings are uncertain, such as in aggregated air conditioning units. We show how the uncertainty in the aggregated protection characteristics can be formulated as a stochastic differential equation with process noise. We cast the parameter inversion within a Bayesian parameter estimation framework, and we present methods to include process noise. We demonstrate the benefits of considering stochasticity in the parameter estimation and the risks of ignoring it., Comment: 7 pages, 9 figures, to be published in Electric Power Systems Research PSCC 2020

In this paper we present a probabilistic analysis framework to estimate behind-the-meter photovoltaic generation in real time. We develop a forward model consisting of a spatiotemporal stochastic process that represents the photovoltaic generation and a stochastic differential equation with jumps that represents the demand. We employ this model to disaggregate the behind-the-meter photovoltaic generation using net load and irradiance measurements.

Randomized linear system solvers have become popular as they have the potential to reduce floating point complexity while still achieving desirable convergence rates. One particularly promising class of methods, random sketching solvers, has achieved the best known computational complexity bounds in theory, but is blunted by two practical considerations: there is no clear way of choosing the size of the sketching matrix apriori; and there is a nontrivial storage cost of the sketched system. In this work, we make progress towards addressing these issues by implicitly generating the sketched system and solving it simultaneously through an iterative procedure. As a result, we replace the question of the size of the sketching matrix with determining appropriate stopping criteria; we also avoid the costs of explicitly representing the sketched linear system; and our implicit representation also solves the system at the same time, which controls the per-iteration computational costs. Additionally, our approach allows us to generate a connection between random sketching methods and randomized iterative solvers (e.g., randomized Kaczmarz method, randomized Gauss-Seidel). As a consequence, we exploit this connection to (1) produce a stronger, more precise convergence theory for such randomized iterative solvers under arbitrary sampling schemes (i.i.d., adaptive, permutation, dependent, etc.), and (2) improve the rates of convergence of randomized iterative solvers at the expense of a user-determined increases in per-iteration computational and storage costs. We demonstrate these concepts on numerical examples on forty-nine distinct linear systems., Comment: 46 pages