In this talk, we will present a new globally convergent model-based stochastic search method for optimal hardening of power lines to improve power grid weather resilience. The new method integrates a recently developed model-free screening method to statistically rank the criticality of power lines using cascading outage simulation data. The dimension reduction achieved by model-free screening enables the new stochastic search method to efficiently identify optimal hardening plans in a high-dimensional search space that is previously intractable by discrete optimization via simulation methods. The new search method also exploits the budget constrained integral search space to design a simple but yet highly efficient sampling distribution model to search the solution space. The proposed method is among the first to perform efficient search of a high-dimensional integral solution space using model-free dimension reduction, and shows promising numerical performance in our preliminary simulation study.

In this presentation, we summarize our completed projects and depict some future research plans. We have explored new statistics and learning techniques including robust statistics, functional linear regression models, and LASSO based approaches. These projects have assisted us to gain a deep understanding of the intrinsic properties and structures, and to develop effective algorithms to make the smart grid more secure and efficient. During the investigating of these projects, we found that the new methodology from the recently developed neural network can be applied to functional data analysis. It can be further applied to load and power generation forecasting in the smart grid. We plan to explore this new research direction to enrich the development of the power system.

This talk focuses on using a machine learning framework to locate power line outages. The same framework is used for the prediction of both single line and multiple line outages. We investigate a range of machine learning algorithms and feature extraction methods. The algorithms are designed to capture the essential dynamic characteristics of the power system when the topology change occurs abruptly. We tested the proposed methods on their prediction performance under different levels of noise and missingness. It is shown that the proposed methods have better tolerance for noisy data and incomplete data compared to the previous work.

We will present an algorithm for the isolation of line outages that relies on a transient stability analysis of the power system, as well as an algorithm for the joint detection and isolation of lines outages in the presence of measurement error. Their performances will be evaluated using synthetic data under various fault parameters, such as the pre-fault operating point and fault location on a line.

We will review the work our team has done to develop an algorithm aimed at detecting and then localizing a fault in a simulated power grid covering the western United States. We will introduce the regional grid simulator operated by the University of Wyoming, the types of faults it can simulate and the signatures of these faults (time series of voltage, current and phase). We will then explain how our algorithm works and discuss our plans to improve it.

Modeling and predicting solar events, in particular, the solar ramping event, is critical for improving situational awareness for solar power generation systems. However, this is a difficult task since solar ramping events are significantly impacted by weather conditions such as temperature, humidity, and cloud density. We develop a novel method to model and predict ramping events based on a spatio-temporal interactive Bernoulli process and demonstrate our approach's good performance on real solar radiation datasets.

Modeling and predicting solar events, in particular, the solar ramping event, is critical for improving situational awareness for solar power generation systems. However, this is a difficult task since solar ramping events are significantly impacted by weather conditions such as temperature, humidity, and cloud density. We develop a novel method to model and predict ramping events based on a spatio-temporal interactive Bernoulli process and demonstrate our approach's good performance on real solar radiation datasets.

We examine synchronization of large-scale power grid models and how synchrony can be used for model reduction and parameter selection for stable power networks.

A dynamic neural network (NN) based multi-class classifier is proposed for improving online prediction of co-herent generator groups (CGGs), following the occurrences of various contingencies in the power grid. This is motivated by the increasing availability of the measurements from phasor measurement units (PMUs) and the number of grouping schemes is limited. The proposed method consists of three steps. First, by performing offline simulations, a library of system dynamic responses characterized by post-contingency rotor angles and speeds of individual generators is obtained. To generate sufficient data, up to N-2 contingencies and the uncertain parameters associated with the power grid including type and location of disturbance and fault clearing times are modeled. Secondly, the training dataset is produced by generating labels for individual contingencies using a hierarchical clustering method based on rotor angle and speed data. Finally, the dynamic NN models are trained for online applications such as emergency controls and controlled islanding. The proposed method is tested on the standard 16-generator 68-bus system to demonstrate its performance. Furthermore, the impact of the sample data lengths on the CGG numbers is evaluated. It is interesting to observe that the time domain stability behaviors can be determined by examining the changes in the CGG numbers.

I will discuss our recent work on the interplay between time-of-use rates, network reliability and DER adoption & dispatch.

In a recent publication we demonstrated the effectiveness of stochastic collocation with sparse grids for the quantification of uncertainty of non-linear random power flow networks. Given sufficient complex analytic regularity properties, (fast) sub-exponential convergence rates have been derived in theory and shown in practice for the 39 Bus 10 Generator New England network. Nevertheless, if regularity is low, even at a single point, the fast convergence rates of sparse grids are lost. This motivates the application of Radial Basis Function (RBF) interpolation, where high and low regularity regions can be more accurately approximated. However, solving the RBF system can be slow and numerically unstable, thus challenging its effectiveness for stochastic collocation. A multilevel basis is constructed to be adapted to a kD-tree partitioning of the observations. Numerically unstable RBF systems are mapped into well conditioned multilevel matrices. It is shown that this transformation is exact without any loss of accuracy. The multilevel method is tested on numerically unstable problems of up 25 dimensions. The numerical efficiency is increased many of orders of magnitude for the same accuracy. Moreover, unsolvable RBF problems on a double precision computer can be accurately solved with this multilevel method.

This is a joint work with Mark Kon.

Time-varying optimization is a non-traditional type of online optimization that assumes the objective function changes in a small but arbitrary amount over time. In particular, we assume the problem is large enough, and the change fast enough, that one cannot solve the problem completely within one discrete time step. Notably, this model captures power systems that have fast dynamics. We seek algorithms that come as close as possible to tracking the true solution. Under certain assumptions, we prove a lower bound on the complexity of a class of these algorithms. Online gradient descent does not achieve this bound, nor does standard Nesterov-style acceleration. We develop a "long-step" Nesterov style method which does achieve the lower bound and is thus optimal. We also discuss future directions about using deep learning for power systems, where we rigorously analyze generalization bounds using algorithmic stability. Joint work with Emiliano Dall'Anese (U. Colorado, electrical engineering) and Liam Madden (U. Colorado, applied math).

We propose a new modeling framework for highly multivariate stochastic processes that synthesizes ideas from recent multiscale and spectral approaches with graphical models. Krock et al. (2020) writes a univariate Gaussian process as a linear combination of basis functions weighted with entries of a gaussian graphical vector whose graph is estimated from optimizing an L1 penalized likelihood. This work extends the setting to a multivariate process where the basis functions are weighted with Gaussian graphical vectors. We develop a basis model where the graphical vectors are assumed to be independent by level. Using an orthogonal basis grants linear complexity and memory usage in the number of locations, the number of basis functions, and the number of realizations. An additional fusion penalty encourages a parsimonious conditional independence structure in the multilevel graphical model. We illustrate our method on a large climate ensemble from the National Center for Atmospheric Research's Community Atmosphere Model that involves 40 spatial processes, many of which are critical for understanding renewable power energy availability in the future.

We will present a proximal algorithm that implements a variational recursion on the space of joint probability measures to propagate the stochastic uncertainties in power system dynamics over high dimensional state space. The proposed algorithm takes advantage of the exact nonlinearity structures in the trajectory-level dynamics of the networked power systems, and shows that by lifting the dynamics in the space of measures, it is possible to design a scalable algorithm that obviates the discretization of the high dimensional state space or function approximation. We will provide the theoretical details, convergence guarantees, and numerical examples on realistic test systems.

Dynamical systems representing power networks have inherent structure reflecting underlying physics. When constructing reduced-order models for these systems it is vital to retain this structure so that the reduced models can be interpreted as physically meaningful surrogates. We first develop a structure-preserving model reduction approach for parametric linearized swing equations. In this setting, parameters represent variations in operating conditions and the parametric reduced model is capable of providing a high-fidelity approximation over a wide range of operating conditions. These capabilities are expanded to include nonlinear swing dynamics, which we reformulate (without approximation error) as a quadratically nonlinear system. This, in turn, allows us to employ a wide range of input-independent model reduction techniques that retain also critical nonlinear structural features of the original system.

Complementary data-driven approaches are considered as well. Indeed, synchrophasor data provide unprecedented opportunities for inferring power system dynamics, i.e., estimating voltage angles, frequencies, and accelerations along with power injection at all buses. Aligned to this goal, we offer a novel framework for learning dynamics induced by small-signal disturbances leveraging the toolbox of Gaussian processes (GPs). We extend results on inferring the input and output of a linear time-invariant system using GPs to the multi-input multi-output setup by exploiting swing dynamics. This physics-aware learning technique captures time derivatives in continuous time, accommodates datastreams sampled potentially at different rates, and is able to cope with missing data and heterogeneous levels of accuracy. While Kalman filter-based approaches require knowledge of all system inputs, the framework we introduce handles readings of system inputs, outputs, their derivatives, and combinations thereof on an arbitrary subset of buses. Relying on minimal system information, it allows for uncertainty quantification in addition to point estimates of system dynamics. GP model parameters are computed from data using a scalable method-of-moments approach. Numerical tests verify that this technique can monitor frequencies at non-metered buses, impute and predict synchrophasor data, and infer power deviations, all under approximate and more detailed power system dynamic models.

Failure in brittle materials led by the evolution of micro- to macro-cracks under repetitive or increasing loads is often catastrophic with no significant plasticity to advert the onset of fracture. Early failure detection with respective location are utterly important features in any practical application, both of which can be effectively addressed using artificial in- telligence. In this paper, we develop a supervised machine learning (ML) framework to predict failure in an isothermal, linear elastic and isotropic phase-field model for damage and fatigue of brittle materials. Time-series data of the phase- field model is extracted from virtual sensing nodes at different locations of the geometry. A pattern recognition scheme is introduced to represent time-series data/sensor nodes responses as a pattern with a corresponding label, integrated with ML algorithms, used for damage classification with identified patterns. We perform an uncertainty analysis by superposing random noise to the time-series data to assess the robustness of the framework with noise-polluted data. Results indicate that the proposed framework is capable of predicting failure with acceptable accuracy even in the presence of high noise levels. The findings demonstrate satisfactory performance of the supervised ML framework, and the applicability of artificial intelligence and ML to a practical engineering problem, i.,e, data-driven failure prediction in brittle materials.

Phasor measurement units (PMUs) provide synchronized phasor measurements at a sampling rate of 30 or 60 samples per second. This data provides insight into the operation of the electrical grid. However, there may be missing data or errors in the data. The ground truth matrix of PMU data is typically of low rank, so it is desired to develop methods to fit a low-rank matrix to the observed data. Mathematically, we wish to decompose a matrix into a sum of a low rank matrix (to capture the underlying structure) and a sparse matrix (to capture the missing data). We have developed methods based on nonconvex regularizers, which are functions of the singular values of the low rank matrix and the nonzeroes of the sparse matrix. The use of nonconvex regularizers has some practical and theoretical advantages over the use of convex approaches, and has been successfully implemented. We have also developed a real-time algorithm that exploits the low-rank Hankel structure of PMU data to fill in missing data and correct bad data in PMU measurements.

Joint work with April Sagan, Joe Chow, and Meng Wang.

The transition into renewable energy sources -with limited or no inertia- is seen as potentially threatening to classical methods for achieving grid synchronization. A widely embraced approach to mitigate this problem is to mimic inertial response using grid-connected inverters. That is, introduce virtual inertia to restore the stiffness that the system used to enjoy. In this talk, we seek to challenge this approach and advocate towards taking advantage of the system’s low inertia to restore frequency steady-state without incurring excessive control efforts. With this aim in mind, we develop an analysis and design framework for inverter-based frequency control. We define several performance metrics of practical relevance for power engineers and systematically evaluate the performance of standard control strategies, such as virtual inertia and droop control, in the presence of power disturbances and measurement noise. Our analysis unveils the relatively limited role of inertia on improving performance as well as the inability of droop control to improve performance without incurring large steady-state control efforts. To solve this problem, we propose a novel dynamic droop control (iDroop) for grid-connected inverters -exploiting classical lead/lag compensation from control theory- that can significantly outperform existing solutions with comparable control efforts.