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1. Network Reduction in Transient Stability Models using Partial Response Matching

   https://doi.org/10.1109/NAPS46351.2019.9000378  

   Francis, Benjamin L.; Nuttall, Jacob R.; Transtrum, Mark K.; Saric, Andrija T.; Stankovic, Aleksandar M (October 2019, 2019 North American Power symposium)

   Abstract—We describe a method for simultaneously identifying and reducing dynamic power systems models in the form of differential-algebraic equations. Often, these models are large and complex, containing more parameters than can be identified from the available system measurements. We demonstrate our method on transient stability models, using the IEEE 14-bus test system. Our approach uses techniques of information geometry to remove unidentifiable parameters from the model. We examine the case of a networked system with 58 parameters using full observations throughout the network. We show that greater reduction can be achieved when only partial observations are available, including reduction of the network itself. Index Terms—Parameter Estimation, Reduced Order Systems, System Identification. 

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2. Qualitative inverse problems: mapping data to the features of trajectories and parameter values of an ODE model

   https://doi.org/10.1088/1361-6420/acd414  

   Duan, X.; Rubin, J. E.; Swigon, D. (May 2023, Inverse Problems)

   Abstract Our recent work on linear and affine dynamical systems has laid out a general framework for inferring the parameters of a differential equation model from a discrete set of data points collected from a system being modeled. It introduced a new class of inverse problems where qualitative information about the parameters and the associated dynamics of the system is determined for regions of the data space, rather than just for isolated experiments. Rigorous mathematical results have justified this approach and have identified common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka–Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter and dynamical features in the data space that we call theP_n-diagram, which is particularly useful for visualization of the qualitative dependence of the system dynamics on data for low-dimensional (smalln) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associatedP₂-diagrams. 

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3. Generalizing Graph ODE for Learning Complex System Dynamics across Environments

   https://doi.org/10.1145/3580305.3599362  

   Huang, Zijie; Sun, Yizhou; Wang, Wei (August 2023, Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD ’23))

   Learning multi-agent system dynamics has been extensively studied for various real-world applications, such as molecular dynamics in biology, multi-body system in physics, and particle dynamics in material science. Most of the existing models are built to learn single system dynamics, which learn the dynamics from observed historical data and predict the future trajectory. In practice, however, we might observe multiple systems that are generated across different environments, which differ in latent exogenous factors such as temperature and gravity. One simple solution is to learn multiple environment-specific models, but it fails to exploit the potential commonalities among the dynamics across environments and offers poor prediction results where per-environment data is sparse or limited. Here, we present GG-ODE (Generalized Graph Ordinary Differential Equations), a machine learning framework for learning continuous multi-agent system dynamics across environments. Our model learns system dynamics using neural ordinary differential equations (ODE) parameterized by Graph Neural Networks (GNNs) to capture the continuous interaction among agents. We achieve the model generalization by assuming the dynamics across different environments are governed by common physics laws that can be captured via learning a shared ODE function. The distinct latent exogenous factors learned for each environment are incorporated into the ODE function to account for their differences. To improve model performance, we additionally design two regularization losses to (1) enforce the orthogonality between the learned initial states and exogenous factors via mutual information minimization; and (2) reduce the temporal variance of learned exogenous factors within the same system via contrastive learning. Experiments over various physical simulations show that our model can accurately predict system dynamics, especially in the long range, and can generalize well to new systems with few observations. 

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4. Global analysis of synchronization performance for power systems: bridging the theory-practice gap

   https://doi.org/10.1109/TAC.2019.2942536  

   Paganini, Fernando; Mallada, Enrique (September 2019, IEEE Transactions on Automatic Control)

   The issue of synchronization in the power grid is receiving renewed attention, as new energy sources with different dynamics enter the picture. Global metrics have been proposed to evaluate performance, and analyzed under highly simplified assumptions. In this work we extend this approach to more realistic network scenarios, and more closely connect it with metrics used in power engineering practice. In particular, our analysis covers networks with generators of heterogeneous ratings and richer dynamic models of machines. Under a suitable proportionality assumption in the parameters, we show that the step response of bus frequencies can be decomposed in two components. The first component is a system-wide frequency that captures the aggregate grid behavior, and the residual component represents the individual bus frequency deviations from the aggregate. Using this decomposition, we define --and compute in closed form-- several metrics that capture dynamic behaviors that are of relevance for power engineers. In particular, using the system frequency, we define industry-style metrics (Nadir, RoCoF) that are evaluated through a representative machine. We further use the norm of the residual component to define a synchronization cost that can appropriately quantify inter-area oscillations. Finally, we employ robustness analysis tools to evaluate deviations from our proportionality assumption. We show that the system frequency still captures the grid steady-state deviation, and becomes an accurate reduced-order model of the grid as the network connectivity grows. Simulation studies with practically relevant data are included to validate the theory and further illustrate the impact of network structure and parameters on synchronization. Our analysis gives conclusions of practical interest, sometimes challenging the conventional wisdom in the field. 

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5. Optimizing differential equations to fit data and predict outcomes

   https://doi.org/10.1002/ece3.9895  

   Frank, Steven A. (March 2023, Ecology and Evolution)

   Abstract Many scientific problems focus on observed patterns of change or on how to design a system to achieve particular dynamics. Those problems often require fitting differential equation models to target trajectories. Fitting such models can be difficult because each evaluation of the fit must calculate the distance between the model and target patterns at numerous points along a trajectory. The gradient of the fit with respect to the model parameters can be challenging to compute. Recent technical advances in automatic differentiation through numerical differential equation solvers potentially change the fitting process into a relatively easy problem, opening up new possibilities to study dynamics. However, application of the new tools to real data may fail to achieve a good fit. This article illustrates how to overcome a variety of common challenges, using the classic ecological data for oscillations in hare and lynx populations. Models include simple ordinary differential equations (ODEs) and neural ordinary differential equations (NODEs), which use artificial neural networks to estimate the derivatives of differential equation systems. Comparing the fits obtained with ODEs versus NODEs, representing small and large parameter spaces, and changing the number of variable dimensions provide insight into the geometry of the observed and model trajectories. To analyze the quality of the models for predicting future observations, a Bayesian‐inspired preconditioned stochastic gradient Langevin dynamics (pSGLD) calculation of the posterior distribution of predicted model trajectories clarifies the tendency for various models to underfit or overfit the data. Coupling fitted differential equation systems with pSGLD sampling provides a powerful way to study the properties of optimization surfaces, raising an analogy with mutation‐selection dynamics on fitness landscapes. 

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