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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Active Learning of Microgrid Frequency Dynamics Using Neural Ordinary Differential Equations

Accurate frequency modelling of inverter‐based resource (IBR)‐dominated power systems is crucial for ensuring stable, reliable and resilient operations, particularly given their inherent low‐inertia characteristics and fast dynamics that traditional swing equation‐based models inadequately capture. This paper explores neural ordinary differential equations (Neural ODEs) as a computationally efficient, data‐driven framework for modelling power system frequency dynamics, specifically within microgrids integrating high penetrations of distributed energy resources (DERs). The developed neural ODEs framework incorporates a neural network architecture designed to capture input dynamics. By actively perturbing the system with a known signal, the Python‐based neural ODEs framework was trained using measured system states and inputs, without the need for detailed system information. The framework, tested on a model of the Cordova, AK, microgrid, achieved a goodness of fit ranging from 60% to 99% across different state variables and maintained a mean square error in the 10 -6 p.u. range under square and step excitation signals. The proposed approach demonstrated robustness to measurement noise and initial condition variations while maintaining low computational complexity suitable for real‐time power system control applications. Furthermore, transfer learning enabled the neural ODEs model to adapt to the following changes in system topology or generator dispatch, highlighting its effectiveness for dynamic microgrids with frequently evolving configurations and diverse DERs.

Aryal, Tara [South Dakota State Univ., Brookings, ↗
DOE OSTI2022

Dynamic Parameter Estimation with Physics-based Neural Ordinary Differential Equations

Accurate estimation of dynamic parameters of gen-erators is crucial to building a reliable model for dynamical studies and reliable operation of the power system. This paper develops a physics-based neural ordinary differential equations (ODE) approach to learn the parameters of generator dynamic model using phasor measurement units (PMU) data. We design a physics-based neural network to represent the swing equations of the power system dynamics. A loss function is defined as the difference between dynamic simulation results from the physics-based neural networks and pseudo PMU measurements. The parameters of generator dynamic model are iteratively updated using the neural ODEs and the adjoint method. By exploiting the mini-batch scheme in neural ODE training, the parameter estimation performance is significantly improved. Numerical study results on a 3-machine 9-bus system show that the proposed algorithm outperforms state-of-the-art baseline method in both computation time and dynamic parameter estimation accuracy.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Physics-Informed Gaussian Process Regression for States Estimation and Forecasting in Power Grids

Real-time state estimation and forecasting are critical for the efficient operation of power grids. In this paper, a physics-informed Gaussian process regression (PhI-GPR) method is presented and used for forecasting and estimating the phase angle, angular speed, and wind mechanical power of a three-generator power grid system using sparse measurements. In standard data-driven Gaussian process regression (GPR), parameterized models for the prior statistics are fit by maximizing the marginal likelihood of observed data. In the PhI-GPR method, we propose to compute the prior statistics offline by solving stochastic differential equations (SDEs) governing the power grid dynamics. The short-term forecast of a power grid system dominated by wind generation is complicated by the stochastic nature of the wind and the resulting uncertainty in wind mechanical power. Here, we assume that the power grid dynamics are governed by swing equations, with the wind mechanical power fluctuating randomly in time. We solve these equations for the mean and covariances of the power grid states using the Monte Carlo simulation method. We demonstrate that the proposed PhI-GPR method can accurately forecast and estimate observed and unobserved states. For the considered problem, PhI-GPR has computational advantages over the ensemble Kalman filter (EnKF) method: In PhI-GPR, ensembles are computed offline and independently of the data acquisition process, whereas for EnFK, ensembles are computed online with data acquisition, rendering real-time forecast more challenging. We also demonstrate that the PhI-GPR forecast is more accurate than the EnKF forecast when the random mechanical wind power is non-Markovian. In contrast, the two methods produce similar forecasts for the Markovian mechanical wind power. For observed states, we show that PhI-GPR provides a forecast comparable to the standard data-driven GPR; both forecasts are significantly more accurate than the autoregressive integrated moving average (ARIMA) forecast. We also show that the ARIMA forecast is more sensitive to observation frequency and measurement errors than the PhI-GPR forecast.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Optimal operational controls for power grid

Power grid operation is facing increased penetration of renewables and distributed energy resources at both transmission and distribution levels. This presents an increased level of randomness and uncertainty for its operational quality, where the challenges are mainly how controls and optimizations can be performed so that the operational quality of the power grid can be maintained. An interesting feature is also that we need to make a good use of uncertainties embedded in the swing equation from the renewable contributions rather than simply trying to minimize the impact of uncertainties. Furthermore, this requires effective solutions on system estimation, control, and optimization. Indeed, in past decades, work has been carried out on the development of advanced controls and optimization for such complicated dynamic systems. Examples are optimal operational control for both performance and stability enhancement considering random renewables penetration, frequency, and active power control for microgrids and parameter estimation for various parts of the power grid.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

Decentralized Data-Driven Estimation of Generator Rotor Speed and Inertia Constant Based on Adaptive Unscented Kalman Filter

This paper proposes an online decentralized and data-driven method for synchronous generator rotor speed and inertia estimation. First, an analytical relationship between the generator terminal voltage/current phasors and its internal rotor speed is derived using the Thevenin equivalent. The parameters of the latter are obtained via a recursive estimator, which allows us to estimate the generator's internal rotor speed from voltage/current phasors without the need for any generator information. Second, we reformulate the equivalent swing equation into the Kalman filter form, which enables us to use a priori-estimated internal rotor speed along with measured real and reactive power injections to obtain an estimate of the inertia for each online synchronous generator. This is achieved through an adaptive unscented Kalman filter. We demonstrate that the use of terminal bus frequency to approximate the rotor speed for inertia estimation by existing methods leads to large estimation biases/errors. Numerical simulations carried out on the IEEE 39-bus and Texas 2000-bus systems reveal that our proposed method is consistently more accurate than existing methods. Moreover, our proposed method is insensitive to both the type and location of system disturbances, and it is robust to different levels of measurement noise.

bus frequency↗
DOE OSTI2023

Quantification of storage required for preserving frequency security in wind‐integrated systems

Abstract The penetration of wind power generation into the power grid has been accelerated in recent times due to the aggressive emission targets set by governments and other regulatory authorities. Although wind power has the advantage of being environment‐friendly, wind as a resource is intermittent in nature. In addition, wind power contributes little inertia to the system as most wind turbines are connected to the grid via power electronic converters. These negative aspects of wind power pose serious challenges to the frequency security of power systems as penetration increases. In this work, an approach is proposed where an energy storage system (ESS) is used to mitigate frequency security issues of wind‐integrated systems. ESSs are well equipped to supply virtual inertia to the grid due to their fast‐acting nature, thus replenishing some of the energy storage capability of displaced inertial generation. In this work, a probabilistic approach is proposed to estimate the amount of inertia required by a system to ensure frequency security. Reduction in total system inertia due to the displacement of conventional synchronous generation by wind power generation is considered in this approach, while also taking into account the loss of inertia due to forced outages of conventional units. Monte Carlo simulation is employed for implementing the probabilistic estimation of system inertia. An ESS is then sized appropriately, using the system swing equation, to compensate for the lost inertia. The uncertainty associated with wind energy is modeled into the framework using an autoregressive moving average technique. Effects of increasing the system peak load and changing the wind profile on the expected system inertia are studied to illustrate various factors that might affect system frequency security. The proposed method is validated using the IEEE 39‐bus test system.

17 WIND ENERGY↗
DOE OSTI2022

Enhanced Frequency Support Scheme of Generic Inverter-Based Resource Models for Renewable-Dominated Power Grids

The frequency response of SG-dominated power grids is predictable ahead of an occurrence of a frequency event because the frequency response of SGs is consistent, and it can be inferred from the swing equation [1]. However, increasing the portion of IBRs in an SG-dominated power grid might make the characteristics of the conventional power grids no longer valid because this changing resource mix affects grid dynamics and controls [2]. Thus, maintaining these characteristics greatly benefits the control and operation of the power grids with high penetration of IBRs. To maintain these characteristics in IBR-dominated power grids, IBRs should have frequency response capability similar to that of an SG. The WECC modeling validation subcommittee has developed generic IBR models for large system planning [3]-[5]. These models can represent various vendors' dynamic behavior for WTG, PV, and ESS [5]. The current generic IBR models approved by WECC can provide frequency response only from droop control loops in REPC models [6], [7]. The contribution of the loops is proportional to the frequency deviation from the nominal frequency. Thus, it presents an insufficient contribution to arrest frequency variation compared to the frequency response of SGs because it allows a high ROCOF in the early stage of frequency events. This shortfall will become greater as the PL of IBRs increases in power grids. Controller enhancement for the generic IBR models is required to secure the frequency stability under high PL of IBRs as in the SG-dominated power grids. This paper proposes a control extension for the generic IBR models to enhance the frequency support capabilities and discusses the classification of frequency support for the different types of IBR considering their operating constraints. An inertial control scheme is implemented in the REPC and REEC models of the generic IBR models to achieve these objectives. The inertial control scheme includes the following stages: Control area data acquisition, inertia time constant estimation, IBR-related constraint check, IBR contribution determination, and inertial response provision. In the scheme, a REPC acquires control area data from a system operator and estimates a total inertia time constant for the control area the applicable IBR power plant belongs. Then, the estimated inertial time constant is transferred to each IBR controller—REEC—within the power plant. Each REEC checks the availability of applicable IBR for inertial response participation. If the IBR is available, the REEC amplifies the estimated inertial time constant to utilize it for inertial response provision. In this way, the proposed inertial response scheme extends the functionality of the generic IBR models to provide SG-like frequency response within their constraints. Various scenarios considering different IBR types, IBR penetration levels, and frequency control schemes were simulated and compared in an IEEE 39-bus system using PSCAD simulator to verify the effectiveness of the proposed scheme.

Kim, Jinho↗
DOE OSTI2024

Towards the Flexibility of HVDC-Interconnected Systems: A Novel Emergency Freqeuncy Response Model

In this paper, we propose a novel multi-time scale emergency frequency response model by unlocking the flexibility of High Voltage Direct Current (HVDC) systems. Unlike assigning power ramping rates for FACTS to regulate frequency in traditional methods, this paper designs a step-change electromagnetic power frequency response (EPFR) scheme, by leveraging the temporal over/under DC voltage capability of HVDC. Wherein the Kullback-Leibler Divergence is adopted to convexify the modified swing equation after integrating the step-change power. Further, to avoid the complicated differential equations, we equivalently reformulate the duration limits of DC voltage deviation into the HVDC decreasing power ramping rates, which participate in the system primary frequency response. Finally, the new steady-state operation level of HVDC is involved with the secondary frequency response. As a result, an improved three-level algorithm is developed to solve the model, wherein the instant step-change, primary, and secondary frequency response are coordinated together. After applying the proposed frequency response scheme on the test system, the EPFR is validated to effectively provide the most instantaneous supports when faced with bulk power loss due to extreme contingencies, and the resilience is ensured within acceptable expenditures.

Jiang, Sufan↗
DOE OSTI2023

Developing VSC-HVDC Oscillation Damping Control Constraints in Unit Commitment

High-voltage direct current (HVDC) systems within the existing power grids will play a pivotal role in enabling high share of renewable power integration and transportation electrification. There is a need for transmission planning models to accurately capture the impact of VSC-HVDC oscillation damping control on the system planning and model the constraints in the security constrained unit commitment (SCUC) problem. In this paper, the SCUC problem in a hybrid DC-AC grid considering damping on electromechanical inter-area oscillation is studied. An electromechanical oscillation-driven transmission capability constraints are derived from system swing equation and incorporated into the SCUC. Multiple system damping control scenarios are set based on the extracted plausible bounds of damping coefficient and inertia constant to linearize the nonlinear expressions. The SCUC problem with proposed constraints is tested on the Reliability Test System Grid Modernization Lab Consortium (RTS-GMLC) via PLEXOS. The impact of the proposed constraints on regional generation, inertia, power flow, annual system cost, and renewable energy curtailment are fully assessed.

damping control↗
DOE OSTI2025

Open-Source Ambient-Signal-Based Inertia Monitoring Tool for Power systems [SWR-25-59]

This script, developed by NREL, implements an inertia estimation algorithm using a sliding window approach. The inertia (H) is estimated based on frequency deviations and active power deviations, utilizing the swing equation from power system dynamics. An example application is provided using the WECC 240-bus system.

Peng, Jiangkai [National Renewable Energy Laborato↗
DOE OSTI2020

Scalable, Physical Effects Measurable Microgrid for Cyber Resilience Analysis (SPEMMCRA)

The ability to advance state of the art automated protections for industrial control systems (ICS) has as a precursor in the ability to understand the tradeoff space. That is, to enable a cyber feedback loop in a control system environment you must first consider both the security mitigation available, the benefits and the impacts to the control system functionality when the mitigation is used. More damaging impacts could be precipitated that the mitigation was intended to rectify. This paper details networked ICS that controls a simulation of the frequency response represented with the swing equation. The microgrid loads and base generation can be balanced through the control of an emulated battery and power inverter. The simulated plant, which is implemented in Raspberry Pi computers, provides an inexpensive platform to realize the physical effects of cyber attacks to show the tradeoffs of available mitigatoins. This network design can include a commercial ICS controller to introduce real world implementation of feedback controls, and provides a scalable, physical effects measurable Microgrid for cyber resilience analysis (SPEMMCRA).

42 ENGINEERING↗
DOE OSTI2023

Developing VSC-HVDC Oscillation Damping Control Constraints in Unit Commitment

High-voltage direct current (HVDC) systems within the existing power grids will play a pivotal role in enabling high share of renewable power integration and transportation electrification. There is a need for transmission planning models to accurately capture the impact of VSC-HVDC oscillation damping control on the system planning and model the constraints in the security constrained unit commitment (SCUC) problem. In this paper, the SCUC problem in a hybrid DC-AC grid considering damping on electromechanical inter-area oscillation is studied. An electromechanical oscillation-driven transmission capability constraints are derived from system swing equation and incorporated into the SCUC. Multiple system damping control scenarios are set based on the extracted plausible bounds of damping coefficient and inertia constant to linearize the nonlinear expressions. The SCUC problem with proposed constraints is tested on the IEEE 73-bus reliability test system (RTS) case via PLEXOS. The impact of the proposed constraints on regional generation, inertia, power flow, annual system cost, and renewable energy curtailment are fully assessed.

damping control↗
DOE OSTI2024

Power System Frequency Dynamics Modeling, State Estimation, and Control using Neural Ordinary Differential Equations (NODEs) and Soft Actor-Critic (SAC) Machine Learning Approaches

With the global energy transition of the electric power system, grid control, supervision, and protection is becoming more challenging. With the increasing integration of renewable energy sources (RES), the system dynamics are changing, causing traditional power system dynamic modeling with swing equation-based modeling approaches to fail. Additionally, the converter-dominated power grid is decreasing the system inertia, making the power system more fragile to the frequency swings. This paper first investigates and compares the application of a model-based Kalman filter state estimation approach with (i) a model-free machine learning approach --- neural ordinary differential equations (NODEs) --- and (ii) a data-driven system identification (SysId) approach to model and infer critical state values of the power system frequency dynamics. Then a model predictive control (MPC) framework is compared to a model-free Soft Actor-Critic (SAC) reinforcement learning (RL) control algorithm in providing efficient fast frequency response (FFR) to the power system frequency dynamics. The approaches are compared in terms of their performance goals as well as their per-timestep computational efficiency. Furthermore, the comparative study for state estimation shows that for the model-free requirement, both NODEs and SysId can provide accurate state estimates; however, with increasing model complexity, NODEs can be a better choice for model identification. Similarly, the results from the FFR comparative study show that the SAC RL-based FFR, once trained, outperforms MPC with better control signals and faster computation time, making the SAC RL-based FFR better option for providing FFR to the power system.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Comprehensive Analysis of PINNs for Power System Transient Stability

The integration of machine learning in power systems, particularly in stability and dynamics, addresses the challenges brought by the integration of renewable energies and distributed energy resources (DERs). Traditional methods for power system transient stability, involving solving differential equations with computational techniques, face limitations due to their time-consuming and computationally demanding nature. This paper introduces physics-informed Neural Networks (PINNs) as a promising solution for these challenges, especially in scenarios with limited data availability and the need for high computational speed. PINNs offer a novel approach for complex power systems by incorporating additional equations and adapting to various system scales, from a single bus to multi-bus networks. Our study presents the first comprehensive evaluation of physics-informed Neural Networks (PINNs) in the context of power system transient stability, addressing various grid complexities. Additionally, we introduce a novel approach for adjusting loss weights to improve the adaptability of PINNs to diverse systems. Our experimental findings reveal that PINNs can be efficiently scaled while maintaining high accuracy. Furthermore, these results suggest that PINNs significantly outperform the traditional ode45 method in terms of efficiency, especially as the system size increases, showcasing a progressive speed advantage over ode45.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Comprehensive Analysis of PINNs for Power System Transient Stability

The integration of machine learning in power systems, particularly in stability and dynamics, addresses the challenges brought by the integration of renewable energies and distributed energy resources (DERs). Traditional methods for power system transient stability, involving solving differential equations with computational techniques, face limitations due to their time-consuming and computationally demanding nature. This paper introduces physics-informed Neural Networks (PINNs) as a promising solution for these challenges, especially in scenarios with limited data availability and the need for high computational speed. PINNs offer a novel approach for complex power systems by incorporating additional equations and adapting to various system scales, from a single bus to multi-bus networks. Our study presents the first comprehensive evaluation of physics-informed Neural Networks (PINNs) in the context of power system transient stability, addressing various grid complexities. Additionally, we introduce a novel approach for adjusting loss weights to improve the adaptability of PINNs to diverse systems. Our experimental findings reveal that PINNs can be efficiently scaled while maintaining high accuracy. Furthermore, these results suggest that PINNs significantly outperform the traditional ode45 method in terms of efficiency, especially as the system size increases, showcasing a progressive speed advantage over ode45.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2026

Inertia estimation for power grids: A review of methods, challenges, and future prospects

The electric power grid is undergoing a significant transformation, shifting from traditional synchronous generators to inverter-based resources (IBRs) such as solar photovoltaics, wind turbines, and energy storage systems. This evolution leads to a reduction in system inertia, a critical attribute for maintaining frequency stability in response to disturbances. Consequently, the ability to monitor and estimate system inertia has become increasingly essential. This paper provides a comprehensive review of existing inertia estimation methodologies, analyzing them from multiple perspectives, including the types of data utilized, underlying estimation principles, operational modes, and system-wide applicability. A comparative summary table is included to distill commonalities and key characteristics across various studies. In addition, the paper examines practical implementations of inertia estimation across several major power systems worldwide, including the U.S. interconnections, the Nordic power system, and the U.K. grid. Key challenges are identified, particularly in estimating contributions from virtual inertia sources and load-induced inertia in increasingly converter-dominated networks. To address these emerging challenges, the paper proposes an integrated framework for real-time inertia estimation and monitoring. This framework encompasses critical components such as data acquisition, inertia estimation from both synchronous and non-synchronous sources, load-induced effects, optimization techniques, forecasting, and virtual inertia scheduling. Collectively, these elements enable dynamic, system-wide monitoring and adaptive control of grid inertia.

Inertia estimation↗
DOE OSTI2025

Consensus-Based Control and Optimization of Power System Inertia

The integration of distributed energy resources (DERs), such as solar, wind, and energy storage systems, into power grids through inverter-based resources reduces power system inertia, leading to faster frequency dynamics and potential grid instability. To address this challenge, this paper proposes a distributed, consensus-based approach for the real-time control and optimization of inertia sources (synchronous generators and/or DERs) during system disturbances, enhancing both system stability and economic performance. The distributed control and optimization approach assumes each inertia source exchanges information solely with its neighbor ones, making it easily scalable to large power grid networks. The impacts of the communication connectivity among the inertia sources as well as their generation capacity limits on the distributed approach are investigated. We also demonstrate the approach’s robustness in scenarios involving communication time delays and packet losses, validating its effectiveness through numerical simulations on a 4-bus test system and a two-area 8-bus test system.

Power system inertia↗