General Lab Information

Energy Systems Division

Renewables and Grid Modernization

Advanced Grid Modeling

Formal Analysis for Dynamic Stability Assessment


Completed on September 30, 2021.


The increased penetration of distributed energy resources (DERs) in the power grid can lead to new and unprecedented challenges in maintaining reliable operation. Power electronic devices are usually used to interface with DERs, such as FACTS, energy storage systems, and HVDC links with the grid. Although they enable ultra-fast grid control and load changes, the high penetration of power electronic components will reduce the grid inertia significantly, making the grid highly sensitive to disturbances and threatening the power system stability. This study developed new techniques for assessing the stability of transmission and distribution systems containing DERS.


To address this concern, BNL developed an innovative and tractable method for assessing the stability of both transmission and distribution systems in the power grid under uncertainties from heterogeneous sources, such as DERs. A formal theory with mathematical rigor was developed for computing the bounds of all possible (infinitely many) trajectories and estimating the stability margin for the entire system including the integrated transmission and distribution network. A new open-source Formal Analysis tool based on reachable set computations was developed that can be used for real-time dynamic analysis and stability margin calculations. It is applicable for not only forecasting and monitoring grid performance but also formally verifying various resiliency enhancement strategies such as new schemes for system integrity protection and automation to adapt to this evolution of electric networks.


  • Data mining techniques have been developed to capture the behaviors of the dynamic systems, as well as the uncertainties, without using the element-based dynamic equivalent.
  • Reachability analysis modules have been completed in a set of Matlab scripts.
  • The QDG method has been implemented in a Matlab script.
  • After reachable sets of system states were calculated, the corresponding Geršgorin disks for extracted edges of the reachable sets were sequentially evaluated to assess the stability condition. This has also been implemented in a Matlab script as part of the overall FA toolset.
  • Based on the proposed scheme of distributed formal analysis (DFA), a prototype local cloud computing platform using the concept of virtual machine has been developed to explore the DFA capability. The prototype platform is very promising based a proof-of-concept study using an example networked microgrid system.
  • Introduces a scalable reachability analysis technique into power grid applications to overcome issues with existing direct and numerical simulation methods and enable mathematically rigorous evaluation of the impacts of inherent uncertainties on the system trajectories based on reachable sets and eigen analysis
  • Developed distributed reachability analysis (DRA) or distributed formal analysis (DFA) scheme for ordinary differential equation (ODE) representation of dynamic systems and can be applied to power systems for better convergency and scalability.
  • Developed a data driven reachability analysis scheme based on real-time measurement to reduce system complexity and focus on the portion of the system of particular interest.
  • Developed ODE representation, i.e., electro-magnetic transient (EMT) type models, of power grids by modeling major components and their associated controls in transmission and distribution networks.
  • Developed a MATLAB®-based grid reachability analysis (GRA) tool, available by request, that can be used to compute reachable sets for generic power systems including the integrated transmission and distribution systems using centralized, distributed, and data-driven formal analysis.


M. Yue and A. Yogarathnam, “Formal Analysis for Dynamic Stability Assessment of Large Interconnected Grids under Uncertainties,” Technical report submitted to DOE Office of Electricity, AGM Program, November 2020.

S. Zhang, A. Yogarathinam, J. Zhan, M. Yue, and G. Lin, “A Step Towards Machine Learning-based Coherent Generator Grouping for Emergency Control Applications in Modern Power Grid,” accepted by IEEE PES General Meeting 2020.

 Y. Zhou, P. Zhang, and M. Yue, “An ODE-Enabled Distributed Transient Stability Analysis for Networked Microgrids,” was submitted to IEEE PES GM 2020, and is under second review.

 Y. Li, P. Zhang, M. Althoff, and M. Yue, “Distributed Formal Analysis for Power Networks with Deep Integration of Distributed Energy Resources”, IEEE Transactions on Power Systems, Vol. 34, No. 6, November 2019.

Y. Li, P. Zhang, and M. Yue, “Networked Microgrid Stability through Distributed Formal Analysis,” Applied Energy, Vol. 228, October 2018, pp279-288.