| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | August 13, 2019 |
| Latest Amendment Date: | August 13, 2019 |
| Award Number: | 1923099 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Tomek Bartoszynski
tbartosz@nsf.gov (703)292-4885 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 15, 2019 |
| End Date: | July 31, 2023 (Estimated) |
| Total Intended Award Amount: | $105,281.00 |
| Total Awarded Amount to Date: | $105,281.00 |
| Funds Obligated to Date: |
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| Recipient Sponsored Research Office: |
7430 EAST DR LIBRARY TOWER 700 MONTGOMERY AL US 36117 (334)244-3249 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
P.O. Box 244023 Montgomery AL US 36124-4023 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Program Element Code(s): | |
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
Power networks are critical infrastructures for generating, transferring, and consuming electric energy, and they are of fundamental importance to every aspect of modern life. Improving the efficiency, stability, and resilience of power networks is therefore of great interest to our society. A single power network can operate on many different modes --- some lead to efficient operations while others lead to catastrophic failures. Understanding all such operation modes and the possible transition among them for large and complex power networks remains a difficult mathematical question that could have important real-world consequences. This project aims to develop new methodology and computational tools for solving this problem by utilizing recent discoveries in mathematics. The project will provide training to graduate and undergraduate students, and open source software to the larger community.
At the heart of power network analysis lies the mathematical problem of solving power-flow equations: systems of nonlinear equations that describe the intricate balancing conditions of electric power in a power network. The solutions to power-flow equations describe the set of theoretically possible operating modes for a power network, which are of crucial importance in the rigorous analysis of power networks, especially in the problem of assessing network stability. Despite many decades of active research, the complete analysis of power-flow solutions is still a difficult and often computationally impractical task. By leveraging new tools developed in convex geometry, tropical geometry, and homotopy methods, this project aims to develop a flexible divide-and-conquer approach for completely solving power-flow equations and to create practical software implementations. The resulting theoretical framework and software packages would allow researchers to conduct a complete analysis of the full set of power-flow solutions and thus understand the stability and resilience of power networks. Moreover, this project provides opportunities for broadening our understanding of the role of convex polytopes in the study of emergent phenomena in complex networks that may be of value in a much broader context.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Power grids are experiencing unprecedented transformations under the pressure of population growth, urbanization, and the introduction of uncontrollable renewable sources. Our ability to control and design increasingly complicated power grids will no doubt be constrained by our understanding of the complex and nonlinear interactions among their components. The power-flow equations are nonlinear equations that describe the intricate balancing conditions on the active and reactive power injections in a power network. Their real solutions give operating points for the underlying power network. There can be more than one potential operating point due to the inherent nonlinearity. The problem of finding some or all of them has been an active research topic, and it is the focus of this project.
The central goal of this project is to gain a new perspective in the study of nonlinear power-flow equations by leveraging previously under-utilized mathematical tools from the field of convex geometry. It is well known that infinitesimal changes in the parameters of a power network result in infinitesimal changes in the non-degenerate power-flow solutions. This is the basis of the idea known as "continuation", which is familiar to electrical engineers. This project generalized the idea of local continuation to several different kinds of "global" continuation and thus linked the study of power-flow equations to convex geometry. In a series of papers, we showed that by changing power network parameters continuously along certain paths, non-degenerate power-flow solutions also move continuously, and they asymptotically stablize into special degenerate power-flow solutions on much smaller and simpler "facet" systems corresponding to faces of a high-dimensional convex polytope, known as the "symmetric edge polytope". Simply put, this project has established a concrete dictionary by which knowledge about a type of polytope can be translated into knowledge about power network balancing conditions.
Following this general theme, this project explored several theoretical and practical directions. The new insights gained here are combined with numerical homotopy continuation methods and consolidated into a new framework for studying and solving power-flow equations. Efficient and scalable solvers for power-flow equations are being developed based on this framework.
Last Modified: 08/02/2023
Modified by: Tianran Chen
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