A Padé-Weierstrass technique for the rigorous enforcement of control limits in power flow studies


The study of electrical power flow in interconnected systems is of great significance in power engineering. It uses simple notations like per-unit systems and majorly focuses on important power parameters such as the voltages, real and reactive powers and voltage angles. Power flow analysis allows engineers to determine the best operating procedures for the existing systems as well as the expansion of the power systems.

Presently, power flow analysis involves specifying loads and sources in terms of power constants to determine the steady state of the power networks which forms the basis of other analysis and tools. Unfortunately, enforcing control limits in power flow analysis which is deemed as the ideal way of enhancing the accuracy of the analysis has been very difficult.

Among the features considered during the analysis, voltage regulation by synchronous generators is very vital. Consequently, the bus voltage problems are represented by nonlinear algebraic equations. Furthermore, control devices are utilized to restrict the regulated magnitudes and addition of new variables to the system. The variables help in balancing the equations and the unknowns. However, all the controls have limits, therefore, the control behavior change when the limits are reached must be taken into consideration. It also consists of a switching behavior where the regulated magnitudes can be varied from the given setpoints thus resulting in more complex network modules that consumes a lot of time to analyses.

Several methods for enforcing the limits have been developed. For instance, the type switching and feedback adjustments methods are associated with rebounds due to nonlinear interactions of the controls that lead to slow or prevention of convergence. Therefore, researches have been looking for alternatives for preventing interactions and convergence problems and have identified the rigorous solution of control limits problem as a promising solution.

Dr. Antoni Trias and Dr. Jose Luis Marín at Aplicaciones en Informática Avanzada developed a new general-purpose power method for the rigorous solution of the control limits based on the Lagrangian equations. The method takes into consideration the Mvar limits in regulating generator voltages which are based on the Holomorphic Embedded Loadflow Method. They purposed to achieve a direct, deterministic and constructive method for efficient and precision analysis of power flow in various systems. The work is published in the journal, International Journal of Electrical Power and Energy Systems.

The authors observed that the developed method directly avoided the convergence problems emanating from mutual interactions of the controls as opposed to the initial traditional approaches. Consequently, it resulted in more accurate results with enhanced numerical stability. This was attributed to the possibility of using several steps to calculate the analytic continuation of the power series using the Padé-Weierstrass technique.

The development of a Padé-Weierstrass technique was very significant in the research as it enables exploration of the nature of power flow equations under variable. This allowed for high numerical precision because it can be used even if no limits are being enforced. According to the authors, the method can be extended to other types of regulating devices with limits such as the sequential shunt unit. Therefore, the study will help advance analysis of power flows in electrical systems which will further lead to better use of electrical systems to enhance efficiency and conserve energy.

About the author

Antonio Trias


  • 1974 – 1976 University of California. Post Doctoral Research Fellow High Energy Physics. Lawrence Berkeley Laboratory.
  • 1969 – 1974 Autonomous University of Barcelona. PhD Physics. Cum Laude.
  • 1964 – 1969 University of Barcelona. Physics. Cum Laude.




HELM patents are under his name or assigned to one of the group companies he has ownership in.

Trias, A. 2009. “System and method for monitoring and managing electrical power transmission and distribution networks”. USPTO Patent 7519506.

Trias, A. 2011. “System and method for monitoring and managing electrical power transmission and distribution networks”. USPTO Patent 7979239.

Trias, A. 2014. “System and method for monitoring and managing three-phase power flows in electrical transmission and distribution networks”. USPTO Patent 8849614.

Trias, A. 2017. “Sigma algebraic approximants as a diagnostic tool in power networks”. USPTO Patent 9563722 B2.

Trias, A. 2018. “System and method for obtaining the power flow in DC grids with constant power loads and devices with algebraic nonlinearities”. USPTO Patent Allowance pending number.


Over 30 years of Industry Experience.

He is a founding member of Aplicaciones en Informática Avanzada, as well as the R&D Vice President.

Under his direction the Innovation division has produced a series of algorithms suited to solve a broad variety of technical problems of great complexity. The key for this success has been his ability to combine the mathematical techniques and models from the theoretical physics field with optimization algorithms, theoretical economics and artificial intelligence techniques and apply them effectively in the industrial world.

He has been a member of the Board of Directors of AIA group companies specially in EleQuant, Inc. and Gridquant España, S.L.U.

Recent past has involved him in solving challenging problems for electrical grids and Regional and National Transmission system organizations focused on the North American Market (México, US and Canada). He has actively been participated in teaching the new paradigm for the electrical field based on the knowledge that his break though patented innovation is creating. His work is focused on the sustainability of the company´s new vision: Algorithms for a Better World. His work has also been a centerpiece in the earlier strategic development of the innovation in the Music Industry participated companies which were sold.

Most recently his work has been centered on HELM different extensions within NASA funded research on space exploration.


1988– Present AIA. Aplicaciones en Informática Avanzada, S.L./EleQuant, Inc. / EQKIDS LLC

2012–2018 He leads the R&D of the group with NASA SBIR Programs.

SBIR PHASE I awarded June 2014; successfully concluded December 2014

TOPIC S3, Spacecraft and Platform Subsystems; SUBTOPIC S3.03 Power Electronics and Management, and Energy Storage.Subcontracted by Gridquant Technologies, LCC.

  • Proposal Title: Holomorphic Embedded Load Flow for autonomous spacecraft power systems.
  • Focus of the project: adapting the new HELM (Holomorphic Embedding Load Flow Method) power flow to the relevant basic devices of DC micro-grids present on current and projected power systems of spacecraft and space stations.
  • Project summary:
  • Final summary chart


Trias, A., & Marín, J. (2018). A Padé-Weierstrass technique for the rigorous enforcement of control limits in power flow studies. International Journal of Electrical Power & Energy Systems, 99, 404-418.

Go To International Journal of Electrical Power & Energy Systems

Check Also

Multi-stage active management of renewable-rich power distribution network to promote the renewable energy consumption and mitigate the system uncertainty - Advances in Engineering

Multi-stage active management of renewable-rich power distribution network to promote the renewable energy consumption and mitigate system uncertainty