Robust Koopman EMPC for optimal frequency regulation of VSC-MTDC systems

Yubin Jia; Chaojie Li; Hongming Yang

doi:10.1016/j.apenergy.2026.127639

Robust Koopman EMPC for optimal frequency regulation of VSC-MTDC systems

Yubin Jia^*
, Chaojie Li
, Hongming Yang

^*Corresponding author for this work

Department of Electrical Engineering

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

This paper proposes a data-driven output feedback economic model predictive control (EMPC) strategy for load dispatch and frequency regulation in VSG multi-terminal HVDC (VSC-MTDC) systems. The deep Koopman operator is represented for modeling the nonlinear system. EMPC strategy is implemented to realize the economic load dispatch (ELD) and frequency regulation. To deal with the system uncertainties and obtain better control and optimization, a robust output feedback EMPC is utilized to solve the optimal laws for the feedback control and state observer. The asymptotic stability of the nominal closed-loop system is strictly ensured through the EMPC framework, and all trajectories converge uniformly to a predetermined neighborhood of the origin under uncertainty. The proposed method is verified through simulation of VSC-MTDC systems. The simulation results validate that the system achieves a more optimized load dispatch using this algorithm while enhancing the system's robustness and stability. © 2026 Elsevier Ltd.

Original language	English
Article number	127639
Number of pages	11
Journal	Applied Energy
Volume	412
Online published	9 Mar 2026
DOIs	https://doi.org/10.1016/j.apenergy.2026.127639
Publication status	Published - 1 Jun 2026

Funding

This work was supported by the National Natural Science Foundation of China (52577107, 62103101, 72061147004, 72342001), the Natural Science Foundation of Jiangsu Province under Grant BK20210217, the Shandong Energy Institute (SEI U202310), and the Research Project of Department of Science and Technology of Hunan Province Under (2025JJ10009, 2022RC4025, 2023JJ50312, 2023JJ50010, 2024RC9012).

Research Keywords

Economic model predictive control
Frequency regulation
Koopman operator
Load dispatch
Multi-terminal HVDC
Output feedback control

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10.1016/j.apenergy.2026.127639

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title = "Robust Koopman EMPC for optimal frequency regulation of VSC-MTDC systems",

abstract = "This paper proposes a data-driven output feedback economic model predictive control (EMPC) strategy for load dispatch and frequency regulation in VSG multi-terminal HVDC (VSC-MTDC) systems. The deep Koopman operator is represented for modeling the nonlinear system. EMPC strategy is implemented to realize the economic load dispatch (ELD) and frequency regulation. To deal with the system uncertainties and obtain better control and optimization, a robust output feedback EMPC is utilized to solve the optimal laws for the feedback control and state observer. The asymptotic stability of the nominal closed-loop system is strictly ensured through the EMPC framework, and all trajectories converge uniformly to a predetermined neighborhood of the origin under uncertainty. The proposed method is verified through simulation of VSC-MTDC systems. The simulation results validate that the system achieves a more optimized load dispatch using this algorithm while enhancing the system's robustness and stability. {\textcopyright} 2026 Elsevier Ltd.",

keywords = "Economic model predictive control, Frequency regulation, Koopman operator, Load dispatch, Multi-terminal HVDC, Output feedback control",

author = "Yubin Jia and Chaojie Li and Hongming Yang",

year = "2026",

month = jun,

day = "1",

doi = "10.1016/j.apenergy.2026.127639",

language = "English",

volume = "412",

journal = "Applied Energy",

issn = "0306-2619",

publisher = "Elsevier B.V.",

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TY - JOUR

T1 - Robust Koopman EMPC for optimal frequency regulation of VSC-MTDC systems

AU - Jia, Yubin

AU - Li, Chaojie

AU - Yang, Hongming

PY - 2026/6/1

Y1 - 2026/6/1

N2 - This paper proposes a data-driven output feedback economic model predictive control (EMPC) strategy for load dispatch and frequency regulation in VSG multi-terminal HVDC (VSC-MTDC) systems. The deep Koopman operator is represented for modeling the nonlinear system. EMPC strategy is implemented to realize the economic load dispatch (ELD) and frequency regulation. To deal with the system uncertainties and obtain better control and optimization, a robust output feedback EMPC is utilized to solve the optimal laws for the feedback control and state observer. The asymptotic stability of the nominal closed-loop system is strictly ensured through the EMPC framework, and all trajectories converge uniformly to a predetermined neighborhood of the origin under uncertainty. The proposed method is verified through simulation of VSC-MTDC systems. The simulation results validate that the system achieves a more optimized load dispatch using this algorithm while enhancing the system's robustness and stability. © 2026 Elsevier Ltd.

AB - This paper proposes a data-driven output feedback economic model predictive control (EMPC) strategy for load dispatch and frequency regulation in VSG multi-terminal HVDC (VSC-MTDC) systems. The deep Koopman operator is represented for modeling the nonlinear system. EMPC strategy is implemented to realize the economic load dispatch (ELD) and frequency regulation. To deal with the system uncertainties and obtain better control and optimization, a robust output feedback EMPC is utilized to solve the optimal laws for the feedback control and state observer. The asymptotic stability of the nominal closed-loop system is strictly ensured through the EMPC framework, and all trajectories converge uniformly to a predetermined neighborhood of the origin under uncertainty. The proposed method is verified through simulation of VSC-MTDC systems. The simulation results validate that the system achieves a more optimized load dispatch using this algorithm while enhancing the system's robustness and stability. © 2026 Elsevier Ltd.

KW - Economic model predictive control

KW - Frequency regulation

KW - Koopman operator

KW - Load dispatch

KW - Multi-terminal HVDC

KW - Output feedback control

UR - https://www.scopus.com/pages/publications/105031912046

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-105031912046&origin=recordpage

U2 - 10.1016/j.apenergy.2026.127639

DO - 10.1016/j.apenergy.2026.127639

M3 - RGC 21 - Publication in refereed journal

SN - 0306-2619

VL - 412

JO - Applied Energy

JF - Applied Energy

M1 - 127639

ER -

Robust Koopman EMPC for optimal frequency regulation of VSC-MTDC systems

Abstract

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Research Keywords

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