MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS
The rapid integration of renewable energy sources (RES), such as wind power plants (WPPs) and solar power plants (SPPs), into modern Integrated Power Systems (IPS) has introduced new challenges and opportunities for grid stability and control. One critical aspect of power system operation is the reg...
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General Energy Institute of the National Academy of Sciences of Ukraine
2025
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| author | Denysov, Viktor |
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| author_institution_txt_mv | [
{
"author": "Viktor Denysov",
"institution": null
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| description | The rapid integration of renewable energy sources (RES), such as wind power plants (WPPs) and solar power plants (SPPs), into modern Integrated Power Systems (IPS) has introduced new challenges and opportunities for grid stability and control. One critical aspect of power system operation is the regulation of controlling active emergency frequency and power regulators (EFPR). EFPR plays a pivotal role in this process, and its effective operation is increasingly dependent on advanced mathematical models that account for the dynamic and intermittent nature of RES. The article emphasizes the importance of interdisciplinary research, combining power system engineering, applied mathematics, and data science, to develop innovative solutions for the challenges of modern power systems. The development of advanced mathematical models for EFPRs is essential for ensuring the stability and reliability of power systems in the era of renewable energy. An overview of mathematical models and approaches used to control active emergency frequency and power regulators in power systems with potential participation of wind and solar power plants is presented. Priority areas for possible research and development are highlighted. |
| doi_str_mv | 10.15407/srenergy2025.03.056 |
| first_indexed | 2026-03-24T02:03:34Z |
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Системні дослідження в енергетиці. 2025. 3(83) 56
ISSN 2786-7102 (Online), ISSN 2786-7633 (Print)
https://doi.org/10.15407/srenergy2025.03.056
УДК 621.311:502.131
Viktor Denysov, PhD (Engin.), https://orcid.org/0000-0002-3297-1114
General Energy Institute of NAS of Ukraine, 172, Antonovycha St., Kyiv, 03150, Ukraine
e-mail: visedp@gmail.com
__________________________________________________________________________________
MATHEMATICAL MODELS FOR CONTROLLING ACTIVE
EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER
SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR
POWER PLANTS. PRIORITY DIRECTIONS
Abstract. The rapid integration of renewable energy sources (RES), such as wind power plants (WPPs)
and solar power plants (SPPs), into modern Integrated Power Systems (IPS) has introduced new challenges
and opportunities for grid stability and control. One critical aspect of power system operation is the
regulation of controlling active emergency frequency and power regulators (EFPR). EFPR plays a pivotal
role in this process, and its effective operation is increasingly dependent on advanced mathematical models
that account for the dynamic and intermittent nature of RES. The article emphasizes the importance of
interdisciplinary research, combining power system engineering, applied mathematics, and data science,
to develop innovative solutions for the challenges of modern power systems. The development of advanced
mathematical models for EFPRs is essential for ensuring the stability and reliability of power systems in
the era of renewable energy. An overview of mathematical models and approaches used to control active
emergency frequency and power regulators in power systems with potential participation of wind and solar
power plants is presented. Priority areas for possible research and development are highlighted.
Keywords: renewable energy sources, Integrated Power Systems, grid stability and control, active
emergency frequency and power regulators, mathematical models.
1. Introduction
The integration of renewable energy sources (RES), such as wind power plants (WPPs) and solar power
plants (SPPs), into power systems has introduced new complexities in maintaining grid stability, particularly
during emergency conditions. Frequency and power regulation are critical aspects of power system operation,
ensuring that supply and demand remain balanced even during disturbances. Emergency frequency and power
regulators (EFPRs) are essential for mitigating the effects of sudden imbalances, which can be exacerbated by
the intermittent and variable nature of RES (Fig. 1) [1−3].
Fig. 1. Topology of the power system with energy storage clusters (ESC) [1]
This article explores the mathematical models used for controlling active EFPRs in power systems with
the potential integration of WPPs and SPPs, highlighting priority directions for research and development.
EFPRs are designed to respond rapidly to frequency deviations and power imbalances caused by sudden
changes in generation or load (Fig. 2).
https://orcid.org/0000-0002-3297-1114
mailto:visedp@gmail.com
Системні дослідження в енергетиці. 2025. 3(83) 57
Fig. 2. Framework of the optimal-droop-based frequency emergency control (FEC) [1]
These systems typically involve:
– Primary Control: Immediate response from governors and inertial responses to stabilize frequency.
– Secondary Control: Automatic Generation Control (AGC) to restore frequency to its nominal value.
– Tertiary Control: Economic dispatch and long-term adjustments to optimize system operation.
The aim of the research is a review of the mathematical models used for controlling active emergency
frequency and power regulators in power systems, with the potential involvement of WPPs and SPPs.
Highlighting priority directions for research and development.
With the increasing penetration of WPPs and SPPs, which lack inherent inertia and are subject to rapid
fluctuations, the role of EFPRs has become more challenging. Advanced mathematical models are required to
ensure effective control under these conditions.
Mathematical Models for EFPRs in RES-Integrated Power Systems
The integration of RES necessitates the development of sophisticated mathematical models [4−10], that
account for their unique characteristics, such as variability, uncertainty, and lack of inertia. Below are key
modeling approaches for EFPRs:
Dynamic Frequency Response Models
Dynamic frequency response models are critical for understanding how EFPRs can mitigate frequency
deviations caused by RES variability. These models typically include:
– Differential equations representing the dynamics of synchronous generators, loads, and RES [2].
– Inertia emulation models for WPPs and SPPs to simulate synthetic inertia (Fig. 6,7) [11].
– Stochastic models to account for the uncertainty in wind speed and solar irradiance (Fig. 3) [12].
Model Predictive Control (MPC)
MPC is a powerful tool for optimizing EFPR operation over a finite time horizon [13]. Key features
include:
– Prediction of RES generation and load demand using time-series data [14].
– Incorporation of system constraints and operational limits.
– Real-time adjustment of control actions to minimize frequency deviations and power imbalances
(Fig. 4) [15].
Системні дослідження в енергетиці. 2025. 3(83) 58
Fig. 3. Classification of scenario generation methods [12]
Fig. 4. Activation of frequency containment reserve (FCR), frequency restoration reserve (FRR) and replacement
reserve (RR) after power imbalance [15]
Optimal Power Flow (OPF) with Emergency Constraints
OPF models [16] are extended to include emergency constraints for EFPRs. These models involve:
– Nonlinear constraints representing power flow equations during disturbances.
– Objective functions that minimize frequency deviations, load shedding, or operational costs.
– Integration of RES forecasts to improve the accuracy of emergency control actions.
Data-Driven and Machine Learning Models
Data-driven approaches leverage historical and real-time data to enhance EFPR performance. These
models include:
– Prediction of frequency deviations and power imbalances using machine learning algorithms (Fig. 5)
[17].
– Identification of optimal control strategies using reinforcement learning [18].
– Enhanced fault detection and diagnosis in EFPR systems.
Системні дослідження в енергетиці. 2025. 3(83) 59
Fig. 5. Neural network architecture [17]
Priority Directions for Research and Development
To address the challenges posed by RES integration, several priority directions have emerged in the
development of mathematical models for EFPRs:
Synthetic Inertia and Fast Frequency Response
WPPs and SPPs lack the inherent inertia of conventional generators, making frequency regulation more
challenging. Research should focus on:
– Developing models for synthetic inertia and fast frequency response from RES [11].
– Integrating these models into EFPR control strategies.
Fig. 6. Proposed synthetic inertia from wind turbines [11]
Fig. 7. Schematic diagram of full-rated power converter (FRPC) – permanent magnet synchronous generator (PMSG)
wind turbine implemented in Simulink [11]
Системні дослідження в енергетиці. 2025. 3(83) 60
Enhanced Uncertainty Modeling
The variability of WPPs and SPPs introduces significant uncertainty into power system operation.
Advanced probabilistic and robust optimization techniques are needed to account for this uncertainty in EFPR
models.
Coordination Between EFPRs and RES Inverters
Modern WPPs and SPPs are equipped with power electronic inverters that can provide fast frequency
response. Developing models that enable seamless coordination between EFPRs and RES inverters is crucial
for maximizing grid stability.
Paper [19] presents an innovative approach for real-time emergency voltage control strategies for
transient stability enhancement through the integration of edge-graph convolutional networks with
reinforcement learning. Thus, a method is proposed that allows you to transform the traditional problem of
optimizing emergency management, replacing it with a sequential decision-making process.
Real-Time Implementation
The transition from offline models to real-time control systems is a key challenge. Research efforts
should focus on developing computationally efficient algorithms that can be implemented in real-time EFPR
systems. The study [20] first analyzes the optimal solution for a single-zone system and then extends it to a
two-zone interconnected system. In many cases, traditional control methods are applied without considering
the potential advantages of using Genetic Algorithm (GA) for various Load Frequency Control (LFC) in
interconnected networks. By examining the effectiveness of these intelligent optimization methods in both
single and interconnected power systems, this study aims to enhance the understanding of their applicability
and performance in real-world scenarios. The research methodology includes a comprehensive analysis of
various LFC strategies, including traditional methods and those optimized using GA and Particle Swarm
Optimization (PSO).
Resilience to Cyber Threats
As EFPRs become more reliant on digital communication and control systems, ensuring their resilience
to cyber threats is critical. Mathematical models should incorporate cybersecurity considerations to protect
against potential attacks. Intrusion Detection and Prevention (IDP) systems are very important for Cyber-
Physical System (CPS) security because they watch for and respond to cyber dangers in real time (Fig. 8) [21].
Fig. 8. Intrusion Detection and Prevention (IDP) System [21]
Системні дослідження в енергетиці. 2025. 3(83) 61
Integration with Energy Storage Systems
Energy storage systems (ESS) can provide additional flexibility for frequency and power regulation.
Developing models that integrate EFPRs with ESS is a promising area of research (Fig. 9) [22, 23].
Fig. 9. Comparison of the state of charge for some energy storage devices for different ratios of their capacity [22]
4. Discussion
The increasing penetration of renewable energy sources (RES), particularly wind power plants (WPPs)
and solar power plants (SPPs), has significantly transformed power system dynamics. This transition has
necessitated the development of advanced mathematical models for controlling active emergency frequency
and power regulators (EFPRs) to maintain grid stability. This discussion highlights key findings, implications,
and future research directions based on the review of mathematical models for EFPRs.
One of the most critical challenges in EFPR control is the variability and intermittency of RES. Unlike
conventional generators, WPPs and SPPs lack inherent inertia, making power systems more susceptible to
frequency deviations and sudden imbalances. The development of synthetic inertia and fast frequency response
mechanisms has emerged as a promising solution. However, these solutions require further refinement to
enhance their accuracy and effectiveness in real-time applications.
The inclusion of Model Predictive Control (MPC) and Optimal Power Flow (OPF) techniques has
demonstrated significant potential in improving EFPR response under emergency conditions. However, these
methods require high computational capabilities and real-time data integration. Future research should focus
on developing more efficient algorithms that reduce computational complexity while maintaining high
accuracy in predicting system responses.
The increasing reliance on data-driven approaches, including machine learning and artificial
intelligence, presents both opportunities and challenges. Machine learning models, such as neural networks
and reinforcement learning, can enhance EFPR decision-making by predicting frequency deviations and
optimizing control strategies. However, the reliability of these models depends on the quality and availability
of real-time data. To improve robustness, hybrid approaches that integrate traditional control methods with
machine learning should be explored.
Additionally, the cybersecurity vulnerabilities associated with data-driven EFPR systems require further
investigation. As EFPRs become more reliant on digital communication and automated control, they are
increasingly susceptible to cyber threats. Future research should prioritize the development of intrusion
detection and prevention (IDP) systems that can safeguard EFPR operations from potential cyberattacks.
Energy storage systems (ESS) have been identified as a crucial component in enhancing EFPR
performance. By providing additional flexibility, ESS can help mitigate the impact of RES variability and
improve the reliability of emergency frequency response mechanisms. However, optimizing the coordination
between EFPRs and ESS remains a challenge. Advanced mathematical models should incorporate ESS
Системні дослідження в енергетиці. 2025. 3(83) 62
characteristics, such as state of charge, response time, and degradation factors, to maximize their effectiveness
in real-time frequency control.
Based on the reviewed mathematical models, the following priority research directions have been
identified:
1. Development of Hybrid Control Strategies: Combining traditional EFPR methods with machine
learning and optimization algorithms can enhance response accuracy and adaptability under dynamic grid
conditions.
2. Enhanced Probabilistic and Stochastic Modeling: Addressing the uncertainty in RES generation
through advanced probabilistic models can improve the robustness of EFPR strategies.
3. Cybersecurity-Integrated Control Mechanisms: Strengthening cybersecurity measures within EFPR
systems to protect against cyber threats and ensure secure grid operation.
4. Real-Time Implementation and Computational Efficiency: Reducing computational burdens while
maintaining the accuracy of EFPR models for real-time applications.
5. Seamless Coordination Between EFPRs, RES Inverters, and ESS: Developing integrated models that
enable efficient interaction between EFPRs, RES inverters, and energy storage solutions to optimize power
system stability.
The successful implementation of these research directions will significantly contribute to the
development of resilient and adaptive EFPR systems, ensuring the stability and reliability of modern power
grids in the era of renewable energy.
5. Conclusion
The integration of renewable energy sources, particularly WPPs and SPPs, has introduced new
challenges for maintaining power system stability. The role of EFPRs in mitigating frequency deviations and
power imbalances has become increasingly critical in modern power systems. This article reviewed various
mathematical models that enhance the effectiveness of EFPRs, including dynamic frequency response models,
Model Predictive Control, Optimal Power Flow approaches, and data-driven techniques.
Advanced mathematical modeling is essential to address the variability and uncertainty of RES. The
inclusion of synthetic inertia, enhanced probabilistic modeling, and machine learning-based control strategies
can significantly improve EFPR performance. Additionally, the integration of ESS and cybersecurity measures
will further enhance system resilience.
Future research should focus on real-time implementation, computational efficiency, and seamless
coordination between EFPRs, RES inverters, and energy storage systems. By advancing these areas, power
system operators can ensure reliable and stable grid operation in the face of increasing renewable energy
penetration. The continued development of sophisticated mathematical models will be instrumental in
achieving a sustainable and secure energy future.
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Системні дослідження в енергетиці. 2025. 3(83) 64
МАТЕМАТИЧНІ МОДЕЛІ КЕРУВАННЯ АКТИВНИМИ
АВАРІЙНИМИ РЕГУЛЯТОРАМИ ЧАСТОТИ ТА ПОТУЖНОСТІ
В ЕНЕРГОСИСТЕМАХ З ПОТЕНЦІЙНОЮ ІНТЕГРАЦІЄЮ
ВІТРОВИХ ТА СОНЯЧНИХ ЕЛЕКТРОСТАНЦІЙ.
ПРІОРИТЕТНІ НАПРЯМКИ
Віктор Денисов, канд. техн. наук, https://orcid.org/0000-0002-3297-1114
Інститут загальної енергетики НАН України, вул. Антоновича, 172, Київ, 03150, Україна
e-mail: visedp@gmail.com
Анотація. Швидка інтеграція відновлюваних джерел енергії (ВДЕ), таких як вітрові
електростанції (ВЕС) та сонячні електростанції (СЕС), у сучасні об’єднані енергетичні системи
(ОЕС) створила нові виклики та можливості для стабільності та контролю мережі. Одним з
найважливіших аспектів роботи енергосистеми є управління активними аварійними регуляторами
частоти і потужності (АРЧП). АРЧП відіграють ключову роль у цьому процесі, і їх ефективна
робота все більше залежить від передових математичних моделей, які враховують динамічний та
переривчастий характер ВДЕ. У статті наголошується на важливості міждисциплінарних
досліджень, що поєднують інженерію енергетичних систем, прикладну математику та науку про
дані, для розробки інноваційних рішень для подолання викликів сучасних енергосистем. Розробка
передових математичних моделей для АРПЧ має важливе значення для забезпечення стабільності
та надійності енергосистем в епоху відновлюваної енергетики. Представлено огляд математичних
моделей та підходів, що використовуються для керування активними аварійними регуляторами
частоти та потужності в енергосистемах за потенційної участі вітрових та сонячних
електростанцій. Виділено пріоритетні напрями можливих досліджень і розробок.
Ключові слова: відновлювані джерела енергії, інтегровані енергосистеми, стабільність та
управління мережею, активні аварійні регулятори частоти та потужності, математичні моделі.
Надійшла до редколегії: 05.05.2025
https://orcid.org/0000-0002-3297-1114
mailto:visedp@gmail.com
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| id | systemreorg-article-906 |
| institution | System Research in Energy |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:23:47Z |
| publishDate | 2025 |
| publisher | General Energy Institute of the National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | systemreorg/cf/591fd8c9065aa6183156ad7988adcdcf.pdf |
| spelling | systemreorg-article-9062026-07-18T12:57:49Z MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS Математичні моделі керування активними аварійними регуляторами частоти та потужності в енергосистемах з потенційною інтеграцією вітрових та сонячних електростанцій. Пріоритетні напрямки Denysov, Viktor renewable energy sources, Integrated Power Systems, grid stability and control, active emergency frequency and power regulators, mathematical models. відновлювані джерела енергії, інтегровані енергосистеми, стабільність та управління мережею, активні аварійні регулятори частоти та потужності, математичні моделі. The rapid integration of renewable energy sources (RES), such as wind power plants (WPPs) and solar power plants (SPPs), into modern Integrated Power Systems (IPS) has introduced new challenges and opportunities for grid stability and control. One critical aspect of power system operation is the regulation of controlling active emergency frequency and power regulators (EFPR). EFPR plays a pivotal role in this process, and its effective operation is increasingly dependent on advanced mathematical models that account for the dynamic and intermittent nature of RES. The article emphasizes the importance of interdisciplinary research, combining power system engineering, applied mathematics, and data science, to develop innovative solutions for the challenges of modern power systems. The development of advanced mathematical models for EFPRs is essential for ensuring the stability and reliability of power systems in the era of renewable energy. An overview of mathematical models and approaches used to control active emergency frequency and power regulators in power systems with potential participation of wind and solar power plants is presented. Priority areas for possible research and development are highlighted. Швидка інтеграція відновлюваних джерел енергії (ВДЕ), таких як вітрові електростанції (ВЕС) та сонячні електростанції (СЕС), у сучасні об’єднані енергетичні системи (ОЕС) створила нові виклики та можливості для стабільності та контролю мережі. Одним з найважливіших аспектів роботи енергосистеми є управління активними аварійними регуляторами частоти і потужності (АРЧП). АРЧП відіграють ключову роль у цьому процесі, і їх ефективна робота все більше залежить від передових математичних моделей, які враховують динамічний та переривчастий характер ВДЕ. У статті наголошується на важливості міждисциплінарних досліджень, що поєднують інженерію енергетичних систем, прикладну математику та науку про дані, для розробки інноваційних рішень для подолання викликів сучасних енергосистем. Розробка передових математичних моделей для АРПЧ має важливе значення для забезпечення стабільності та надійності енергосистем в епоху відновлюваної енергетики. Представлено огляд математичних моделей та підходів, що використовуються для керування активними аварійними регуляторами частоти та потужності в енергосистемах за потенційної участі вітрових та сонячних електростанцій. Виділено пріоритетні напрями можливих досліджень і розробок. General Energy Institute of the National Academy of Sciences of Ukraine 2025-08-26 Article Article application/pdf https://systemre.org/index.php/journal/article/view/906 10.15407/srenergy2025.03.056 System Research in Energy; No. 3 (83) (2025): System Research in Energy; 56-64 Системні дослідження в енергетиці; № 3 (83) (2025): Системні дослідження в енергетиці; 56-64 2786-7102 2786-7633 en https://systemre.org/index.php/journal/article/view/906/812 Copyright (c) 2025 Viktor Denysov https://creativecommons.org/publicdomain/zero/1.0 |
| spellingShingle | renewable energy sources Integrated Power Systems grid stability and control active emergency frequency and power regulators mathematical models. Denysov, Viktor MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS |
| title | MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS |
| title_alt | Математичні моделі керування активними аварійними регуляторами частоти та потужності в енергосистемах з потенційною інтеграцією вітрових та сонячних електростанцій. Пріоритетні напрямки |
| title_full | MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS |
| title_fullStr | MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS |
| title_full_unstemmed | MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS |
| title_short | MATHEMATICAL MODELS FOR CONTROLLING ACTIVE EMERGENCY FREQUENCY AND POWER REGULATORS IN POWER SYSTEMS WITH POTENTIAL INTEGRATION OF WIND AND SOLAR POWER PLANTS. PRIORITY DIRECTIONS |
| title_sort | mathematical models for controlling active emergency frequency and power regulators in power systems with potential integration of wind and solar power plants. priority directions |
| topic | renewable energy sources Integrated Power Systems grid stability and control active emergency frequency and power regulators mathematical models. |
| topic_facet | renewable energy sources Integrated Power Systems grid stability and control active emergency frequency and power regulators mathematical models. відновлювані джерела енергії інтегровані енергосистеми стабільність та управління мережею активні аварійні регулятори частоти та потужності математичні моделі. |
| url | https://systemre.org/index.php/journal/article/view/906 |
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