State-space design and analysis of load frequency control with advanced modeling and controller strategies for single, two, and three-area power systems

Kartik Chandra  Patra; Asutosh  Patnaik

doi:10.59400/adecp4247

State-space design and analysis of load frequency control with advanced modeling and controller strategies for single, two, and three-area power systems

Kartik Chandra Patra
Department of Electrical Engineering, C.V. Raman Global University, Bhubaneswar 752054, India
Asutosh Patnaik
Department of Electrical Engineering, C.V. Raman Global University, Bhubaneswar 752054, India

Article ID: 4247

Keywords: load frequency control (LFC); signal stabilization; linear quadratic regulator (LQR); digital deadbeat controller (DDC); limit cycles (LC); Linear Quadratic Gaussian (LQG)

Abstract

Load Frequency Control (LFC) is a fundamental issue in modern power systems, aimed at maintaining system frequency at its nominal value (50/60 Hz) while ensuring accurate regulation of tie-line power exchanges in interconnected areas. This study develops a comprehensive state-space modeling framework for single-area, two-area, and three-area power systems to support dynamic analysis and controller design. Integral controllers and optimal control strategies based on the Riccati equation—namely, Linear Quadratic Regulator (LQR) and Linear Quadratic Gaussian (LQG)—are implemented to enhance system performance. These controllers effectively minimize frequency and voltage deviations under varying load conditions, thereby improving power quality. To achieve faster and more precise control, a Digital Deadbeat Controller (DDC) is also proposed, ensuring rapid convergence to steady-state conditions. A comparative analysis across different system configurations highlights the steady-state frequency deviations and overall performance improvements. The study also addresses persistent challenges caused by memory-dependent nonlinearities such as backlash/deadband, which induce limit cycle (LC) oscillations and degrade system stability. Backlash nonlinearity, inherent in speed governors used in LFC, plays a significant role in these oscillations. Two methods are proposed to mitigate LC behavior: signal stabilization using deterministic or random inputs. Simulation results obtained using MATLAB/Simulink demonstrate the effectiveness of these approaches. The findings confirm that both DDC and signal stabilization techniques achieve the desired performance (Δf = 0, t→0), indicating robust and efficient control suitable for practical power system applications.

Published

2026-06-25

How to Cite

Patra, K. C., & Patnaik, A. (2026). State-space design and analysis of load frequency control with advanced modeling and controller strategies for single, two, and three-area power systems. Advances in Differential Equations and Control Processes, 33(2). https://doi.org/10.59400/adecp4247

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Vol. 33 No. 2 (2026)

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

[1]Jain D, Saxena D. Comprehensive review on control schemes and stability investigation of hybrid AC-DC microgrid. Electric Power Systems Research. 2023; 218: 109182. doi: 10.1016/j.epsr.2023.109182

[2]Yang S, Fan X, Zhao Y, et al. A novel method for transient stability margin evaluation of the post-disturbance power system. In: Proceedings of the 15th International Conference on Developments in Power System Protection (DPSP 2020); 9–12 March 2020; Liverpool, UK. doi: 10.1049/cp.2020.0048

[3]Dong X, Hao X, Chen Q, et al. A Distributed Power Transfer Limit Calculation Method for Multi-Area Interconnection Power Networks. IEEE Transactions on Power Systems. 2021; 36(5): 4723–4732. doi: 10.1109/TPWRS.2021.3062374

[4]Tungadio DH, Sun Y. Load frequency controllers considering renewable energy integration in power system. Energy Reports. 2019; 5: 436–453. doi: 10.1016/j.egyr.2019.04.003

[5]Ashfaq T, Mumtaz S, Ahmad S, et al. Automatic Generation Control in Renewables-Integrated Multi-Area Power Systems: A Comparative Control Analysis. Sustainability. 2024; 16(13): 5735. doi: 10.3390/su16135735

[6]Fosha C, Elgerd O. The Megawatt-Frequency Control Problem: A New Approach Via Optimal Control Theory. IEEE Transactions on Power Apparatus and Systems. 1970; PAS-89(4): 563–577. doi: 10.1109/TPAS.1970.292603

[7]Ali T, Malik SA, Hameed IA, et al. Load Frequency Control and Automatic Voltage Regulation in a Multi-Area Interconnected Power System Using Nature-Inspired Computation-Based Control Methodology. Sustainability. 2022; 14(19): 12162. doi: 10.3390/su141912162

[8]Loka R, Parimi AM, Srinivas STP, et al. Region of convergence by parameter sensitivity constrained genetic algorithm-based optimization for coordinated load frequency control in multi-source distributed hybrid power system. Sustainable Energy Technologies and Assessments. 2022; 54: 102887. doi: 10.1016/j.seta.2022.102887

[9]Tripathi S, Singh VP, Kishor N, et al. Load frequency control of power system considering electric Vehicles’ aggregator with communication delay. International Journal of Electrical Power & Energy Systems. 2023; 145: 108697. doi: 10.1016/j.ijepes.2022.108697

[10]Alshammari B, Salah RB, Kahouli O, et al. Design of Fuzzy TS-PDC Controller for Electrical Power System via Rules Reduction Approach. Symmetry. 2020; 12(12): 2068. doi: 10.3390/sym12122068

[11]Nguyen BM, Tran-Huynh N, Kawanishi M, et al. Load frequency control design in state space. In: Encyclopedia of Electrical and Electronic Power Engineering. Elsevier; 2023. pp. 314–333. doi: 10.1016/B978-0-12-821204-2.00047-7

[12]Shah NN, Pandit AR, Shah MT. Automatic Load Frequency Control of Two Area system Using L-Q-R Method. International Journal Of Current Engineering And Scientific Research. 2016; 3(6): 54–66. Available online: https://troindia.in/journal/ijcesr/vol3iss6/54-66.pdf

[13]Kundur P. Power System Stability & Control. Tata McGraw Hill; 2008. pp. 581–626.

[14]Saadat H. Power system Analysis. McGraw–Hill Book Company; 1999.

[15]Aldeen M, Marsh JF. Decentralised proportional-plus-integral design method for interconnected power systems. IEE Proceedings C Generation, Transmission and Distribution. 1991; 138(4): 263. doi: 10.1049/ip-c.1991.0033

[16]Bevrani H, Golpîra H, Messina AR, et al. Power system frequency control: An updated review of current solutions and new challenges. Electric Power Systems Research. 2021; 194: 107114. doi: 10.1016/j.epsr.2021.107114

[17]Ahmed M, Magdy G, Khamies M, et al. An efficient coordinated strategy for frequency stability in hybrid power systems with renewables considering interline power flow controller and redox flow battery. Journal of Energy Storage. 2022; 52: 104835. doi: 10.1016/j.est.2022.104835

[18]Cavdar B, Sahin E, Sesli E, et al. Cascaded fractional order automatic generation control of a PV-reheat thermal power system under a comprehensive nonlinearity effect and cyber-attack. Electrical Engineering. 2023; 105(6): 4339–4360. doi: 10.1007/s00202-023-01943-y

[19]Gulzar MM, Sibtain D, Alqahtani M, et al. Load frequency control progress: A comprehensive review on recent development and challenges of modern power systems. Energy Strategy Reviews. 2025; 57: 101604. doi: 10.1016/j.esr.2024.101604

[20]Khan IA, Mokhlis H, Mansor NN, et al. New trends and future directions in load frequency control and flexible power system: A comprehensive review. Alexandria Engineering Journal. 2023; 71: 263–308. doi: 10.1016/j.aej.2023.03.040

[21]Tavakoli S, Zamani AA, Khajehoddin A. Efficient load frequency control in multi-source interconnected power systems using an innovative intelligent control framework. Energy Reports. 2024; 11: 2805–2817. doi: 10.1016/j.egyr.2024.02.037

[22]Yamashita K, Miyagi H. Multivariable self-tuning regulator for load frequency control system with interaction of voltage on load demand. IEE Proceedings D Control Theory and Applications. 1991; 138(2): 177. doi: 10.1049/ip-d.1991.0025

[23]Aldeen M. A Fresh Approach to the LQR Problems with Application to Power Systems. Proceedings of International Conference on Power Engineering; 1993: 374–379.

[24]Pan CT, Liaw CM. An adaptive controller for power system load-frequency control. IEEE Transactions on Power Systems. 1989; 4(1): 122–128. doi: 10.1109/59.32469

[25]Djukanovic M, Novicevic M, Sobajic DJ, et al. Two-Area Load Frequency Control with Neural Nets. In: Proceedings of the 1993 North American Power Symposium; October 1993; Washington, DC, USA. pp. 161–169.

[26]Birch AP. An enhanced neural network load frequency control technique. In: Proceedings of the International Conference on Control ’94; 21–24 March 1994; Coventry, UK. pp. 409–415. doi: 10.1049/cp:19940167

[27]Li C, Feng C, Li J, et al. Comprehensive frequency regulation control strategy of thermal power generating unit and ESS considering flexible load simultaneously participating in AGC. Journal of Energy Storage. 2023; 58: 106394. doi: 10.1016/j.est.2022.106394

[28]Guo W, Wu F. Stability behavior of load adjustment and primary frequency control of pumped storage power plant with upstream and downstream surge tanks. Journal of Energy Storage. 2023; 60: 106626. doi: 10.1016/j.est.2023.106626

[29]Yu J, Liao S, Xu J. Frequency control strategy for coordinated energy storage system and flexible load in isolated power system. Energy Reports. 2022; 8: 966–979. doi: 10.1016/j.egyr.2022.02.133

[30]Lian Z, Wen C, Guo F, et al. Decentralized secondary control for frequency restoration and power allocation in islanded AC microgrids. International Journal of Electrical Power & Energy Systems. 2023; 148: 108927. doi: 10.1016/j.ijepes.2022.108927

[31]Gulzar MM, Iqbal A, Sibtain D, et al. An Innovative Converterless Solar PV Control Strategy for a Grid Connected Hybrid PV/Wind/Fuel-Cell System Coupled With Battery Energy Storage. IEEE Access. 2023; 11: 23245–23259. doi: 10.1109/ACCESS.2023.3252891

[32]Perninge M, Eriksson R. Optimal Tertiary Frequency Control in Power Systems with Market-Based Regulation. IFAC-PapersOnLine. 2017; 50(1): 4374–4381. doi: 10.1016/j.ifacol.2017.08.881

[33]Carere F, Gatta FM, Geri A, et al. Sardinia as a 100% renewable power system: A frequency stability study. Sustainable Energy, Grids and Networks. 2022; 32: 100899. doi: 10.1016/j.segan.2022.100899

[34]Sambariya D, Nath V. Load Frequency Control Using Fuzzy Logic Based Controller for Multi-area Power System. British Journal of Mathematics & Computer Science. 2016; 13(5): 1–19. doi: 10.9734/BJMCS/2016/22899

[35]Indulkar CS, Raj B. Application of fuzzy controller to automatic generation control. Electric Machines & Power Systems. 1995; 23(2): 209–220. doi: 10.1080/07313569508955618

[36]Chidambaram IA, Velusami S. Decentralized Biased Controllers for Load-Frequency Control of Interconnected Power Systems Considering Governor Dead Band Non-Linearity. In: Proceedings of the 2005 Annual IEEE India Conference—Indicon; 11–13 December 2005; Chennai, India. pp. 521–525. doi: 10.1109/INDCON.2005.1590225

[37]Patra KC, Kar N, Patnaik A. Suppression Limit Cycles in 2x2 Nonlinear Systems with Memory Type Nonlinearities. Design, Construction, Maintenance. 2024; 4: 169–185. Doi: 10.37394/232022.2024.4.18

[38]Patra KC, Patnaik A. Load Frequency Control by Speed Governors of Two-Area Power System using Digital Controller with Deadbeat Approach. International Journal of Electrical Engineering and Computer Science. 2025; 7: 200–217. doi: 10.37394/232027.2025.7.19

[39]Parmar KPS, Majhi S, Kothari DP. Automatic generation control of an interconnected hydrothermal power system. In: Proceedings of the 2010 Annual IEEE India Conference (INDICON); 17–19 December 2010; Kolkata, India. pp. 1–5. doi: 10.1109/INDCON.2010.5712663

[40]Kothari DP, Nagrath IJ. Modern Power System Analysis. McGraw-Hill; 2003.

[41]Elgerd OI. Electric Energy Systems Theory: An Introduction. Tata McGraw-Hill; 1983.

[42]Stefani RT, Shahian B, Savant CJ, et al. Design of Feedback Control Systems. Oxford University Press; 2002.

[43]Llewis FL, Syrmos VL. Optimal Control. John Wiley & Sons; 1995.

[44]Azzam M. Robust automatic generation control. Energy Conversion and Management. 1999; 40(13): 1413–1421. doi: 10.1016/S0196-8904(99)00040-0

[45]Sivanagaraju S, Sreenivasan G. Power System Operation and Control. Pearson Education India; 2009.

[46]Kakilli A, Oguz Y, Çalik, H. The Modelling of Electric Power Systems on the State Space and Controlling of Optimal LQR Load Frequency. Journal of Electrical & Electronics Engineering. 2009; 9(2). Available online: https://electricajournal.org/public/pdfs/63/6.pdf

[47]Patra KC, Patnaik A. Estimation of Limit Cycles and Signal Stabilization with Deadbeat approach in three Dimensional Nonlinear systems. International Journal of Control Systems and Robotics. 2025; 10: 22–38.

[48]Patra KC, Patnaik A. Investigation of Limit Cycles and Signal Stabilization by Deterministic and Gaussian Random Signals Based on Deadbeat Approach in 3×3 Nonlinear Systems with Memory Nonlinearities. International Journal of Electrical Engineering and Computer Science. 2025; 7: 159–188. doi: 10.37394/232027.2025.7.17

[49]Patra KC, Kar N. Suppression limit cycles in 2 × 2 nonlinear systems with memory type nonlinearities. International Journal of Dynamics and Control. 2022; 10(3): 721–733. doi: 10.1007/s40435-021-00860-x

[50]Oldenburger R, Nakada T. Signal stabilization of self-oscillating systems. IRE Transactions on Automatic Control. 1961; 6(3): 319–325. doi: 10.1109/TAC.1961.1105218

[51]Gelb A, Velde WEV. Multiple-Input describing functions and nonlinear system design. In: BiblioBoard Library Catalog.

[52]McGraw-Hill; 1968.

[53]Atherton DP, Towill DR. Nonlinear Control Engineering-Describing Function Analysis and Design. IEEE Transactions on Systems, Man, and Cybernetics. 1977; 7(9): 678–678. doi: 10.1109/TSMC.1977.4309808

[54]Tsay TS. Load–frequency control of interconnected power system with governor backlash nonlinearities. International Journal of Electrical Power & Energy Systems. 2011; 33(9): 1542–1549. doi: 10.1016/j.ijepes.2011.06.005

[55]Sah SV, Prakash V, Pathak PK, et al. Fractional Order AGC Design for Power Systems via Artificial Gorilla Troops Optimizer. In: Proceedings of the 2022 IEEE International Conference on Power Electronics, Drives and Energy Systems (PEDES); 14–17 December 2022; Jaipur, India. pp. 1–6. doi: 10.1109/PEDES56012.2022.10079975

[56]Hussain J, Zou R, Wu Z, et al. Design and Performance Analysis of Walrus Optimization Algorithm (WaOA)—Based Cascade Controller for Load Frequency Control of a Multi-Area Power System With Renewable Sources. International Journal of Numerical Modelling: Electronic Networks, Devices and Fields. 2025; 38(2): e70046. doi: 10.1002/jnm.70046

[57]Kuo BC. Digital Control Systems. Oxford University Press; 2009.

[58]Ogata K. Discrete-Time Control Systems. Prentice-Hall; 2009.

[59]Taub H, Schilling DL, Saha G. Principles of Communication Systems. Tata McGraw-Hill; 2008. p. 107.

[60]Wang C, Yang M, Zheng W, et al. Analysis and Suppression of Limit Cycle Oscillation for Transmission System With Backlash Nonlinearity. IEEE Transactions on Industrial Electronics. 2017; 64(12): 9261–9270. doi: 10.1109/TIE.2017.2711564

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