**Authors**

Department of Electrical Engineering, Shahed University, Tehran, Iran

**Abstract**

This paper presents a new methodology for uncertain polytopic linear parameter-varying (LPV) modeling of power systems based on parameter set mapping (PSM) with principle component analysis (PCA). At first, an LPV representation of the system dynamics is generated by linearization of its usual differential-algebraic equations about the transient operating points. Then, the PCA-based PSM algorithm is used to reduce the number of models and generate a reduced polytopic LPV model. Because of the system nonlinearity and approximations of model reduction, some uncertainties are considered for each model. A robust pole placement controller is designed to assign the poles of polytopic model in a linear matrix inequality (LMI) region such that the response of the system has a proper damping ratio. A sufficient condition is also proposed to guarantee the asymptotic stability of the closed loop model against the uncertainties. Finally, the proposed controller is synthesized as a power system stabilizer (PSS). It is considered for a single-machine power system and then it is simulated in multi-machine case and compared its performance with a tuned standard conventional PSS and other cases of the controller. The results show the robust performance of the proposed controller especially in different operation conditions and faults.

**Keywords**

[1] سعید تیمورزاده، فرخ امینیفر و مجید صنایع پسند، «میراسازی نوسانات بین ناحیهای: طرح گسترده هماهنگی حذف بار و تولید مبتنی بر منطق Fuzzy » مجله مهندسی برق دانشگاه تبریز، دوره 47، شماره1، سال 1396.

[2] عادل اکبری مجد، حسین شایقی، حمید محمدنژاد و عبداله یونسی، «کنترل کننده مقاوم تطبیقی بار فرکانس مبتنی بر یادگیری تقویتی برای یک سیستم قدرت به هم پیوسته شامل SMES»، مجله مهندسی برق دانشگاه تبریز، دوره 47، شماره 2، سال 1396.

[3] M. Marzband, S. S. Ghazimirsaeid, H. Uppal, and T. Fernando, “A real-time evaluation of energy management systems for smart hybrid home Microgrids,” Electr. Power Syst. Res., vol. 143, pp. 624–633, Feb. 2017.

[4] M. Marzband, M. M. Moghaddam, M. F. Akorede, and G. Khomeyrani, “Adaptive load shedding scheme for frequency stability enhancement in microgrids,” Electr. Power Syst. Res., vol. 140, pp. 78–86, Nov. 2016.

[5] M. Marzband, R. R. Ardeshiri, M. Moafi, and H. Uppal, “Distributed generation for economic benefit maximization through coalition formation-based game theory concept,” Int. Trans. Electr. Energy Syst., vol. 27, no. 6, e2313, 2017.

[6] M. Marzband, M. Javadi, J. L. Domínguez-García, and M. Mirhosseini Moghaddam, “Non-cooperative game theory based energy management systems for energy district in the retail market considering DER uncertainties,” IET Gener. Transm. Distrib., vol. 10, no. 12, pp. 2999–3009, Sep. 2016.

[7] M. Marzband, F. Azarinejadian, M. Savaghebi, and J. M. Guerrero, “An optimal energy management system for islanded microgrids based on multiperiod artificial bee colony combined with markov chain,” IEEE Syst. J., vol. 100, no. 99, pp. 1–11, 2015.

[8] F. Amato, F. Garofalo, L. Glielmo, and A. Pironti, “Robust and Quadratic Stability Via Polytopic Set,” Int. J. Robust Nonlinear Control, vol. 5, no. 8, pp. 745–756, 1995.

[9] G. Cai, C. Hu, B. Yin, H. He, and X. Han, “Gain-Scheduled ℋ 2 Controller Synthesis for Continuous-Time Polytopic LPV Systems,” Math. Probl. Eng., vol. 2014, no. 2014, pp. 1–14, 2014.

[10] C. C. Ku and C. I. Wu, “Gain-scheduled H∞ control for linear parameter varying stochastic systems,” ASME J. Dyn. Syst. Meas. Control, vol. 137, no. 11, pp. 1–12, 2015.

[11] A. Hajiloo and W. F. Xie, “The Stochastic Robust Model Predictive Control of Shimmy Vibration in Aircraft Landing Gears,” Asian J. Control, vol. 17, no. 2, pp. 476–485, Mar. 2015.

[12] R. H. Ordóñez-Hurtado and M. A. Duarte-Mermoud, “Finding common quadratic Lyapunov functions for switched linear systems using particle swarm optimisation,” Int. J. Control, vol. 85, no. 1, pp. 12–25, 2012.

[13] Y. Tong, L. Zhang, P. Shi, and C. Wang, “A common linear copositive Lyapunov function for switched positive linear systems with commutable subsystems,” Int. J. Syst. Sci., vol. 44, no. 11, pp. 1994–2003, 2013.

[14] W. Xiang and J. Xiao, “Finite-time stability and stabilisation for switched linear systems,” Int. J. Syst. Sci., pp. 1–17, 2011.

[15] B. P. Rasmussen and Y. J. Chang, “Stable controller interpolation and controller switching for LPV systems,” ASME J. Dyn. Syst. Meas. Control, vol. 132, no. 1, pp. 1–12, 2009.

[16] S. D. Ramos, A. C. J. Domingos, and E. Vazquez Silva, “An algorithm to verify asymptotic stability conditions of a certain family of systems of differential dquations,” Appl. Math. Sci., vol. 8, no. 31, pp. 1509–1520, 2014.

[17] H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear systems: A survey of recent results,” IEEE Trans. Automat. Contr., vol. 54, no. 2, pp. 308–322, 2009.

[18] J. Xiong, J. Lam, Z. Shu, and X. Mao, “Stability Analysis of Continuous-Time Switched Systems With a Random Switching Signal,” IEEE Trans. Autom. Contr., vol. 59, no. 1, pp. 180–186, 2014.

[19] Z. She and B. Xue, “Discovering multiple Lyapunov functions for switched hybrid systems,” SIAM J. Control Optim., vol. 52, no. 5, pp. 3312–3340, 2014.

[20] W. a. De Souza, M. C. M. Teixeira, M. P. a Santim, R. Cardim, and E. Assunção, “On switched control design of linear time-invariant systems with polytopic uncertainties,” Math. Probl. Eng., vol. 2013, 2013.

[21] C. Hoffmann and H. Werner, “A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations,” IEEE Trans. Control Syst. Technol., 2014.

[22] J. S. Shamma, “An overview of LPV systems,” in Control of Linear Parameter Varying Systems with Applications, J. Mohammadpour and C. W. Scherer, Eds. Springer US, pp. 3–26, 2012.

[23] C. Hoffmann and H. Werner, “LFT-LPV modeling and control of a Control Moment Gyroscope,” in 2015 54th IEEE Conference on Decision and Control (CDC), pp. 5328–5333, 2015.

[24] F. R. López-Estrada, J.-C. Ponsart, D. Theilliol, Y. Zhang, and C.-M. Astorga-Zaragoza, “LPV Model-Based Tracking Control and Robust Sensor Fault Diagnosis for a Quadrotor UAV,” J. Intell. Robot. Syst., vol. 84, no. 1–4, pp. 163–177, Nov. 2016.

[25] F. Blanchini, D. Casagrande, S. Miani, and U. Viaro, “Robust linear parameter-varying control of induction motors,” Int. J. Robust Nonlinear Control, vol. 25, no. 12, pp. 1783–1800, Aug. 2015.

[26] Y. Huang, C. Sun, and C. Qian, “Linear Parameter Varying Switching Attitude Tracking Control for a Near Space Hypersonic Vehicle Via Multiple Lyapunov Functions,” Asian J. Control, vol. 17, no. 2, pp. 523–534, Mar. 2015.

[27] C. Hoffmann, S. M. Hashemi, H. S. Abbas, and H. Werner, “Benchmark problem - nonlinear control of a 3-DOF robotic manipulator,” in 52nd IEEE Conference on Decision and Control, pp. 5534–5539, 2013.

[28] W. J. Rugh and J. S. Shamma, “Research on gain scheduling,” Automatica, vol. 36, no. 10, pp. 1401–1425, 2000.

[29] W. Xie, “Multi-objective H2/L2 performance controller synthesis for LPV systems,” Asian J. Control, vol. 14, no. 5, pp. 1273–1281, Sep. 2012.

[30] B. Pal and B. Chaudhuri, Robust Control in Power Systems. London, UK: Springer, 2005.

[31] R. A. Jabr, B. C. Pal, and N. Martins, “A Sequential Conic Programming Approach for the Coordinated and Robust Design of Power System Stabilizers,” IEEE Trans. Power Syst., vol. 25, no. 3, pp. 1627–1637, 2010.

[32] M. Soliman, a. L. Elshafei, F. Bendary, and W. Mansour, “Robust decentralized PID-based power system stabilizer design using an ILMI approach,” Electr. Power Syst. Res., vol. 80, no. 12, pp. 1488–1497, 2010.

[33] A. Pal, J. S. Thorp, S. S. Veda, and V. A. Centeno, “Applying a robust control technique to damp low frequency oscillations in the WECC,” Int. J. Electr. Power Energy Syst., vol. 44, no. 1, pp. 638–645, 2013.

[34] H. M. Soliman, M. H. Soliman, and M. F. Hassan, “Resilient guaranteed cost control of a power system,” J. Adv. Res., vol. 5, no. 3, pp. 377–385, 2014.

[35] R. Bos, X. Bombois, and P. M. J. Van den Hof, “Accelerating simulations of computationally intensive first principle models using accurate quasi-linear parameter varying models,” J. Process Control, vol. 19, no. 10, pp. 1601–1609, 2009.

[36] A. Kwiatkowski and H. Werner, “PCA-based parameter set mappings for LPV models with fewer parameters and less overbounding,” IEEE Trans. Control Syst. Technol., vol. 16, no. 4, pp. 781–788, 2008.

[37] M. B. A. Jabali and M. H. Kazemi, “A new LPV modeling approach using PCA-based parameter set mapping to design a PSS,” J. Adv. Res., vol. 8, no. 1, pp. 23–32, 2017.

[38] I. T. Jolliffe, Principal Component Analysis, 2nd ed. New York: Springer, 2002.

[39] M. B. A. Jabali and M. H. Kazemi, “Uncertain Polytopic LPV Modelling of Robot Manipulators and Trajectory Tracking,” Int. J. Control. Autom. Syst., vol. 15, no. 2, pp. 883-891, 2017.

[40] M. Chilali and P. Gahinet, “H∞ design with pole placement constraints: An LMI approach,” IEEE Trans. Automat. Contr., vol. 41, no. 3, pp. 358–367, 1996.

[41] J. Shin, S. Nam, J. Lee, S. Baek, Y. Choy, and T. Kim, “A Practical Power System Stabilizer Tuning Method and its Verification in Field Test,” J. Electr. Eng. Technol., vol. 5, no. 3, pp. 400–406, 2010.

[42] DIgSILENT, “39 Bus New England System.” DIgSILENT GmbH, Gomaringen, Germany, 2015.