Abstract
This chapter presents the development and the analysis of a scheme for aggregate power tracking control of heterogeneous populations of thermostatically controlled loads (TCLs) based on the theory and techniques for partial differential equation (PDE) control. By employing a thermostat-based deadband control with forced switching in the operation of individual TCLs, the aggregated dynamics of TCL populations are governed by a pair of Fokker-Planck equations coupled via the actions located both on the boundaries and in the domain. The technique of input-output feedback linearization is used for the design of aggregate power tracking control, which results in a nonlinear system in a closed loop. As the considered setting is a problem with time-varying boundaries, well-posedness assessment and stability analysis are carried out to confirm the validity of the developed control scheme. A simulation study is conducted to evaluate the effectiveness and the performance of the proposed approach.
This work is supported in part by NSFC under grant NSFC-11901482 and in part by NSERC under grant RGPIN-2018-04571.
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References
D. Angeli, P.-A. Kountouriotis, A stochastic approach to “dynamic-demand” refrigerator control. IEEE Trans. Control Syst. Technol. 20(3), 581–592 (2012)
M.J. Balas, Active control of flexible dynamic systems. J. Optim. Theory Appl. 25(3), 415–436 (1978)
S. Bashash, H.K. Fathy, Modeling and control of aggregate air conditioning loads for robust renewable power management. IEEE Trans. Control Syst. Technol. 21(4), 1318–1327 (2013)
D.S. Callaway, Tapping the energy storage potential in electric loads to deliver load following and regulation, with application to wind energy. Energy Conv. Manag. 50(5), 1389–1400 (2009)
P.D. Christofides, Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes (Birkhuser, Boston, 2001)
P.D. Christofides, P. Daoutidis, Feedback control of hyperbolic PDE systems. AIChE J. 42(11), 3063–3086 (1996)
R. Deng, Z. Yang, M.-Y. Chow, J. Chen, A survey on demand response in smart grids: mathematical models and approaches. IEEE Trans. Ind. Informat. 11(3), 570–582 (2015)
A. Ghaffari, S. Moura, M. Krstić, Modeling, control, and stability analysis of heterogeneous thermostatically controlled load populations using partial differential equations. J. Dyn. Syst. Meas. Control 137, 101009 (2015)
M. Ghanavati, A. Chakravarthy, Demand-side energy management by use of a design-then-approximate controller for aggregated thermostatic loads. IEEE Trans. Control Syst. Technol. 26(4), 1439–1448 (2018)
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, 3rd edn. (Yale University Press, New Haven, 1923) Reprinted (Dover Publications, New York, 1953)
H. Hao, B.M. Sanandaji, K. Poolla, T.L. Vincent, Aggregate flexibility of thermostatically controlled loads. IEEE Trans. Power Syst. 20(1), 189–198 (2015)
S. Ihara, F.C. Schweppe, Physically based modelling of cold load pickup. IEEE Trans. Power Appart. Syst. PAS-100(9), 4142–4150 (1981)
H.K. Khalil, Nonlinear Systems, 3rd edn. (Prentice-Hall, Englewood Cliffs, 2002)
S. Koch, J.L. Mathieu, D.S. Callaway, Modeling and control of aggregated heterogeneous thermostatically controlled loads for ancillary services, in Proceedings of Power Systems Computation Conference, Stockholm (2011), pp. 1–8
M. Krstic, I. Kanellakopoulos, P. Kokotovic, Nonlinear and Adaptive Control Design (Wiley, New York, 1995)
O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type (American Mathematical Society, Providence, 1968)
G. Laparra, M. Li, G. Zhu, Y. Savaria, Desynchronized model predictive control for large populations of fans in server racks of datacenters. IEEE Trans. Smart Grid 1(1), 411–419 (2020)
J. Lévine, Analysis and Control of Nonlinear Systems: A Flatness-based Approach (Springer, Berlin, 2009)
M. Liu, Y. Shi, Model predictive control of aggregated heterogeneous second-order thermostatically controlled loads for ancillary services. IEEE Trans. Power Syst. 31(3), 1963–1971 (2016)
M. Liu, Y. Shi, X. Liu, Distributed MPC of aggregated heterogeneous thermostatically controlled loads in smart grid. IEEE Trans. Ind. Electron. 63(2), 1120–1129 (2016)
N. Mahdavi, J.H. Braslavsky, M.M. Seron, S.R. West, Model predictive control of distributed air-conditioning loads to compensate fluctuations in solar power. IEEE Trans. Smart Grid 8(6), 3055–3065 (2017)
A. Maidi, J.-P. Corriou, Distributed control of nonlinear diffusion systems by input-output linearization. Int. J. Robust Nonlinear Control 24(3), 389–405 (2014)
R. Malhamé, C.-Y. Chong, Electric load model synthesis by diffusion approximation of a high-order hybrid-state stochastic system. IEEE Trans. Autom. Control 30(9), 854–860 (1985)
S. Moura, V. Ruiz, J. Bendtsen, Modeling heterogeneous populations of thermostatically controlled loads using diffusion-advection PDEs, in ASME Dynamic Systems and Control Conference, Palo Alto (2013), p. V002T23A001
S. Moura, J. Bendtsen, V. Ruiz, Parameter identification of aggregated thermostatically controlled loads for smart grids using PDE techniques. Int. J. Control 87(7), 1373–1386 (2014)
P. Palensky, D. Dietrich, Demand side management: demand response, intelligent energy systems, and smart loads. IEEE Trans. Ind. Informat. 7(3), 381–388 (2011)
A. Radaideh, U. Vaidya, V. Ajjarapu, Sequential set-point control for heterogeneous thermostatically controlled loads through an extended Markov chain abstraction. IEEE Trans. Smart Grid 10(4), 4095–4106 (2019)
S.E.Z. Soudjani, A. Abate, Aggregation and control of populations of thermostatically controlled loads by formal abstractions. IEEE Trans. Control Syst. Technol. 23(3), 975–990 (2015)
S.H. Tindemans, V. Trovato, G. Strbac, Decentralized control of thermostatic loads for flexible demand response. IEEE Trans. Control Syst. Technol. 23(5), 1685–1700 (2015)
L.C. Totu, R. Wisniewski, J. Leth, Demand response of a TCL population using switching-rate actuation. IEEE Trans. Control Syst. Technol. 25(5), 1537–1551 (2017)
W. Zhang, J. Lian, C.-Y. Chang, K. Kalsi, Aggregated modeling and control of air conditioning loads for demand response. IEEE Trans. Power Syst. 28(4), 4655–4664 (2013)
L. Zhao, W. Zhang, A unified stochastic hybrid system approach to aggregate modeling of responsive loads. IEEE Trans. Autom. Control 63(12), 4250–4263 (2018)
J. Zheng, G. Laparra, G. Zhu, M. Li, Aggregate power control of heterogeneous TCL populations governed by Fokker–Planck equations. IEEE Trans. Control Syst. Technol. 28(5), 1915–1927 (2020)
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Zheng, J., Zhu, G., Li, M. (2023). A PDE-Based Aggregate Power Tracking Control of Heterogeneous TCL Populations. In: Fathi, M., Zio, E., Pardalos, P.M. (eds) Handbook of Smart Energy Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-97940-9_11
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DOI: https://doi.org/10.1007/978-3-030-97940-9_11
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