Research Papers

Modeling, Control, and Stability Analysis of Heterogeneous Thermostatically Controlled Load Populations Using Partial Differential Equations

[+] Author and Article Information
Azad Ghaffari

Department of Mechanical &
Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92039-0411
e-mail: aghaffari@eng.ucsd.edu

Scott Moura

Department of Civil and
Environmental Engineering,
University of California, Berkeley,
Berkeley, CA 94720
e-mail: smoura@berkeley.edu

Miroslav Krstić

Department of Mechanical &
Aerospace Engineering,
University of California, San Diego,
La Jolla, CA 92039-0411
e-mail: krstic@ucsd.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 4, 2014; final manuscript received June 6, 2015; published online July 27, 2015. Assoc. Editor: Umesh Vaidya.

J. Dyn. Sys., Meas., Control 137(10), 101009 (Jul 27, 2015) (9 pages) Paper No: DS-14-1197; doi: 10.1115/1.4030817 History: Received May 04, 2014

Thermostatically controlled loads (TCLs) account for more than one-third of the U.S. electricity consumption. Various techniques have been used to model TCL populations. A high-fidelity analytical model of heterogeneous TCL (HrTCL) populations is of special interest for both utility managers and customers (that facilitates the aggregate synthesis of power control in power networks). We present a deterministic hybrid partial differential equation (PDE) model which accounts for HrTCL populations and facilitates analysis of common scenarios like cold load pick up, cycling, and daily and/or seasonal temperature changes to estimate the aggregate performance of the system. The proposed technique is flexible in terms of parameter selection and ease of changing the set-point temperature and deadband width all over the TCL units. We investigate the stability of the proposed model along with presenting guidelines to maintain the numerical stability of the discretized model during computer simulations. Moreover, the proposed model is a close fit to design feedback algorithms for power control purposes. Hence, we present output- and state-feedback control algorithms, designed using the comparison principle and Lyapunov analysis, respectively. We conduct various simulations to verify the effectiveness of the proposed modeling and control techniques.

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Bashash, S. , and Fathy, H. K. , 2013, “Modeling and Control of Aggregate Air Conditioning Loads for Robust Renewable Power Management,” IEEE Trans. Control Syst. Technol., 21(4), pp. 1318–1327. [CrossRef]
Moura, S. , Ruiz, V. , and Bendtsen, J. , 2013, “Modeling Heterogeneous Populations of Thermostatically Controlled Loads Using Diffusion-Advection PDEs,” ASME Dynamic Systems and Control Conference, ASME Paper No. DSCC2013-3809.
Moura, S. , Bendtsen, J. , and Ruiz, V. , 2013, “Observer Design for Boundary Coupled PDEs: Application to Thermostatically Controlled Loads in Smart Grids,” IEEE Conference on Decision and Control, pp. 6286–6291.
Zhang, W. , Lian, J. , Chang, C.-Y. , and Kalsi, K. , 2013, “Aggregated Modeling and Control of Air Conditioning Loads for Demand Response,” IEEE Trans. Power Syst., 28(4), pp. 4655–4664. [CrossRef]
Lu, N. , and Chassin, D. P. , 2004, “A State-Queueing Model of Thermostatically Controlled Appliances,” IEEE Trans. Power Syst., 19(3), pp. 1666–1673. [CrossRef]
Chassin, D. P. , and Fuller, J. C. , 2011, “On the Equilibrium Dynamics of Demand Response in Thermostatic Loads,” 44th Hawaiian International Conference on System Sciences, pp. 1–6.
Kalsi, K. , Chassin, F. , and Chassin, D. , 2011, “Aggregate Modeling of Thermostatic Loads in Demand Response: A Systems and Control Perspective,” IEEE Conference on Decision and Control and European Control Conference, pp. 15–20.
Callaway, D. S. , 2009, “Tapping the Energy Storage Potential in Electric Loads to Deliver Load Following and Regulation, With Application to Wind Energy,” Energy Convers. Manage., 50(5), pp. 1389–1400. [CrossRef]
Soudjani, S. E. Z. , and Abate, A. , 2013, “Aggregation of Thermostatically Controlled Loads by Formal Abstractions,” European Control Conference, pp. 4232–4237.
Koch, S. , Mathieu, J. L. , and Callaway, D. S. , 2011, “Modeling and Control of Aggregated Heterogeneous Thermostatically Controlled Loads for Ancillary Services,” 17th Power Systems Computation Conference, pp. 961–968.
Mortensen, R. E. , and Haggerty, K. P. , 1990, “Dynamics of Heating and Cooling Loads: Models, Simulation, and Actual Utility Data,” IEEE Trans. Power Syst., 5(1), pp. 243–249. [CrossRef]
Malhamé, R. , and Chong, C.-Y. , 1985, “Electrical Load Model Synthesis by Diffusion Approximation of a High-Order Hybrid-State Stochastic System,” IEEE Trans. Autom. Control, AC-30 (9), pp. 854–860. [CrossRef]
Ihara, S. , and Schweppe, F. C. , 1981, “Physically Based Modeling of Cold Load Pickup,” IEEE Trans. Power Appar. Syst., PAS-100 (9), pp. 4142–4150. [CrossRef]
Chong, C. Y. , and Debs, A. S. , 1979, “Statistical Synthesis of Power System Functional Load Models,” IEEE Conference on Decision and Control, pp. 264–269.
Sanandaji, B. M. , Hao, H. , and Poolla, K. , 2014, “Fast Regulation Service Provision Via Aggregation of Thermostatically Controlled Loads,” 47th Hawaii International Conference on System Sciences, pp. 2388–2397.
Hao, H. , Sanandaji, B. M. , Poolla, K. , and Vincent, T. L. , 2015, “Aggregate Flexibility of Thermostatically Controlled Loads,” IEEE Trans. Power Syst., 30(1), pp. 189–198. [CrossRef]
Khalil, H. K. , 1996, Nonlinear Systems, 2nd ed., Englewood Cliffs, Prentice Hall, NJ.
Wunderground, 2013, “Weather History for Phoenix,” http://www.wunderground.com/history/airport/KPHX/2013/7/13/DailyHistory.html


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Fig. 1

Equivalent electrical circuit of a TCL unit

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Fig. 2

Characteristic of the switch modeled as a Schmitt trigger

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Fig. 3

Distribution functions of TCLs in three regions

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Fig. 4

Population variation of on units over an infinitesimal time–temperature window. The graph is presented for a homogeneous population.

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Fig. 5

The method of finite difference applied for discretization of the transport PDEs to use in numerical simulations

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Fig. 6

Normalized physical parameters

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Fig. 8

System response to a step change in the set-point temperature from 24.5 °C to 24 °C with σ = 1 °C for (solid) our proposed PDE-based and (dashed) the MC model for 40,000 HrTCL units

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Fig. 7

Variation of error between PDE and MC model versus TCL population

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Fig. 9

TCL population uniformly distributes after transient is passed

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Fig. 10

For a constant N, normalized pole locus remains the same regardless of parameter variation as shown in Fig. 6

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Fig. 14

Evolution of the deadband and TCL population versus time. White line shows set-point evolution.

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Fig. 11

Reference tracking for step changes in power level

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Fig. 12

Spatial distribution of TCL units remains practically uniform in the deadband. White line shows set-point evolution.

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Fig. 13

(Top) Hourly variation of environmental temperature in Phoenix, AZ, from July 13th 6:00 a.m. until July 14th 6:00 a.m., 2013 [18]. (Bottom) Variation of power versus time for (dashed) open-loop and (solid) closed-loop system. The customer designs the power steps according to his/her priorities.

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Fig. 15

Power outage happens from t = 1 hr to t = 1.5 hr. Power consumption for (dashed) open-loop and (solid) closed-loop system.

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Fig. 16

Evolution of the deadband and TCL population during power control. White line shows set-point evolution.

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Fig. 17

Static deadband and TCL population without power control. A large TCL population is turning on as power is restored.



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