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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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References

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Figures

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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. 5

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

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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. 3

Distribution functions of TCLs in three regions

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Fig. 6

Normalized physical parameters

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Fig. 7

Variation of error between PDE and MC model versus TCL population

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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. 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. 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. 14

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

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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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