Source-linked AI summary
Modeling and control of thermostatically controlled loads
Soumya Kundu, Nikolai Sinitsyn, Scott Backhaus, Ian Hiskens
TL;DR
The paper addresses the need for analytical models of aggregate TCL power consumption to support load following under intermittent generation. It derives a homogeneous-population response model and LQR controller, showing reference tracking while proposing probing for systems with uncertain load characteristics.
Problem
Intermittent generation creates fast power fluctuations, while existing analytical models for aggregate TCL consumption are needed to design load-following controllers.
Method
The paper derives a transfer function from uniform TCL setpoint changes, linearizes the aggregate response, designs an LQR, and proposes probing for noise and heterogeneity.
Results
The LQR enables aggregate TCL power demand to track reference signals, and the controller suppresses lengthy oscillations while keeping load distributions near steady state.
Takeaways & Limitations
The aggregate response model could support load tracking of renewable-generation fluctuations, while probing can balance over- or under-production when model assumptions fail.
Takeaways & Limitations
The analytical model assumes a homogeneous TCL population with insignificant noise; incorporating heterogeneity and noise remains future work.
Abstract
from arXiv · showhide
As the penetration of intermittent energy sources grows substantially, loads will be required to play an increasingly important role in compensating the fast time-scale fluctuations in generated power. Recent numerical modeling of thermostatically controlled loads (TCLs) has demonstrated that such load following is feasible, but analytical models that satisfactorily quantify the aggregate power consumption of a group of TCLs are desired to enable controller design. We develop such a model for the aggregate power response of a homogeneous population of TCLs to uniform variation of all TCL setpoints. A linearized model of the response is derived, and a linear quadratic regulator (LQR) has been designed. Using the TCL setpoint as the control input, the LQR enables aggregate power to track reference signals that exhibit step, ramp and sinusoidal variations. Although much of the work assumes a homogeneous population of TCLs with deterministic dynamics, we also propose a method for probing the dynamics of systems where load characteristics are not well known.
1 INTRODUCTION
Intermittent renewable generation creates a need for fast load-side balancing, and TCL populations are promising candidates. The paper develops an analytical aggregate-response model for uniform TCL setpoint changes and uses it for control design.
- Electrical loads can compensate for renewable-generation imbalance faster than conventional generators constrained by physical ramp rates.
- TCL populations are well matched to load following because their aggregate behavior can provide controllable power response.
- Prior work modeled TCL populations using coupled ordinary and partial differential equations, discrete-time temperature dynamics, and extensions for noise and heterogeneity.
- Hysteresis-based control manipulates a common thermostat setpoint while estimating aggregate ON/OFF-state probabilities rather than tracking individual loads.
- The paper derives a transfer function for homogeneous TCL populations, linearizes the aggregate response, designs an LQR, and explores noise and heterogeneity numerically.
2 STEADY STATE DISTRIBUTION OF LOADS
The steady-state model describes TCL temperature evolution and represents aggregate populations through ON- and OFF-state probability densities. For homogeneous, low-noise loads, these densities are linked to cooling and heating residence times and agree with simulation evidence.
- A thermostatically controlled cooling load has temperature dynamics that differ between its ON and OFF states when noise is absent.
- In steady state, the numbers of ON and OFF loads are proportional to their cooling and heating periods, with Nh + Nc = N under negligible noise.
- All loads share the same θamb, C, R, and P values under the homogeneous-population assumption.
- The model defines ON and OFF probability densities, with cumulative distributions representing the probabilities of loads in each state below a given temperature.
- Figure 2 compares analytically calculated densities with simulations of 10,000 TCLs containing small noise, and the results support the modeling assumptions.
3 SETPOINT VARIATION
The paper derives aggregate power response to a uniform TCL setpoint change by integrating responses across four post-disturbance load conditions. A linearized, damped approximation is compared with simulation under small setpoint shifts and low-noise, homogeneous-population assumptions.
- Setpoint disturbance: A setpoint disturbance shifts the TCL deadband, requiring separate analysis of four starting conditions across the ON- and OFF-state density curves.Individual power responses are computed with Laplace transforms and integrated over the state distributions to obtain total power consumption.
- Setpoint disturbance: The aggregate response is formed by averaging power waveforms for loads at four representative density-curve points and summing their contributions across the population.The four cases include loads remaining in or switching between the ON and OFF states after the deadband shift.
- Linearized model: Small deadband width and small setpoint shifts relative to deadband width support a series expansion that yields a linearized aggregate power model.The latter condition keeps the load densities near their steady-state forms.
- Linearized model: The analytical model is undamped because it assumes homogeneity and low noise, whereas the actual system is damped by heterogeneity and noise.A damping term σ is added to capture this effect, with its value estimated or selected to match the observed response.
- Model comparison: A damping coefficient of 0.002 min^-1 produced a close match between the approximate model and the simulated response to a setpoint step.The comparison uses the same setpoint disturbance as the aggregate-power response shown for the step change.
4 CONTROL LAW
The controller uses an LQR with integral error action to adjust the common TCL deadband shift, enabling aggregate power to track step, ramp, and sinusoidal references while keeping load distributions near steady state.
- The TCL controller is expressed in state-space form, with deadband shift u(t) as input and aggregate power deviation y(t) as output.
- An integral controller is included because the system has an open-loop zero very close to the imaginary axis.
- The LQR minimizes a cost function to obtain an optimal control law, with a pre-compensator selected for unity DC gain.
- The controller tracks step, ramp, and sinusoidal aggregate-power reference signals through TCL setpoint shifts.
- With control, the load distribution remains close to steady state and lengthy oscillations are suppressed relative to the uncontrolled response.
5 HETEROGENEITY AND NOISE
The analytical controller is limited to nearly homogeneous TCL populations with low noise. When those assumptions fail, the paper proposes probing with separated setpoint pulses to balance energy over time.
- The developed tracking model assumes a homogeneous population of loads with deterministic dynamics and very low noise.
- When those assumptions fail, the model cannot support design of a tracking controller, so the paper proposes a probing method for energy balancing.
- The probing method raises the temperature setpoint briefly, returns it to its original value, and repeats separated short pulses.
- Probing monitors energy delivered during the pulses relative to nominal consumption, which can indicate net energy delivery.
6 CONCLUSION
The paper derives an aggregate TCL response model and designs an LQR for reference tracking. Its scope remains limited by homogeneity and negligible noise, motivating probing and future extensions.
- The paper analytically derives a transfer function from uniform TCL setpoint changes to changes in aggregate power demand.
- The designed LQR enables aggregate power demand to track reference signals, suggesting use for fluctuations in renewable generation.
- The analysis assumes a homogeneous TCL population and insignificant noise; further work is needed to incorporate heterogeneity and noise.
- The paper proposes probing for energy balance when the modeling assumptions do not hold and identifies plug-in electric vehicles as another candidate for compensating renewable variability.
ga(t)
The section contains the label “Temerature Setpoint (degC)” repeated across four passages.
- “Temerature Setpoint (degC)” is the repeated label in the section.
- The label identifies temperature setpoint in degrees Celsius.
- The same temperature-setpoint label appears in four listed passages.