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Characterizing Demand Response Capability of Thermostatically Controlled Loads with Reach and Hold Sets

Mazen Elsaadany, Hamid R. Ossareh, Mads R. Almassalkhi

TL;DR

The paper addresses ex-ante quantification of demand-response flexibility in fleets of thermostatically controlled loads (TCLs) under setpoint control. It develops a second-order Markov-chain aggregate model and introduces the reach-and-hold set to describe how much power can be shifted and for how long, formulating a linear-programming approach to compute an inner approximation of this set. To make computation tractable, it proposes a broadcast proportional control policy with a tilde-α transformation that yields a linear relation between inputs and the aggregate response, enabling a small, solvable LP and practical open-loop schedules. Robustness analyses assess sensitivity to initial-condition and parameter uncertainty, showing that while instantaneous reach estimates can be affected, energy performance remains reliable and short-horizon predictions are most sensitive. The framework provides grid operators with a practical, implementable tool for planning and scheduling DR from TCL fleets, with future work on resilience and user comfort considerations.

Abstract

Aggregations of thermostatically controlled loads (TCLs), such as air conditioners, offer valuable flexibility to the power grid. The aggregate power consumption of a TCL fleet can be controlled by adjusting thermostat setpoints. An ex-ante quantification of the flexibility that results from such setpoint change can inform grid operator decisions. This paper develops a rigorous, yet practical method to quantify flexibility in terms of the `reach-and-hold' set of TCL aggregations, which defines how much power can be shifted (reach) and for how long (hold). To quantify the reach-and-hold set, we employ a Markov-chain-based model of the TCL aggregation that captures second-order TCL dynamics, enabling accurate characterization of reach-and-hold sets. A tractable optimization problem is then formulated to numerically compute an inner approximation of these sets. Simulation results validate that our method accurately characterizes the fleet's flexibility and effectively controls its power consumption. Furthermore, a robustness analysis is carried out to investigate the effects of uncertainty in initial conditions and TCL parameters.

Characterizing Demand Response Capability of Thermostatically Controlled Loads with Reach and Hold Sets

TL;DR

The paper addresses ex-ante quantification of demand-response flexibility in fleets of thermostatically controlled loads (TCLs) under setpoint control. It develops a second-order Markov-chain aggregate model and introduces the reach-and-hold set to describe how much power can be shifted and for how long, formulating a linear-programming approach to compute an inner approximation of this set. To make computation tractable, it proposes a broadcast proportional control policy with a tilde-α transformation that yields a linear relation between inputs and the aggregate response, enabling a small, solvable LP and practical open-loop schedules. Robustness analyses assess sensitivity to initial-condition and parameter uncertainty, showing that while instantaneous reach estimates can be affected, energy performance remains reliable and short-horizon predictions are most sensitive. The framework provides grid operators with a practical, implementable tool for planning and scheduling DR from TCL fleets, with future work on resilience and user comfort considerations.

Abstract

Aggregations of thermostatically controlled loads (TCLs), such as air conditioners, offer valuable flexibility to the power grid. The aggregate power consumption of a TCL fleet can be controlled by adjusting thermostat setpoints. An ex-ante quantification of the flexibility that results from such setpoint change can inform grid operator decisions. This paper develops a rigorous, yet practical method to quantify flexibility in terms of the `reach-and-hold' set of TCL aggregations, which defines how much power can be shifted (reach) and for how long (hold). To quantify the reach-and-hold set, we employ a Markov-chain-based model of the TCL aggregation that captures second-order TCL dynamics, enabling accurate characterization of reach-and-hold sets. A tractable optimization problem is then formulated to numerically compute an inner approximation of these sets. Simulation results validate that our method accurately characterizes the fleet's flexibility and effectively controls its power consumption. Furthermore, a robustness analysis is carried out to investigate the effects of uncertainty in initial conditions and TCL parameters.
Paper Structure (20 sections, 4 theorems, 28 equations, 10 figures, 5 tables)

This paper contains 20 sections, 4 theorems, 28 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Modeling Thermostatically Controlled Loads
  3. Individual HVAC Model
  4. Aggregate Modeling of HVAC Fleet
  5. Setpoint Control
  6. The Reach and Hold Set
  7. Motivation behind the reach-and-hold set
  8. Defining the reach-and-hold Set
  9. Computing the boundary of the reach-and-hold set
  10. Problem Tractability
  11. Tractable Reach and Hold Set Formulation
  12. Problem Formulation
  13. Limiting Overshoot
  14. Reach-and-hold Set Characterization
  15. Validation of the Reach-and-Hold Set
  16. ...and 5 more sections

Key Result

Lemma 1

If $({P}_\text{reach},\tau) \in \mathrm{bd}(\mathcal{R})$ then $({P},\tau) \in \mathcal{R} \,\, \forall P \in [\min\{0,P_\text{reach}\},\max\{0,P_\text{reach}\}]$.

Figures (10)

  • Figure 1: The effects of set-point control on individual HVAC temperature dynamics and aggregate HVAC power consumption obtained from first and second-order ETP models. (a) Air and mass temperature of single HVAC with setpoint change from $20^\circ$C to $22^\circ$C, (b) aggregate power response with setpoint control of HVAC fleet, (c) aggregate power response with direct load control of HVAC fleet.
  • Figure 2: Outdoor temperature profile with shaded region showing considered DR time window.
  • Figure 3: (a) Aggregate HVAC power consumption with setpoint change from 20$^\circ$C to 22$^\circ$C at $t$=13:00 obtained from agent based simulation (ETP model) and Markov models with different temperature discretization resolutions. (b) MSE associated with each Markov model's power characterization.
  • Figure 4: reach-and-hold sets for a 1C battery with initial SOC of 10%, 50% and 90%.
  • Figure 5: Change in aggregate power consumption $P_\text{agg}$ during a DR event with corresponding $P_\text{reach}$ and $T_\text{hold}$ values.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Definition 1: Reach and Hold Set
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof