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Optimal demand response policies for inertial thermal loads under stochastic renewable sources

Gaurav Sharma, P R Kumar

TL;DR

The paper addresses how to leverage intermittent wind power to support inertial thermal loads in microgrids while mitigating reliance on non-renewable generation. It develops and analyzes multiple optimization criteria (CVP, variance, hard thresholds, and STV) to determine the resulting demand-response structure via Hamilton-Jacobi-Bellman equations, showing when optimal policies synchronize or desynchronize the loads. A key contribution is the privacy-preserving threshold policy (Z-policy) that yields tractable, explicit continuum-limit solutions and scalable methods (PMP/Euler-Lagrange) for determining admissible set-point distributions, along with numerical schemes and heuristic policies for non-homogeneous and practical settings. The results offer a principled framework for wind-aware, privacy-preserving DR with scalable planning tools and practical algorithmic guidance for real-world deployment and future grid-wide applications.

Abstract

In this paper, we consider the problem of preferentially utilizing intermittent renewable power, such as wind, optimally to support thermal inertial loads in a microgrid environment. Thermal inertial loads can be programmed to preferentially consume from renewable sources. The flexibility in power consumption of inertial loads therefore can be used to absorb the fluctuations in intermittently available renewable power sources, and promote reduction of fossil fuel based costly non-renewable generation. Under a model which promotes renewable consumption by penalizing the non-renewable, but does not account for variations in the end-user requirements, the optimal solution leads to all the users' temperatures behave in a lockstep fashion, that is the power is allocated in such a fashion that all the temperatures are brought to a common value and they are kept the same after that point, resulting in synchronization among all the loads. In the first part, we showed that under a model which additionally penalizes the comfort range violation, the optimal solution is in-fact of desynchronization nature, where the temperatures are intentionally kept apart to avoid power surges resulting from simultaneous comfort violation from many loads.

Optimal demand response policies for inertial thermal loads under stochastic renewable sources

TL;DR

The paper addresses how to leverage intermittent wind power to support inertial thermal loads in microgrids while mitigating reliance on non-renewable generation. It develops and analyzes multiple optimization criteria (CVP, variance, hard thresholds, and STV) to determine the resulting demand-response structure via Hamilton-Jacobi-Bellman equations, showing when optimal policies synchronize or desynchronize the loads. A key contribution is the privacy-preserving threshold policy (Z-policy) that yields tractable, explicit continuum-limit solutions and scalable methods (PMP/Euler-Lagrange) for determining admissible set-point distributions, along with numerical schemes and heuristic policies for non-homogeneous and practical settings. The results offer a principled framework for wind-aware, privacy-preserving DR with scalable planning tools and practical algorithmic guidance for real-world deployment and future grid-wide applications.

Abstract

In this paper, we consider the problem of preferentially utilizing intermittent renewable power, such as wind, optimally to support thermal inertial loads in a microgrid environment. Thermal inertial loads can be programmed to preferentially consume from renewable sources. The flexibility in power consumption of inertial loads therefore can be used to absorb the fluctuations in intermittently available renewable power sources, and promote reduction of fossil fuel based costly non-renewable generation. Under a model which promotes renewable consumption by penalizing the non-renewable, but does not account for variations in the end-user requirements, the optimal solution leads to all the users' temperatures behave in a lockstep fashion, that is the power is allocated in such a fashion that all the temperatures are brought to a common value and they are kept the same after that point, resulting in synchronization among all the loads. In the first part, we showed that under a model which additionally penalizes the comfort range violation, the optimal solution is in-fact of desynchronization nature, where the temperatures are intentionally kept apart to avoid power surges resulting from simultaneous comfort violation from many loads.

Paper Structure

This paper contains 18 sections, 7 theorems, 50 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Load dynamics, wind model and optimization criteria
  3. Contract violation probability (CVP) model:
  4. Variance minimization model
  5. Hard temperature threshold model
  6. Stochastic threshold variation (STV) model
  7. HJB equation and nature of the optimal demand response
  8. Privacy respecting architecture using a threshold policy
  9. Evaluating distribution under threshold policy
  10. Optimization for finite loads case under Z-policy
  11. Continuum limit for binary wind and comfort setting model
  12. Evaluation of optimal solution using minimum principle
  13. Extensions under multiple comfort level settings
  14. Extension under multiple wind models
  15. Numerical solution for non-homogeneous loads
  16. ...and 3 more sections

Key Result

Theorem 1

The optimal response in the CVP model with $\mathcal{M}(q_0,q_1)$ wind process and linear dynamics is to provide all the wind power to the coolest TCL that is outside the temperature range. Therefore the optimal policy is de-synchronizing.

Figures (12)

  • Figure 1: In a general scheme, the Aggregator is aware of the loads' temperatures and wind process, and decides to optimal power components as a function of temperature and wind availability that should be used by a/c loads for cooling.
  • Figure 2: In a privacy respecting demand response model, the states of the loads are not revealed to the aggregator. The aggregator controls the aggregate power consumption behavior by setting the parameters randomly and broadcasts the wind availability to all the loads to control the ensemble of air-conditioner loads.
  • Figure 3: In the $Z$-policy, a load consumes wind power whenever available to maintain its comfort range. When wind power is not available, it is allowed to heat till its temperature heats up to a value $z$ or the maximum temperature in the comfort range, then it consume just enough power to maintain that temperature level.
  • Figure 4: A particular distribution of the set-point policy for values $(z,\Theta_1,\Theta_2,q_0,q_1,r_0,r_1,c,h )=(100,100,50,.04,.04,.02,.02,1.1,1)$. The probability distribution consists of three point masses and four probability distributions which are continuous everywhere except at $\Theta_1$
  • Figure 5: When the wind is blowing, loads cool at the maximum rate, subject to not going below temperature 0, load temperatures do not rise above minimum of $Z$ and $\Theta_M(t)$
  • ...and 7 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof