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Online Smoothed Demand Management

Adam Lechowicz, Nicolas Christianson, Mohammad Hajiesmaili, Adam Wierman, Prashant Shenoy

TL;DR

This work introduces Online Smoothed Demand Management (OSDM), a general online optimization framework for energy procurement and demand delivery with storage, capturing both inflexible base demand and deferrable flexible demand under storage dynamics and smoothing penalties. It presents PAAD, a unified competitive algorithm that partitions demand into drivers (base and flexible) with per-driver threshold decisions and aggregates them to global purchasing and delivery actions, achieving optimal worst-case competitive ratios. To address pessimism in worst-case analysis, the authors propose PALD, a differentiable, learning-augmented framework that optimizes threshold functions from historical data while enforcing robustness certificates that preserve a bounded competitive ratio; PALD-S and PALD-C illustrate value in nonstationary, data-rich environments. A comprehensive case study on grid-integrated data centers with local storage using real price traces demonstrates substantial average-case gains from PALD over PAAD, highlighting practical implications for grid stability and energy-managed compute. Overall, OSDM bridges online algorithm theory with practical demand-management applications, providing both rigorous guarantees and data-driven enhancements for modern energy-aware infrastructures.

Abstract

We introduce and study a class of online problems called online smoothed demand management $(\texttt{OSDM})$, motivated by paradigm shifts in grid integration and energy storage for large energy consumers such as data centers. In $\texttt{OSDM}$, an operator makes two decisions at each time step: an amount of energy to be purchased, and an amount of energy to be delivered (i.e., used for computation). The difference between these decisions charges (or discharges) the operator's energy storage (e.g., a battery). Two types of demand arrive online: base demand, which must be covered at the current time, and flexible demand, which can be satisfied at any time steps before a demand-specific deadline $Δ_t$. The operator's goal is to minimize a cost (subject to the constraints above) that combines a cost of purchasing energy, a cost for delivering energy (if applicable), and smoothness penalties on the purchasing and delivery rates to discourage fluctuations and encourage ``grid healthy'' decisions. $\texttt{OSDM}$ generalizes several problems in the online algorithms literature while being the first to fully model applications of interest. We propose a competitive algorithm called $\texttt{PAAD}$ (partitioned accounting \& aggregated decisions) and show it achieves the optimal competitive ratio. To overcome the pessimism typical of worst-case analysis, we also propose a novel learning framework that provides guarantees on the worst-case competitive ratio (i.e., to provide robustness against nonstationarity) while allowing end-to-end differentiable learning of the best algorithm on historical instances of the problem. We evaluate our algorithms in a case study of a grid-integrated data center with battery storage, showing that $\texttt{PAAD}$ effectively solves the problem and end-to-end learning achieves substantial performance improvements compared to $\texttt{PAAD}$.

Online Smoothed Demand Management

TL;DR

This work introduces Online Smoothed Demand Management (OSDM), a general online optimization framework for energy procurement and demand delivery with storage, capturing both inflexible base demand and deferrable flexible demand under storage dynamics and smoothing penalties. It presents PAAD, a unified competitive algorithm that partitions demand into drivers (base and flexible) with per-driver threshold decisions and aggregates them to global purchasing and delivery actions, achieving optimal worst-case competitive ratios. To address pessimism in worst-case analysis, the authors propose PALD, a differentiable, learning-augmented framework that optimizes threshold functions from historical data while enforcing robustness certificates that preserve a bounded competitive ratio; PALD-S and PALD-C illustrate value in nonstationary, data-rich environments. A comprehensive case study on grid-integrated data centers with local storage using real price traces demonstrates substantial average-case gains from PALD over PAAD, highlighting practical implications for grid stability and energy-managed compute. Overall, OSDM bridges online algorithm theory with practical demand-management applications, providing both rigorous guarantees and data-driven enhancements for modern energy-aware infrastructures.

Abstract

We introduce and study a class of online problems called online smoothed demand management , motivated by paradigm shifts in grid integration and energy storage for large energy consumers such as data centers. In , an operator makes two decisions at each time step: an amount of energy to be purchased, and an amount of energy to be delivered (i.e., used for computation). The difference between these decisions charges (or discharges) the operator's energy storage (e.g., a battery). Two types of demand arrive online: base demand, which must be covered at the current time, and flexible demand, which can be satisfied at any time steps before a demand-specific deadline . The operator's goal is to minimize a cost (subject to the constraints above) that combines a cost of purchasing energy, a cost for delivering energy (if applicable), and smoothness penalties on the purchasing and delivery rates to discourage fluctuations and encourage ``grid healthy'' decisions. generalizes several problems in the online algorithms literature while being the first to fully model applications of interest. We propose a competitive algorithm called (partitioned accounting \& aggregated decisions) and show it achieves the optimal competitive ratio. To overcome the pessimism typical of worst-case analysis, we also propose a novel learning framework that provides guarantees on the worst-case competitive ratio (i.e., to provide robustness against nonstationarity) while allowing end-to-end differentiable learning of the best algorithm on historical instances of the problem. We evaluate our algorithms in a case study of a grid-integrated data center with battery storage, showing that effectively solves the problem and end-to-end learning achieves substantial performance improvements compared to .

Paper Structure

This paper contains 41 sections, 33 theorems, 154 equations, 12 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Problem Formulation
  5. Online Smoothed Demand Management (OSDM)
  6. Motivating Applications
  7. Modeling Smoothness Penalty
  8. Modeling Delivery Cost
  9. Assumptions and Notation
  10. Warmup and Preliminary Results
  11. A Unified Competitive Algorithm
  12. The PAAD Algorithm
  13. Competitive Analysis
  14. Proof Overview
  15. Fundamental Limits
  16. ...and 26 more sections

Key Result

Theorem 3.1

For the subset of OSDM-S instances described above, the "doubling extension" of RORO is $\zeta$-competitive, where $\alpha_{\texttt{RORO}}$ is the optimal competitive ratio for OCS (see eq:alpha_roro) and $\zeta$ is at least:

Figures (12)

  • Figure 1: Diagram of the OSDM problem at a single (discrete) time step $t$. The demand manager specifies a purchasing decision $x_t$, a delivery decision $z_t$, and (implicitly) a storage state $s_t$. As a function of these decisions, they pay ➊ a purchasing cost $p_t x_t$, ➌ a smoothness penalty $\mathcal{S}(x_t, x_{t-1})$ that discourages fluctuations in the purchasing rate, ➋ a delivery cost of serving demand $\mathcal{D}(z_t, s_{t-1}, p_t)$, and ➍ a switching penalty $\delta \vert z_t - z_{t-1} \vert$ that discourages fluctuations in the delivery rate.
  • Figure 2: Diagram of the intuition behind the PAAD algorithm. Each driver accounts for a single unit of base or flexible demand, making local decisions based on its own state and role that are aggregated into a single global decision. Purchasing decisions $x_t$ govern how much to purchase at the current market price, and delivery decisions $z_t$ dictate how much demand is satisfied at the current time (subject to feasibility constraints).
  • Figure 3: CDFs of ECR for all algorithms in default CAISO experiments.
  • Figure 4: Average ECR for all algorithms in varying regions, with a tracking cost, and varying storage size. In (a), top subfigure is coeff. of var. for each region.
  • Figure 5: Average ECR for all algorithms in CAISO with varying parameters as specified.
  • ...and 7 more figures

Theorems & Definitions (69)

  • Definition 2.1: Competitive ratio
  • Definition 2.2: Switching cost for purchasing decisions
  • Definition 2.3: Tracking cost for purchasing decisions
  • Definition 2.4: Monotone affine price-dependent delivery cost
  • Theorem 3.1
  • Definition 4.1: Base demand threshold function
  • Definition 4.2: Flexible demand threshold functions
  • Theorem 4.1
  • Corollary 4.3
  • Lemma 4.4
  • ...and 59 more