Online Smoothed Demand Management

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 before a demand-specific deadline $Δ_t$. The operator’s goal is to minimize a cost (subject to above constraints) 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 for $\texttt{OSDM}$ 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 $\texttt{PAAD}$.

💡 Research Summary

The paper introduces a novel online optimization problem called Online Smoothed Demand Management (OSDM), motivated by the need for large energy consumers such as data centers to purchase electricity, serve workloads, and operate local battery storage in real time while avoiding abrupt changes in purchasing and delivery rates. At each discrete time step t, the system observes a market price pₜ, an inelastic (base) demand bₜ that must be satisfied immediately, and a flexible demand fₜ that can be deferred until its deadline Δₜ. The operator decides how much energy to purchase (xₜ) and how much to deliver to the workload (zₜ). The difference xₜ – zₜ updates the battery state sₜ, which is constrained by a capacity S. The objective combines four cost components: (i) procurement cost Σ pₜ xₜ, (ii) a delivery cost D(zₜ, sₜ₋₁, pₜ) that may depend on the current price and battery level, (iii) a smoothness penalty S(xₜ, xₜ₋₁) on the purchasing rate, and (iv) a switching penalty δ|zₜ – zₜ₋₁| on the delivery rate. The problem captures four intertwined challenges: time‑varying prices, inventory (battery) constraints, a mix of inelastic and deferrable workloads with heterogeneous deadlines, and penalties that discourage rapid ramps.

Existing online models (e.g., online linear inventory management, online conversion with switching costs, metrical task systems) each capture a subset of these aspects but cannot handle their combination. Consequently, standard competitive‑ratio techniques do not directly apply.

The authors propose PAAD (Partitioned Accounting & Aggregated Decisions), the first competitive algorithm for OSDM. PAAD decomposes the global problem into a collection of “driver” sub‑problems, each associated with a distinct threshold function that encodes price bounds, smoothness coefficients, and deadline information. Within each sub‑problem, the optimal purchase and delivery amounts are computed independently; the results are then aggregated to form a globally feasible solution {xₜ, zₜ}. By treating purchasing and delivery smoothness via separate Lagrange multipliers, PAAD simultaneously exploits price arbitrage through battery charging/discharging and limits abrupt changes.

Theoretical analysis shows that PAAD achieves an α‑competitive ratio where α is sublinear in the price‑range ratio p_max/p_min (specifically O(√(p_max/p_min))) and linear in the magnitudes of the switching and delivery costs. Matching lower‑bound constructions demonstrate that no online algorithm can obtain a better dependence on these parameters under the same model assumptions, establishing PAAD’s optimality.

Recognizing that worst‑case competitive ratios are often overly pessimistic, the paper introduces PALD (Partitioned Accounting & Learned Decisions), a learning‑augmented framework. PALD treats the threshold functions as differentiable parameters and trains them end‑to‑end on historical OSDM instances to minimize empirical cost. Crucially, PALD incorporates a robustness certificate: if the learned parameters deviate beyond a certified region, the algorithm falls back to the provably optimal PAAD thresholds, guaranteeing a bounded competitive ratio even under adversarial inputs. This design sidesteps the need for explicit predictions of prices or demands, which are difficult to obtain reliably in non‑stationary electricity markets.

Experimental evaluation uses real electricity price data and Alibaba workload traces to simulate a grid‑integrated data center with co‑located battery storage. PAAD alone already attains near‑optimal performance in practice, confirming the tightness of the theoretical bound. PALD further reduces average total cost by 15–25 % relative to PAAD, while still respecting the robustness guarantee; under artificially stressed price spikes, PALD’s competitive ratio remains within a small constant factor of PAAD’s bound.

The related‑work discussion situates OSDM among demand‑response heuristics, model‑predictive control, Lyapunov‑based methods, and various online algorithmic problems, highlighting how OSDM uniquely integrates multiple realistic constraints that have previously been studied in isolation.

In summary, the paper makes three major contributions: (1) formalizing OSDM, a comprehensive online model that captures mixed demand, storage dynamics, price variability, and smoothness penalties; (2) designing PAAD, an optimal‑competitive algorithm that solves OSDM via partitioned accounting and aggregation; and (3) presenting PALD, a differentiable learning framework that improves average‑case performance while preserving worst‑case guarantees. The work bridges a gap between theory and practice in energy‑aware computing, and opens avenues for extensions such as multi‑storage systems, non‑linear cost structures, and tighter integration with real‑time forecasting.