Cooling Under Convexity: An Inventory Control Perspective on Industrial Refrigeration

TL;DR

This paper addresses reducing energy consumption in industrial refrigeration by optimally scheduling heat removal (load shifting) under a convex relationship between heat removal and compressor work. It casts suction-pressure optimization as an inventory control problem with convex ordering costs and a zero-holding, infinite-backlog regime, yielding tractable analysis. The authors provide an optimal policy for deterministic demand, derive performance bounds under uncertainty, and propose a practical heuristic with provable near-optimal guarantees, all supported by simulations and a real-data case study. The findings quantify the value of load shifting and offer implementable strategies to reduce energy costs and improve operational efficiency in refrigeration systems. The work advances the understanding of how thermodynamic convexities can be exploited to design better control policies in energy-intensive industrial processes.

Abstract

Industrial refrigeration systems have substantial energy needs, but optimizing their operation remains challenging due to the tension between minimizing energy costs and meeting strict cooling requirements. Load shifting--strategic overcooling in anticipation of future demands--offers substantial efficiency gains. This work seeks to rigorously quantify these potential savings through the derivation of optimal load shifting policies. Our first contribution establishes a novel connection between industrial refrigeration and inventory control problems with convex ordering costs, where the convexity arises from the relationship between energy consumption and cooling capacity. Leveraging this formulation, we derive three main theoretical results: (1) an optimal algorithm for deterministic demand scenarios, along with proof that optimal trajectories are non-increasing (a valuable structural insight for practical control); (2) performance bounds that quantify the value of load shifting as a function of cost convexity, demand variability, and temporal patterns; (3) a computationally tractable load shifting heuristic with provable near-optimal performance under uncertainty. Numerical simulations validate our theoretical findings, and a case study using real industrial refrigeration data demonstrates an opportunity for improved load shifting.

Figures (4)

• Figure 1: The pressure-enthalpy ($P$-$h$) diagram for ammonia, the most common industrial refrigerant. The saturation dome (green) separates phases of matter. In the vapor compression refrigeration cycle (solid black), heat is removed by cycling a refrigerant through the five points on the diagram in numerical order. The suction pressure $P^s$ and discharge pressure $P^d$ uniquely determine all five points. We consider a widely implemented configuration that includes a pressure vessel corresponding to point $4$ which separates the refrigerant into saturated liquid and vapor phases; our model also applies to the standard refrigeration cycle with liquid subcooling in the condenser to point $3'$ (gray). We assume that (i) the system is at thermodynamic steady-state, (ii) the compressor performs isentropic compression, (iii) the expansion valve performs isenthalpic expansion, (iv) the condenser/evaporator return precisely saturated liquid/vapor phases, and (v) pressure drops in the pipes and devices (other than the expansion valve) are neglected.
• Figure 2: Top: The function $G(u)$, which relates specific heat absorption to specific work for ammonia. Bottom: The second derivative $G"(u)$; observe that it is strictly positive. The solid line in each plot corresponds to $P^d = 1.5 \times 10^6$ Pa; the shaded surrounding regions illustrate that a similar convex relationship holds for a wide range of practical discharge pressures.
• Figure 3: Cost comparisons of various policies in deterministic and stochastic settings over 1,000 simulations. Left: the trajectory of the demand support; we consider a discrete uniform probability distribution at every time step. Center: the performance of the optimal and myopic policies in the deterministic setting; the known demand trajectories $\bm{w}$ are generated by sampling the distribution (left) a priori. Right: the performance of the optimal policy, both heuristics, and the myopic policy in the stochastic setting. The known distributions (left) are used to compute the optimal policy via dynamic programming. All four policies are then evaluated via Monte Carlo simulation. The simulations use a state space $x \in [0,1]$, discretization $dx = 0.001$, horizon $N = 50$, cost function $G(u_k) = 100 u_k^2$, and $w_k$ discretized like the state.
• Figure 4: A case study in suction pressure control using real data from February 2025 at an industrial refrigeration facility in Salem, CT. The month of data is broken up into 24-hour windows. Top: original and optimized (from Algorithm \ref{['alg:load_shift']}) suction pressure trajectories over one sample window. Bottom: violin plot of suction pressure data and corresponding optimized values over all windows. Note the reduction in variability after load shifting.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof