The retention factor sets the pre-cooling tradeoff
Compressors are most efficient at full capacity, so running them fully on and storing the excess cooling as thermal inventory seems attractive. The catch is that stored cooling is lossy: a colder space draws more heat from its surroundings, so some of the buffer dissipates before it can absorb a future heat load.
The paper captures this with a single scalar, the retention factor rho, defined as the fraction of thermal inventory that survives after one minute. When rho equals one, no stored cooling is lost and full-load pre-cooling is effective; in practice rho is strictly below one, so full-load operation is not always desirable. The study's two goals follow directly: understand how retention changes the value of pre-cooling, and design control policies that account for lossiness.
Thermal inventory behaves like a lossy stock
The authors model the aggregate stored cooling, in kWh, as inventory in excess of what is needed to hold every room at its temperature limit. Inventory is replenished by the compressors and depleted by random heat loads arriving at the facility, with only the retention fraction carrying over each one-minute step.
Two depletion mechanisms are kept distinct. The external heat load is the demand at the temperature limit, while the additional load induced by carrying inventory is the loss term. The resulting optimization minimizes expected compressor work subject to inventory staying nonnegative, plus a minimum on-time constraint that forces a compressor, once started, to remain on for a set number of consecutive minutes.
Three policies span optimal to simple
The first policy is an approximately optimal dynamic program that minimizes expected compressor energy over discretized states and actions. It is highly performant but hard to interpret and expensive to compute, since the backward recursion needs the full heat-load distribution and a grid whose size grows exponentially with the number of compressors.
Two simpler alternatives are built around full-load efficiency. The full-load policy switches the less efficient compressor between off and full capacity, turning it on whenever inventory cannot cover the worst-case heat load. The partial-load policy, inspired by the structure of the dynamic-programming solution, also runs that compressor at partial load to avoid excess pre-cooling. Both keep the more efficient compressor at full capacity and differ only in how they operate the second one.
Models are fit from real facility telemetry
The policies are evaluated on models fit from one month of telemetry at a real cold storage facility in the United States. Incoming heat is not measured directly, so the authors use the heat absorbed by the evaporators as an indirect proxy, combining cooling status, rated capacity per degree of temperature difference, and the room-to-suction temperature gap.
Compressor electrical power is estimated from measured motor current, line voltage, and power factor, yielding work-heat curves over each compressor's active range. The facility has two compressors, and the less efficient one has a minimum on-time of 60 minutes. The heat-load distribution is built by binning reconstructed observations into twenty equal-width bins.
Full-load pre-cooling degrades as retention falls
Across 10,000 Monte Carlo simulations with 24-hour heat trajectories, energy use rises for all policies as retention decreases. For the dynamic program, cutting rho from 1 to 0.99 raises mean cost by 27.3%, while a further cut to 0.95 adds only 16.7%, a diminishing marginal increase.
The full-load policy suffers most. At rho equal to 1, the partial-load and full-load policies coincide, because inventory carries over without loss and there is no penalty for excess pre-cooling. As rho falls, carrying excess inventory becomes costly: the full-load policy costs 2.5% more than partial-load at rho equal to 0.99 and 22.7% more at rho equal to 0.95. At rho equal to 0, no inventory carries over, pre-cooling provides no benefit, and the full-load policy consumes 36.8% more energy than the other two.
Simple loss-aware policies stay near-optimal
The partial-load policy tracks the dynamic program closely, with mean costs only 1.6%, 1.0%, and 1.8% higher at rho equal to 1, 0.99, and 0.95. The full-load policy's mean costs are 1.6%, 3.5%, and 25.0% higher than the dynamic program at the same retention factors.
A clairvoyant dynamic program with perfect knowledge of future demand provides a benchmark. The practical dynamic program consumes only 3.9% more energy than this benchmark at rho equal to 0.99, and the cost of demand uncertainty grows as retention falls: 1.3%, 3.9%, and 6.3% more at rho equal to 1, 0.99, and 0.95. Lower retention shortens the useful lifetime of pre-cooling, making timing and quantity errors more costly.
Pre-cooling benefit shrinks under severe losses
Relative to a no-pre-cooling benchmark, the dynamic program, partial-load, and full-load policies consume 15.4%, 14.4%, and 10.3% less energy at rho equal to 0.99, so pre-cooling helps even when 1% of inventory dissipates each minute. At rho equal to 0.95, 5% of inventory dissipates each minute, leaving only 4.6% after one hour.
Under those severe losses, the dynamic program and partial-load policy perform close to the no-pre-cooling benchmark, while the full-load policy incurs substantial excess consumption. The authors read this as evidence that load-shifting strategies should be designed with the inventory retention factor in mind, since little stored cooling survives until needed when losses are large.
Where these results do and do not apply
The analysis uses a single aggregate thermal inventory and does not model how evaporators operate or how heat and cooling are disaggregated across rooms, so room-level pre-cooling remains outside the current scope. The simple policies are also derived under an operating regime relating compressor capacities to mean and maximum heat loads, and extending them beyond that regime is left open.
Most importantly, the study demonstrates the importance of the retention factor by analyzing policy costs across settings but does not propose a method for estimating that factor itself. The practical priority is therefore to treat retention as a measured or estimated quantity before committing to aggressive pre-cooling: when retention is near one, full-load operation is attractive, but as it falls, partial-load operation offers substantial savings and the full-load policy's cost penalty grows sharply.
Sources used in this answer
Holding the Cold: Retention-Aware Inventory Control for Industrial Refrigeration
Pre-cooling saves substantial energy when thermal inventory is sufficiently persistent, but ignoring retention makes aggressive full-load pre-cooling increasingly costly as losses grow; simple loss-aware policies can still achieve near-optimal performance.