Efficient Industrial Refrigeration Scheduling with Peak Pricing

TL;DR

The paper addresses efficient scheduling of industrial refrigeration under complex energy pricing by casting the problem in an inventory-control framework with state dynamics $x_t$ (temperature) and an auxiliary peak state $y$, governed by $x_{t+1}=x_t-u_t+q_t$ and $y_{t+1}=\max\{y_t,u_t\}$. It formalizes costs as a stage component $c_t(x_t,u_t)=o_t(u_t)+\mathbb{E}[h_t(x_{t+1})]$ plus a peak charge $P \max\{y,\{u_t\}\}$, and derives a Bellman recursion $V_t(x,y)=\min_{u\ge0}\{ c_t(x,u)+\mathbb{E}_{q_t}[V_{t+1}(x^+,y^+)]\}$ with $V_{T+1}(x,y)=P y$. Under TOU-only costs ($P=0$), the optimal policy is a threshold policy of the form $\pi_t^*(x,y)=x-S_t$ if $x>s_t$, and $0$ otherwise, with $s_t\ge S_t$ (and $s_t=S_t$ when $K=0$). Introducing peak pricing ($P>0$) disrupts threshold optimality; the paper provides a general structural result using a function $g_t(y)$ that partitions the state into three regimes, yielding a policy that may lie in an interval $[y, z^*_t]$ for certain $x$, and proves the value function $V_t$ is convex and nondecreasing in $y$. A simulation study with real facility data demonstrates that dynamic and modified-threshold policies can substantially outperform static designs, validating the practical value of peak-aware control design. Overall, the work highlights the need for augmented-state policies when peak costs are significant and provides theoretical and empirical guidance for efficient refrigeration management. The results contribute a principled understanding of how energy pricing shapes control structure in energy-intensive industrial systems.

Abstract

The widespread use of industrial refrigeration systems across various sectors contribute significantly to global energy consumption, highlighting substantial opportunities for energy conservation through intelligent control design. As such, this work focuses on control algorithm design in industrial refrigeration that minimize operational costs and provide efficient heat extraction. By adopting tools from inventory control, we characterize the structure of these optimal control policies, exploring the impact of different energy cost-rate structures such as time-of-use (TOU) pricing and peak pricing. While classical threshold policies are optimal under TOU costs, introducing peak pricing challenges their optimality, emphasizing the need for carefully designed control strategies in the presence of significant peak costs. We provide theoretical findings and simulation studies on this phenomenon, offering insights for more efficient industrial refrigeration management.

Figures (7)

• Figure 1: We depict a prototypical power consumption profile over different pricing regions.
• Figure 2: We present distributions and respective averages of $\bf{Q}_t$ over the horizon of a day for a particular refrigeration facility.
• Figure 3: We display the power-heat curves of different compressor types as taken from factandfig. We see that for compressors without variable frequency drives (VFDs), the power draw and the respective thermal capacity share an affine relationship.
• Figure 4: We depict the possible optimal inputs for each of the three regimes, dependent on $x$. Note that since $P \geq b$, the optimal $u^*$ must live between $[0, y]$, as delineated by the orange markers.
• Figure 5: We depict the modified policy.

Theorems & Definitions (9)

• Proposition 1: scarf1960optimality
• proof
• Example 1: Peak Pricing
• Proposition 2
• proof
• Proposition 3
• proof
• Remark
• Example 2