Cooling Plant Efficiency Prediction [/ˈkulɪŋ ˈplænt ɪˈfɪʃənsi priˈdɪkʃən/] n - A data center cooling plant is a company of thermodynamic machines, each bearing an efficiency that turns upon its conditions. Chillers run the better when the condenser water is cold. Cooling towers perform the better at low wet-bulb temperatures. Computer room air handlers serve best when their airflow matches the true heat density of the zone. To forecast these efficiencies is to dispatch cooling capacity the more wisely.

Of the Coefficient of Performance and Its Conditions

Chiller efficiency is expressed as the coefficient of performance, the ratio of heat removed to electrical energy consumed:

$$\text{COP}(t)=\frac{Q_{\text{removed}}(t)}{W_{\text{chiller}}(t)}$$

A well-kept centrifugal chiller in favorable conditions may attain a COP above 6; in hot weather, with high condenser water return temperatures, the same machine may fall below 4. The relation of COP to its conditions runs approximately:

$$\text{COP}(t)\approx {\alpha }_0+{\alpha }_1{T}_{\text{cwrt}}(t)+{\alpha }_2{T}_{\text{chws}}(t)+{\alpha }_3\text{PLR}(t)+{\alpha }_4\text{PLR}(t{)}^2$$

where $T_{\text{cwrt}}$ is the condenser water return temperature, $T_{\text{chws}}$ the chilled water supply setpoint, and PLR the part-load ratio. The quadratic PLR term records the degraded efficiency at very low and very high loads.

This regression may be estimated from historical SCADA data, and the coefficients should be re-estimated seasonally, for chiller character drifts with age and fouling. See Linear Factor Models in Trading for the estimation framework; the mechanics transfer without alteration.

Of Tower Capacity and Wet-Bulb Temperature

Cooling tower performance depends almost wholly upon the ambient wet-bulb temperature:

$$T_{\text{cwrt}}(t)=T_{\text{wb}}(t)+{\Delta }_{\text{approach}}$$

The approach temperature is the gap between condenser water return and wet-bulb; a low approach marks a tower laboring well. As fouling grows, or as ambient falls below design, the approach drifts.

To forecast plant efficiency one must first forecast wet-bulb temperature. Local NWP (numerical weather prediction) data yields 24- to 72-hour forecasts; joined to the chiller COP model, these give a forward view of cooling capacity and cost.

PUE as Composite Measure

Power usage effectiveness is:

$$\text{PUE}(t)=\frac{P_{\text{total}}(t)}{P_{\text{IT}}(t)}$$

It gathers several sub-systems: cooling power, lighting, UPS losses, and distribution losses. Cooling is commonly the largest non-IT load, and so drives PUE above all else.

A PUE forecast model:

$$\widehat{\text{PUE}}(t+h)=1+\frac{{\hat{P}}_{\text{cooling}}(t+h)+P_{\text{other}}}{{\hat{P}}_{\text{IT}}(t+h)}$$

requires both an IT load forecast (see Power Load Forecasting in Data Centers) and a cooling plant model. The uncertainties compound: errors in the IT load forecast beget errors in the cooling load forecast through the physical bond between heat dissipation and cooling demand.

Of Optimization Beneath Uncertainty

Once these models stand in place, they support dispatch optimization: which company of chillers to run, at what setpoints, to minimize total cooling power for a given IT load forecast.

In its simple form: minimize ${\sum }_j{W}_j(Q_j,c_j)$ subject to ${\sum }_j{Q}_j=Q_{\text{demand}}$, where $W_j$ is the power consumed by machine $j$ at cooling load $Q_j$ and condition $c_j$.

Forecast uncertainty enters through $Q_{\text{demand}}$. Running the optimization at several quantiles of the load forecast, say the 10th, 50th, and 90th percentile, yields a range of dispatch plans and lets the operator choose by his tolerance for risk. It is the same quantile logic used in Value at Risk and Expected Shortfall, translated into a problem of physical engineering.