- The paper demonstrates a single-parameter relay-based extremum-seeking controller that reduces simulated chilled water plant power by 14.47–15.81% versus fixed full-speed fans and by 9.85% versus idealized PID control in humid Houston.
- The paper develops virtual power meters from chiller performance curves and fan power laws, achieving R² = 0.9611, RMSE of 5.69 kW, and 5.11% normalized error against one year of measured data.
- The paper shows that virtual power trends could enable low-configuration optimization where physical meters are unavailable, but real-world validation is still needed because the VPM was not tested inside the closed-loop controller.
Overview and motivation
This paper by Yu et al. (Pacific Northwest National Laboratory) presents a relay-based extremum seeking control (ESC) strategy for cooling tower fans that minimizes total chilled water plant power consumption, together with a virtual power meter (VPM) that enables deployment on plants lacking physical power meters. The work is motivated by the fact that chillers, though present in only 3% of U.S. buildings, serve 19% of cooled floor space and can account for up to 40% of a building's electricity use. The central optimization problem is the well-known trade-off between cooling tower fan power (which scales roughly with the cube of fan speed) and chiller compressor power (which decreases as condenser water temperature drops). Conventional feedback control regulates temperatures but does not directly minimize energy; model predictive approaches achieve savings but require models that are costly to build and retune. The authors' stated goal is a controller requiring minimal configuration—one time-constant parameter—that operates directly on measured or estimated power.
Control architecture and cost function
The ESC is a SISO relay-based "perturb-and-observe" optimizer layered on top of existing low-level PI loops, so it does not interfere with baseline temperature and pressure regulation. A relay block outputs ϵ∈{−1,+1} based on the sign of the cost gradient, with a dwell time dlim​=τ+τf​ preventing rapid switching. All internal gains are derived from a single parameter: the plant time constant τ, estimated here via an impulse (bump) test on fan speed, yielding τ=183 s—consistent with typical cooling tower response times of a few minutes.
The cost function is the sum of component powers:
J=PCWPump​+PCHWPump​+PChiller​+PCoolTower​+PAHU​
The manipulated variable is the cooling tower fan speed setpoint. Notably, because ESC only requires the gradient of J, accurate absolute power values are not strictly necessary—an observation that underpins the VPM design rationale.
The simulation testbed couples a Python ESC implementation with a Modelica Buildings Library v10.0 chiller plant model (exported as an FMU via FMPy), serving a data-center thermal zone. Modifications included enabling direct fan-speed input, disabling hysteresis-based cycling, and deactivating the water-side economizer.
Simulation results across climates
One-week July simulations were run for Pasco, WA (climate zone 5B, hot-dry summers) and Houston, TX (2A, hot-humid). Key findings:
| Comparison |
Pasco, WA |
Houston, TX |
| ESC vs. fixed 100% fan speed |
15.81% mean savings (95% CI: 15.58–16.04%) |
14.47% mean savings (95% CI: 14.41–14.52%) |
| ESC vs. idealized PID (25°C setpoint) |
0.30% (CI: 0.24–0.36%) |
9.85% (CI: 8.23–11.48%) |
Convexity tests confirm the static map of total power versus fan speed is convex in both climates, with the optimum near 0–20% fan speed in Pasco and near 40% in Houston; ESC operation tracks the convex minimum closely. The contrast against PID is instructive: in dry Pasco, evaporative rejection is effective at low fan speeds, so a 25°C leaving-water setpoint is nearly optimal and PID matches ESC within 0.3%. In humid Houston, holding the same setpoint forces near-100% fan speeds, and ESC's modulation yields nearly 10% savings over even an idealized PID—a strong claim, since real PID implementations with tracking error and poor tuning would widen this gap further.
A sensitivity analysis injecting Gaussian noise (σ=5°C) into chiller temperature sensors showed a systematic downward bias in predicted chiller power, consistent with Jensen's inequality applied to concave performance curves. The authors note that vertical shifts of the static map do not move the optimum, whereas horizontal shifts or deformations could impair convergence; nevertheless, convexity persisted under noise and ESC converged to a near-optimal point.
Virtual power meter development and validation
Because field deployments (with the U.S. Army Reserve) revealed that most chilled water plants lack power meters, the paper develops physics-based VPMs for the chiller and cooling tower fan. The chiller VPM follows the DOE-2.1 electric chiller formulation, using three quadratic/biquadratic performance curves (available capacity vs. temperatures, EIR vs. temperatures, EIR vs. PLR) fitted from manufacturer data using the Copper tool. Fan power follows the standard cubic law scaled from nameplate horsepower.
Validation used one year of 15-minute historical data (May 2023–May 2024) from a test building with a 155-ton air-cooled screw chiller. Two limitations are acknowledged at the outset: chilled water flow rate was unavailable, so a constant flow rate with a correction factor was assumed, and smoothing was applied to the coarse interval data. Despite these simplifications, the VPM achieved:
- R² = 0.9611 on daily average power
- RMSE = 5.69 kW, NRMSE = 5.11%
- Residuals mostly centered on zero, with excursions up to ±15 kW attributed largely to the fixed-flow assumption
The authors argue this accuracy suffices for ESC because the algorithm depends on gradient direction rather than absolute magnitude, and its time-scale separation filters transient estimation noise. This claim is plausible but rests on the assumption that VPM errors do not systematically deform the cost function's shape—the same horizontal-shift concern raised in the sensor-noise analysis—and it remains untested end-to-end, since the VPM was not exercised inside the closed-loop ESC simulations (the simulation testbed uses true model powers).
Limitations and open questions
Several caveats bound the reported results. First, all energy-savings figures derive from simulation of a single data-center load profile during one summer week per climate; annual performance, part-load behavior, and building types other than data centers are unexamined. Second, the water-side economizer was disabled, so interactions with economizer operation are unknown. Third, the comparison PID is idealized (perfect setpoint tracking), which flatters the baseline yet still leaves ESC ahead in Houston. Fourth, the VPM validation covers only the chiller (plus a trivially cubic fan model) at one site with missing flow data; robustness to missing or biased measurements and to longer BAS reporting intervals is explicitly deferred to future work. Finally, no closed-loop demonstration combining VPM-estimated power with ESC has been performed—the central premise that VPM trend accuracy translates to correct ESC convergence awaits field validation.
Conclusion
The paper demonstrates that a single-parameter relay-based ESC, minimizing summed plant power as its cost function, achieves approximately 15% summer energy savings over fixed full-speed fan operation in both dry and humid climates, and up to ~10% over an idealized setpoint-tracking PID in humid conditions. The accompanying VPM, validated against a physical meter with R² above 0.96 and normalized error near 5%, provides a software-only path to deploying this optimizer on the many plants without power metering. The approach's practical significance lies in its minimal configuration burden and reliance on typically available BAS measurements; its principal open question is whether simulation-level savings and VPM-driven convergence carry over to real-world installations, which the authors identify as their next step.