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arxiv: 2605.15431 · v1 · pith:MYQD7ECUnew · submitted 2026-05-14 · 📡 eess.SY · cs.SY

Optimizing Chilled Water Systems with Cooling Towers via Virtual Power Metrics and Extremum-Seeking Control

Min Gyung Yu , Alex Vlachokostas , Karthik Devaprasad , Matt Cornachione , Stephanie Johnson , Tim A. Yoder , Timothy I. Salsbury This is my paper

Pith reviewed 2026-05-19 14:55 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords extremum seeking controlcooling towerchilled water plantvirtual power meterenergy savingsHVAC optimizationfan speed control
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The pith

An extremum-seeking controller for cooling tower fans minimizes total chilled water plant power with virtual power meters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that extremum-seeking control can automatically tune cooling tower fan speed to the point of lowest overall energy use in a chilled water system. It introduces virtual power meters that calculate the necessary power values from routine temperature and flow readings, allowing the technique to work without dedicated power sensors. If correct, this would let existing plants save roughly 15 percent on cooling energy in summer while using only standard instrumentation. Simulations in multiple climates support the savings figure, and the virtual meters track physical ones with 96 percent correlation and low error.

Core claim

The central discovery is that extremum-seeking control applied solely to cooling tower fan speed, fed by virtual power estimates, can converge on the minimum total plant power consumption and deliver approximately 15% energy savings compared to conventional control in summer conditions across different locations.

What carries the argument

Extremum-seeking control that perturbs fan speed to estimate the power gradient and drives the system toward the optimum, enabled by virtual power meters constructed from available sensor data.

Load-bearing premise

Changing only the cooling tower fan speed is sufficient to reach and track the true minimum of the entire plant's power consumption, with other components and loads either fixed or accurately represented in the model.

What would settle it

Running the controller on an actual chilled water plant and measuring whether total power drops to the level predicted by the extremum-seeking algorithm as outdoor conditions vary.

Figures

Figures reproduced from arXiv: 2605.15431 by Alex Vlachokostas, Karthik Devaprasad, Matt Cornachione, Min Gyung Yu, Stephanie Johnson, Tim A. Yoder, Timothy I. Salsbury.

Figure 1
Figure 1. Figure 1: Chiller plant system with the ESC to control cooling tower fan. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Block diagram of real-time optimizer. filter time constant. The relay output ε is then multiplied by a gain factor K and integrated to produce the system’s manipulated vari￾able, x. The manipulated variable is constrained within specified minimum and maximum bounds. The gain fac￾tor K is determined as: K = ∆t(xmax −xmin) 5(τ+τ f) (2) where ∆t is the discrete time step. All internal variables (e.g., K, dlim… view at source ↗
Figure 3
Figure 3. Figure 3: Co-simulation workflow. ESC configuration Implementing the ESC algorithm requires just one sys￾tem parameter, the time constant τ. To estimate this time constant, the system’s dynamic response needs to be tested and, in this work, an impulse response test was used to estimate the time constant. The impulse test is a method where the system’s input is significantly changed from one value to another and held… view at source ↗
Figure 5
Figure 5. Figure 5: Overview of virtual power meter. First, we develop a physics-based model to estimate the power consumption of the energy system. Every sys￾tem has unique system characteristics that can vary based on several factors such as capacity, efficiency, and more. This information can typically be sourced from manu￾facturer specifications or technical documents. Then, the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Weather profiles for two climates. Normalized Root mean square error: NRMSE = RMSE Range(y) = r 1 n n ∑ i=1 (yi −yˆi) 2 (ymax −ymin) (14) where y is the measured value, ˆy is the estimated value from the VPM, ¯y is the mean of the measured value, n is the number of observations, ymax and ymin are the maxi￾mum and minimum value of the measured data. Equation (12) quantifies how well the estimated values fro… view at source ↗
Figure 9
Figure 9. Figure 9: Cooling tower power vs. chiller power by [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Stacked power consumption of components. [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Convexity tests. components throughout the simulation testing period for both climates. Since total power consumption served as the cost function for the ESC algorithm, these plots ef￾fectively act as static maps of the cost function, indicat￾ing which fan speed minimizes total power consumption [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of daily energy savings achieved by ESC relative to Fixed-speed and PID control. Bars represent the mean percentage savings, with error bars indicating 95% confidence intervals. the fact that under dry ambient conditions, evaporative heat rejection in the cooling tower can be highly effec￾tive just through natural evaporation, without requiring mechanical fan operation. This behavior aligns wit… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of timeseries power and cumulative energy: ESC vs Fixed 100% vs PID changed. However, horizontal shifts or deformations that affect the map’s convexity can impair convergence or move the optimizer away from the true optimum. De￾spite these challenges, the static map’s convexity can be found even under noise, and the ESC can still converge reliably to a near-optimal operating point. Moreover, [… view at source ↗
Figure 15
Figure 15. Figure 15: Performance curves of chiller. fications and applied a correction factor to adjust the es￾timated power from the VPM. Additionally, as the data was derived from 15-minute interval historical records, which may not capture smooth changes in temperature and power usage, we applied smoothing techniques to improve the analysis [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 18
Figure 18. Figure 18: Scatter plot of daily average power readings [PITH_FULL_IMAGE:figures/full_fig_p013_18.png] view at source ↗
read the original abstract

This paper presents an extremum seeking control (ESC) method for cooling tower fans to minimize overall power consumption of a chilled water plant system. Simulation studies across different climate locations demonstrate energy savings of approximately 15% compared to conventional control during summer conditions. This paper also proposes a virtual power meter (VPM) to enable use of the strategy in systems that lack physical power meters. Validation tests for the VPMs against physical meters showed good accuracy with a correlation of 96.11% and a normalized error of 5.11%. Coupled with the VPM, the proposed ESC control solution can be implemented on systems using typically available sensor measurements without the need for additional instrumentation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes an extremum-seeking control (ESC) strategy to optimize cooling-tower fan speed for minimizing total chilled-water-plant power consumption. Simulation studies across climate locations report approximately 15% energy savings versus conventional control in summer conditions. A virtual power meter (VPM) is introduced and validated open-loop against physical meters, achieving 96.11% correlation and 5.11% normalized error, to enable deployment without additional instrumentation.

Significance. If the underlying simulation faithfully reproduces the steady-state power map and the VPM error does not degrade ESC gradient estimates, the approach would provide a practical, sensor-light method for HVAC energy optimization with measurable savings. The combination of standard ESC structure with a validated VPM is a modest but useful engineering contribution for systems lacking power metering.

major comments (2)
  1. [Simulation studies] Simulation studies section: the 15% summer savings and ESC convergence claims rest on the assumption that varying only cooling-tower fan speed produces a unique, trackable minimum in total plant power while all other loads, setpoints, and disturbances remain fixed or perfectly known; the manuscript provides no quantitative sensitivity analysis on heat-transfer coefficients, pump curves, or unmodeled dynamics that would shift this minimum.
  2. [VPM validation] VPM validation subsection: the reported 96.11% correlation and 5.11% normalized error are obtained in open-loop tests against physical meters; it is not shown whether this error level preserves the sign and magnitude of the dither-based gradient estimates inside the ESC loop, which is load-bearing for the closed-loop savings claim.
minor comments (2)
  1. [Abstract] The abstract states 'approximately 15%' savings but does not specify the number of independent climate cases, disturbance profiles, or model fidelity metrics; adding these details would strengthen the claim.
  2. [Method] Notation for the VPM output and its integration into the ESC demodulation block should be defined explicitly with an equation reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We respond to each major comment below and indicate planned revisions to address the concerns raised.

read point-by-point responses

  1. Referee: [Simulation studies] Simulation studies section: the 15% summer savings and ESC convergence claims rest on the assumption that varying only cooling-tower fan speed produces a unique, trackable minimum in total plant power while all other loads, setpoints, and disturbances remain fixed or perfectly known; the manuscript provides no quantitative sensitivity analysis on heat-transfer coefficients, pump curves, or unmodeled dynamics that would shift this minimum.

    Authors: We agree that the current manuscript lacks a quantitative sensitivity analysis on parameters such as heat-transfer coefficients, pump curves, and unmodeled dynamics. In the revised version, we will add a dedicated sensitivity study (new subsection in the simulation studies section) that perturbs these parameters within realistic ranges and reports the resulting shifts in the location and trackability of the total plant power minimum. This will directly support the robustness of the reported 15% savings and ESC convergence under the stated assumptions. revision: yes

  2. Referee: [VPM validation] VPM validation subsection: the reported 96.11% correlation and 5.11% normalized error are obtained in open-loop tests against physical meters; it is not shown whether this error level preserves the sign and magnitude of the dither-based gradient estimates inside the ESC loop, which is load-bearing for the closed-loop savings claim.

    Authors: The referee is correct that the VPM accuracy metrics are from open-loop validation only. To close this gap, the revised manuscript will include new closed-loop simulation results in which the observed VPM error statistics are injected into the ESC feedback path. We will show that the dither-based gradient estimates retain the correct sign and adequate magnitude for convergence to the power minimum, thereby confirming that the reported error level does not undermine the closed-loop energy savings. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies standard extremum-seeking control to minimize total plant power via cooling-tower fan speed in a simulation model, with virtual power meters validated directly against independent physical meters (96.11% correlation, 5.11% normalized error). Neither the reported 15% summer savings nor the ESC convergence claims reduce by construction to any fitted parameter, self-defined quantity, or self-citation chain inside the paper; the simulation results and meter comparisons constitute external benchmarks rather than tautological renamings or load-bearing self-references. The derivation therefore remains self-contained against the stated physical model and real-meter validation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on standard extremum-seeking convergence assumptions and on the premise that a data-driven virtual power meter can substitute for physical metering with acceptable error. No new physical constants or ad-hoc entities beyond the VPM are introduced.

axioms (2)
  • standard math Extremum-seeking control converges to the minimum of the plant power map when the dither signal and adaptation gains satisfy standard persistence-of-excitation conditions.
    Invoked by the choice of ESC for fan-speed optimization; location implicit in the method description.
  • domain assumption The virtual power meter mapping from available sensor readings to power is stationary and can be validated on a representative subset of operating conditions.
    Required for the claim that VPM enables ESC without physical meters; stated via the validation correlation and error figures.
invented entities (1)
  • Virtual Power Meter (VPM) independent evidence
    purpose: Estimate fan and pump power from temperature, flow, and other standard measurements to replace physical power meters for ESC implementation.
    New construct proposed in the paper; independent evidence is the reported 96.11% correlation and 5.11% normalized error against physical meters.

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Reference graph

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