Multi-Modal Machine Learning to Predict the Energy Discharge Levels from a Multi-Cell Mechanical Draft Cooling Tower

Abstract

An artificial neural network was developed to augment the accuracy of a physically based computer model in relating heat discharge to visible plume volume of a 12-cell mechanical draft cooling tower. In a previous study, Savannah River National Laboratory developed a 1D model to capture the average power plant discharge levels via analysis of a series of visual images but was unable to accurately predict individual cases, resulting in an overall average error of about $5\%$, but individual comparisons resulted in an $R^2$ of $0.36$. Three optimization algorithms were applied to better fit the entrainment coefficients, and the artificial neural network model was applied to 289 cases of a 12-cell mechanical draft cooling tower power generation facility. Two artificial neural networks configurations consisted of 10 and 47 nodes that used as input readily available plant data, observed cooling tower plume conditions, observed operational conditions, local and regional weather, and the predicted plume volume from the physical model; the individual predictions’ accuracy improved to $R^2>0.95$. This article concludes the sensitivities for the 1D model and additional actions to progress this field of study as well as applications for cooling tower monitoring. This strategy demonstrated an encouraging first step towards using multi-modal artificial neural network machine learning technology for information fusion to estimate power levels from external observations.

1. Introduction

Mechanical draft cooling towers (MDCT) transfer heat and water mass from industrial cooling water to the atmosphere via forced convection driven by fans [1]. The exhaust created by a MDCT is initially a forced jet transitioning to a buoyant plume higher up in the atmosphere. MDCTs operating near full capacity often produce a visible plume of water droplets that form when the saturated exhaust air mixes with cooler ambient air, for which MDCT thermal outputs can be evaluated [2]. Additional entrainment of drier ambient air eventually causes the droplets to evaporate. The exhaust plume continues to rise after all droplets have evaporated until entrainment and expansion in a stable atmosphere eliminate the plume’s buoyancy. The maximum height reached by the visible plume is a function of ambient weather, notably relative humidity, which produces a strong diurnal cycle in plume height [3]. It is assumed that when the visible plume in the digital image disappears, the droplets have completely evaporated.

The thermal energy output as well as weather and specific cooling tower features all affect the volume of the visible plume over the tower. Cizek and Noicka, for example, determined that plume volumes were based on cooling tower diameters as well as the temperature of the exhaust and the ambient air [4]. Furthermore, the amount of water consumption and heat transfer inside the cooling tower forced the evaporation and exhausted larger plume volumes from the facility. Takata et al. used computational fluid dynamics modeling to further investigate the plume volume, length, and width of the plume with and without a wind velocity [5]. The combination of weather conditions with plume dynamics inspired other studies on plume models exhausting out of a single 1D hybrid cooling tower using volume flow rate through a fan, which concluded that for plume abatement, hybridization is preferable to increase wet section airflow [6,7,8].

The purpose of the physics-based numerical model of an MDCT exhaust plume is to form a quantitative link between the enthalpy flux leaving the MDCT and the size of the visible plume of water vapor that condensed over it. Given that an MDCT exhaust plume is an axisymmetric turbulent jet, a steady-state 1D entraining, condensing/evaporating plume model can be selected as the right combination of sufficiently complete physics and rapid execution that allows thousands of simulations to be performed in a few seconds. For example, Fisenko et al. studied the mathematical model of MDCT performance by describing the change in droplet velocity, radii, temperature, and density, whereas Williamson et al. evaluated the heat and mass transfer inside a natural draft cooling tower to analyze nonuniformities [9,10]. Other authors analyzed different cooling towers, such as counter-flow cooling towers, simulated power output predictions and plume visibility at different wind speeds [8,11]. Zargar et al. focused on a 1D hybrid cooling tower model based on its thermal performances, plume abatement, water consumption, and fan power, in which the hybrid model of the cooling tower was desired to increase wet section airflow [12].

The main issue for 1D models of cooling tower plumes in ambient air is that there are no universal entrainment coefficients to solve the equations. One coefficient is the $\varphi$ coefficient, which is based on the velocity difference as the plume is exhausted vertically [13]. Some coefficient values that varied with Hewitt et al.’s coefficient value of $0.15$ included buoyancy driven forces characterized between $0.05$ and $0.12$ [13,14]. Another coefficient is the $\beta$ coefficient, which expressed models that increased entrainment experiences by a turbulent plume rising through an atmosphere with a horizontal wind. Depending on the research, the $\beta$ values varied, with $0.43$ by Slawson and Csanady, $0.71$ by Hewett, and a range 0.5–2.0 from Fay et. al and Hoult et al. [13,15,16,17,18].

Although 1D models are quick and effective tools, there are difficulties in establishing the power output of a specific cooling tower application since there are various coefficients involved in the dynamics of the plume, including lumped parameters that capture multiple dynamics. For this reason, various studies applied machine learning (ML) techniques and artificial neural networks (ANNs) that implemented inputs, outputs, nodes, and training/retraining processes for cooling towers [19,20,21]. Alimoradi et al. applied an ANN-particle swarm optimization algorithm and calculated the wet cooling tower efficiency [22]. Additional ANN particle swarm algorithm studies included a multi-cell-induced draft cooling tower at a coal-fired power station under considerable loads with only measurable values from simulated operations [23]. Other experimental applications predicted the performance of counter flow cooling towers [24,25]. For more industrial applications, Rossa and Borella ran ANNs for their spent fuel assembly studies [26]. Randiligama et al. focused on damage predictions using the feed-forward network with the Bayesian algorithm and Valadbeygi applied deep neural networks to investigate the thermal performance of cooling towers [27,28]. In terms of including weather to predict the performance of cooling towers, Gao et al. focused on cross-wind conditions [29].

Also, when a single cooling tower study was considered, the entrainment coefficients, $\varphi$ and $\beta$, were well known, whereas 12 cooling tower cells were more difficult to accurately simulate. The current study applied a 1D model and quantified 289 power outputs of 12 cooling tower cells at a specific location, based on the measured plume volumes, atmospheric data, and cooling tower operations that included thermal and latent (evaporate) energy discharged to the atmosphere. Afterwards, a few other algorithms, such as the real-Genetic Algorithm (RGA), Dynamic Dimensioned Search (DDS), and Gauss–Marquardt–Levenberg (GML), applied a range of the entrainment coefficients and calculated the mean relative error (MRE) and the root mean squared error (RMSE) using the OSTRICH software. The entrainment coefficients were used in the model, which was compared to the power output. The correlation coefficient remained low, which indicated there were missing components remaining in the simplified 1D model. This inspired the investigation of ML to assess whether a high-fidelity model could be developed. An application of the ANN modeling technique was investigated, with its fidelity validated by prediction comparison to unseen measured power output.

2. Image Analysis and Modelling Cooling Tower Plumes

In Sobecki et al.’s previous journal article, a simulation approach was applied as a 1D Gaussian plume model for a 12-cell MDCT located in the southeastern U.S. as shown in Figure 1 (early in the morning) [30]. Each MDCT cell has a fan with a diameter of $10.06$ m, and the site discharges 400–700 MW thermal exhaust from a natural gas power plant that generates electricity at a range of 500–900 MW. Given the atmospheric and cooling tower operating conditions, visible plume merger occurs as the individual saturated plume (in a parcel of air) is exhausted high up in the air as displayed in Figure 1. Flow through the cell is driven by a fan in the throat of the shroud. The fan sits at top a gearbox, driven by a motor outside the shroud. The cell below the fan contains spray nozzles, fill (where forced convection heat and mass transfer occur), a rain zone, and basin. The horizontal momentum of the wind over the tower causes the plumes to lean in the same direction because of the entrainment. The 1D case is based on a row of individual cooling cells, where the height above the fill was $5.0$ m and the combined area of the twelve cooling tower shrouds was $17.3$ m (with a cell radius of about $5.0$ m each), with a cell spacing of $6.5$ m. Coordinate positions for each tower were also included in the calculation, which then took into account the wind effect based on the direction difference with the azimuth of the row of MDCT relative to the north (maximum and minimum entrainment for a wind blowing perpendicular and parallel to the row of MDCT cells, respectively).

Plume volumes were measured using imagery from eight externally located, stationary calibrated camera stations. The cameras were deployed around the cooling tower as described by Sobecki et al. and Connal et al. [30,31,32]. The plume images were segmented using a neural network-based systems (i.e., Matterport Mask R-CNN [33]), which recognized the plume in each image using manually annotated training data. Afterwards, a space carving technique (which analyzed the sight lines from each camera angle) was used to determine which voxels (small cube boxes into which the domain was divided) were plume or not plume [30,31]. The final plume volume was estimated by counting voxels. Camera angles were originally determined using Structure from Motion on the solid structures in the scene. Using thousands of images, a total of 289 plumes were identified with volumes ranging from 5867 to 1,451,345 m³.

The 1D model and its fundamental equations are described in our prior work [30]. The model inputs included the following: measured plume volumes; meteorological data (taken from an average of 11 weather stations to reduce the average error); entrainment coefficients; height of the cooling tower above the file; radius of a single cell cooling tower; number of cooling tower cells; cooling tower clockwise orientation relative to the north; a bulk drag coefficient used to compute free convection velocities for cells with fans off; and cell spacing between cooling towers. Since 12 plumes have a larger initial “boundary” with the surrounding atmosphere, the entrainment of ambient air into the 12 cells is greater than the entrainment into the equivalent single cell, and the cells are arranged in a row. However, additional entrainment is a function of the wind direction relative to the orientation of the axis of the cell row. Therefore, shielding exists in the downwind cells for a wind direction parallel to the cell axis and is at its lowest value relative to the cell axis.

To summarize, the assumption of the 1D simulation is that the plume is a top-hat model, and the exhaust temperature is at the point of saturation. For the domain setup, the wind is applied at a constant speed and at a uniform direction in a flat terrain where there are no obstructions and forests. The rate at which the cooling tower transfers thermal energy to the atmosphere via the air inside the cooling tower was determined by Sobecki et al., which depended on the parameters of the process cooling water entering and exiting the MDCT [30]. Some variables could not be measured remotely, so the thermal energy transfer rate was estimated by equating it to the liquid heat transfer by the corresponding energy budget equation for the air passing through the cooling tower. It is assumed that the calculation was aside from small radiative and conductive losses as the air and water pass through the tower. Plume visibility was also assumed unless the maximum specific humidity was less than the saturated specific humidity.

For the exhausted and entraining plume model, the simulation ran using lower boundary conditions by applying the enthalpy flux, MDCT exhaust volumetric airflow, and the plume radius. The initial conditions for the plume model used the enthalpy transfer rate (power) as well as the parameters of the air. The exhaust air temperature was increased until a simulated plume size equaled the measured plume from the imaging analysis [30,31].

The standard set of the 1D entraining plume model equations was also taken from Carazzo et al., which included the evaporation and condensation of the plume calculated from the parameters of the atmosphere and the plumes. The differences in static air pressure density with height were accounted for by adding $9.73$ °C/km to the lapse rate used in the equation that was solved from the bottom (starting with the boundary conditions) to the top of the MDCT. As the saturated plume exits the cooling tower, the water vapor initially condenses into minute droplets, and fall velocities carried upward where some liquid water is in the cooling tower updraft, before reaching the top of the stack. For a realistic simulation, the plume must include droplet condensation/evaporation because a plume with condensed water vapor has higher temperature and buoyancy than a plume without condensation (unless the atmosphere is saturated).

A modified entrainment function modeled the ambient air for conditions where there was wind from the dataset [13], and included the entrainment coefficients, $\varphi$ (plume buoyancy) and $\beta$ (plume spreading), and the elevation angle of the plume axis relative to a horizontal surface [30]:

$$a_{em}=R(a_1\varphi |\sqrt{v_{pl}^2+w^2}-v_{air}cos\left(\theta \right)|+b_1\beta |v_{air}sin\left(\theta \right)\left|\right),$$

(1)

where R is the plume radius, $a_1$ is the entrainment for a row of plumes from a cell, $v_{pl}$ is the horizontal component of the plume air velocity, w is the updraft velocity, and $v_{air}$ is the horizontal component of the air velocity at angle $\theta$. The above equation is related to the volumetric flow rate of the plume and therefore the plume size under a specific power output. The first term of the equation is the shear velocity parallel to the plume axis, whereas the second term is the velocity difference perpendicular to the plume axis, both of which are important for determining the power output.

3. Simulated Power vs. Actual Power Results

In a previous study, for 289 case studies, the 1D model’s predicted average power compared well with the observed power output. The percent error decreased to less than $5\%$ after running more than 50 pairs [30]. This statistical estimate dictated that more information is desired for an accurate model, which occurred as the number of cases increased. Conversely, the correlation coefficient ($R^2$) is a linear relationship between individual cases of measured versus model predicted values and should estimate the model’s accuracy if the data collected are well validated. As seen in Figure 2, when comparing the simulated power output to the observed power output, $R^2=0.3616$ with an equation line of $y=0.3025x+277.77$. The error between the simulated power output and the actual power output ranged between $0.0115\%$ and $118.116\%$, which shows that improvement to the analysis is needed if individual cases are studied.

The individual predictions in the scatter plot are spread out and do not lie close to the best fit line. It was hypothesized this was largely due to the uncertainties of the meteorology variability or the empirical values in the entrainment coefficients. An attempt was made to improve the physics-based models by optimizing certain input parameters. This effort employed two global and one local optimization algorithms on the entrainment coefficients. If the automated calibration technique was successful, then the values used for the entrainment coefficients were the primary cause of the low fidelity ($R^2=0.36$) modeling results, and further investigations on AI/ML methods would not be necessary.

4. Optimization Algorithms

The optimization tools in this study were applied by the OSTRICH optimization tool developed by Matott [34]. The purpose of OSTRICH was to implement numerous model-independent optimization and calibration (parameter estimation) algorithms in various applications such as storm water management models [35]. Using tools from multiple coding programs (e.g., C, C++, Fortran, Python, Matlab, etc.), the OSTRICH optimizer allowed the user to develop a script based on the choice of the optimization tool, input and extra files, parameters, etc. Depending on the number of parameters, design of the experiment, and the files applied, OSTRICH calibrated and optimized the study for the best fit result. Although numerous algorithms were considered in this study, the RGA, DDS, and GML, in that order, were chosen to calibrate the entrainment coefficients for a 1D model as their solution techniques vary from each other. As discussed below, the current study focused on a fluid dynamics model. Therefore, RGA and DDS were applied because of their applications in fluid studies, and GML was chosen as a method to further fine tune one of the optimization algorithms that is closest to the solution, i.e., the least collective error compared to the actual power output.

The RGA tool is a simple and straightforward global search algorithm used by Yoon and Shoemaker for two case studies of groundwater bio-remediation [36]. The RGA tool processed natural evolution via selection, recombination, mutation, and replacement, where the higher probability of the offspring(s) survived and computed the next generation with the mean fitness improving. The fittest solution survived to the next generation where a chromosome represented a potential search point and a set of decision variables that consequentially became one real-valued gene in RGA. In the first step of the RGA application, an initial random population with X individuals was generated in a vector along with a generation index. Next, a probability for the explorative search was executed followed by a randomly picked generation of ${\gamma }^i$. If ${\gamma }^i$ was less than the probability, then two parents were automatically selected; otherwise, they went through a tournament selection. The next generation then either went through a recombination or a mutation step. In the recombination step, an offspring was generated by directive recombination. Conversely, in the mutation step, an offspring was generated by mutating via a breeder genetic algorithm continuous mutation. After either step, a screened replacement was applied for a new offspring and repeated in the second step. Finally, if the generation was less than or equal to the maximum number of generations, the next iteration was followed and went to the second step; otherwise, the computation was terminated.

In Yoon’s and Shoemaker’s studies, the RGA with directive recombination outperformed the standard binary-coded GA (BIGA) because RGA solved nonlinear problems such as interactions between degrading microbes, oxygen, and contaminant concentrations. The authors also concluded that RGA found better solutions in less computational time than BIGA and can be applied in other water resources applications such as MDCT plumes.

The DDS tool was developed by Tolson and Shoemaker, where they observed computationally expensive optimization problems such as the watershed model calibration [37]. The DDS tool was applied to problems with many parameters (without algorithmic tuning) and scaled searches to find solutions within the maximum number of user-specified functions. The DDS inputs established the neighborhood perturbation size parameter, maximum number of function evaluations, and lower and upper bounds for the decision variables, as well as initial guess of the solution. Second, the initial iteration evaluated the best objective function and next best solution. Third, the decision variables were randomly selected inside the neighborhood to calculate the probability for each decision variable as a function of the iteration count where the DDS algorithm looped the decision variables to add to the probability if the neighborhood was not empty; otherwise, a decision variable was randomly selected. Fourth, another loop was executed where the decision variable in the neighborhood selected the best solution under a standard normal random variable established by the boundaries. Fifth, the objective function was updated to the best solution under certain conditions. Finally, the iteration count was updated, and the optimization tool completed if a criterion was met.

The GML tool is a local search algorithm developed by Levenberg and Marquardt that solved nonlinear least square problems [38,39]. The GML tool used the Gauss–Newton and steepest (gradient) descent approach to acquire the best of the two solutions in each case, where the Newton method finds the minimum or stationary point of the function. The tool then updated values from point-to-point; however, the solution improved when the starting point was closer to the goal point. Thus, knowledge of the problem was crucial for a converged solution and avoiding recirculation traps. This tool arose in the minimization problems in a least squares curve fitting (i.e., solved nonlinear least squares problems), but only for the local minimum.

These algorithms were implemented as an attempt to improve the power output prediction by finding the ($\varphi$) and ($\beta$) coefficients. The range of the $\varphi$ value Carrizo et al. used was between 0.05 and 0.15, whereas $\varphi =0.15$ was used for Hewitt et al. The upper end value represented calm winds (vertical plumes) and the $\varphi$ value was best represented by a forced jet, not a buoyant plume [13,14]. The value of $\varphi =0.15$ represented calm winds and the upper end of about $0.15$ was a large value considering that $\varphi$ was parallel to the plume and gave a false power output estimation because the calculation ended too early. Conversely, if the value of $\alpha$ was too small, the simulated plume rose rapidly to the top boundary of the simulation (1000 m), which was not considered physically feasible. Conversely, $\beta$ coefficients varied between different authors, with $0.71$ from Hewett et al., a range of $0.3\le \beta \le 1.0$ from Hoult and Fay et al., and $0.5\le \beta \le 2.0$ from Zhang et al. [13,15,17,18]. Given the previous studies, the ranges for the optimization tools were $0.07\le \varphi \le 0.13$ and $0.3\le \beta \le 1.0$, respectively. For the RGA, DDS, and GML, the power observation was set to $5\%$ of the actual power output to find the best $\varphi$ and $\beta$ values.

5. Entrainment Optimization Results

The criteria to measure the network performance relied on $R^2$ values, mean relative errors (MRE), and root mean square errors (RMSE). The measured outcome provided the verification of the assumptions and lists made. To accomplish this, the coefficients were first set to a specified range, an optimization run initialized, and the predicted power output examined against all 289. The optimization algorithm continued until the MRE of the network was minimized. The MRE estimates the percentages between the error and the experimental values:

$$MRE(\%)={\displaystyle \frac{1}{N}}\sum _{i=1}^N|100{\displaystyle \frac{V_{pc}-V_{pa}}{V_{pa}}}|,$$

(2)

where N is the number of points in the given dataset and $V_{pc}$ and $V_{pa}$ are power outputs based on the computation and measurements, respectively. The MRE was followed by the RMSE:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{(V_{pc}-V_{pa})}^2}.$$

(3)

In the current study, a total of 289 simulations were run on the weather and operating data on days when the plume images were collected, and their volumes were measured. The inputs varied between wind speeds, pressure, air and dew point temperatures, exhaust temperature, plume volume, and number of off/on fans (

Table 1. Range of inputs for the 1D simulation.

Parameter Inputs	Ranges
Ambient air temperature	1.61–25.40 °C
Dew point temperature	−4.04–22.77 °C
Wind speed	0.03–4.78 m/s
Pressure	1013.14–1026.96 mb
Exhaust temperature	31.2–40.0 °C
Plume volume	5867–1,451,345 m3
Number of fans “on”	4–12

). The only constant values were the exhaust velocity and the average temperature gradient at $5.8$ m/s and $-6.5$ °C/km, respectively. By using the provided measurements, the power output and entrainment coefficients were evaluated from the equations in the 1D model.

Each optimization method had 1000 runs for a total of $867,000$ cases. For both global optimization methods, the initial $\varphi$ and $\beta$ values for RGA and DDS were $0.085$ and $0.71$, respectively, as set by the 1D model developer. After the DDS algorithm ran, the GML method attempted to fine tune the entrainment coefficients. The entrainment coefficient results and the errors after 1000 runs are shown in

Table 2. Error results from optimization of the entrainment coefficient values and a comparison to SME results.

Algorithm	$\mathit{\varphi }$	$\mathit{\beta }$	RMSE (Minimum)	MRE (Minimum)
SME	$0.085$	$0.71$	$138.0526$	$28.3$
RGA	$0.0894$	$0.7707$	$138.7691$	$28.0288$
DDS	$0.0926$	$0.7367$	$138.7659$	$28.0$
GML (after DDS)	$0.0926$	$0.7366$	$138.7459$	$28.0$

. The entrainment coefficient values not only differed negligibly, but the GML method also did not tune the coefficients, showing that the DDS algorithm exhausted its search. Additionally, the error analysis did not improve, as shown in Figure 3. Compared to the SME coefficient values, some of the methods were able to improve on an individual basis, but the overall physics modeling results were not appreciably improved.

Further analysis (Figure 3) shows a graphical outcome on an individual basis for the DDS algorithm based on the final entrainment values. There was still over/underestimation between cases with some of the cases improving under the DDS/GML method (since they have the same coefficient values), but there was no appreciable improvement. It is concluded that the entrainment coefficients were not the main uncertainties in the 1D model, and therefore, a multi-modal ML method was investigated. While the entrainment coefficients are controlled, the weather and operational data cannot be controlled, and the model sensitivities can over- or underestimate the power output (as observed between observations 25 and 200).

Although all three algorithms have previously been applied in computational fluid dynamic approaches, it remained to be seen if they came to different conclusions; instead, they all came to the same or similar credible estimate of the optimal solutions. The RMSE and SME values in Table 2 did not significantly change along with the $\varphi$ and $\beta$ values, meaning that the weather data and the physics limitations of the 1D model play another important role on the simulation power output. As reported by the current authors in prior work, an average dewpoint temperature error of $1$ °C increases the average error by $17\%$, in which the power output estimated could be grossly over- or underestimated [30]. This is why the weather data were averaged from 11 regional weather stations such that the cooling tower site was centered as much as possible between sites and an analysis on individual and a combination of stations was studied [30]. In regards to the physics model, the 1D model lacks some of the swirling motions and eddies that 3D simulations could perform.

6. Multi-Modal Machine Learning

The ANN method is an ML tool that mimics human brain functions and behaviors in a computerized training mechanism [40]. This method was chosen for its ability to solve a wide array of problem types. By inputting the observed and computed variables into the analysis, the ANN technique adjusts its weights and iterates to find the best predicted output through a training–retraining process. Results are then validated on unseen (held out) data. In the current study, the weather data, plume volumes, cooling tower operations, and power attributes are shown in

Table 3. List of input/output attributes and their sources.

Input/Output Attribute	How the Information Was Obtained
Observed volume	Ground-based images, Mask R-CNN, space carving, cube voxels [30,31]
Simulated power	1D cooling tower model [30]
Taout deg K	Given by cooling tower facility [30]
Pressure	Averaged from NOAA-DOD and regional airport stations [30]
Wind speed	Averaged from NOAA-DOD and regional airport stations [30]
Wind direction	Averaged from NOAA-DOD and regional airport stations [30]
Exhaust velocity	Calculated on average from cooling tower facility [30]
Dewpoint temperature	Averaged from NOAA-DOD and regional airport stations [30]
Temperature gradient	NOAA standard atmospheric lapse rate [30]
Total number of cells on	Given by cooling tower facility [30]
Cell 1 on	Given by cooling tower facility [30]
Cell 2 on	Given by cooling tower facility [30]
Cell 3 on	Given by cooling tower facility [30]
Cell 4 on	Given by cooling tower facility [30]
Cell 5 on	Given by cooling tower facility [30]
Cell 6 on	Given by cooling tower facility [30]
Cell 7 on	Given by cooling tower facility [30]
Cell 8 on	Given by cooling tower facility [30]
Cell 9 on	Given by cooling tower facility [30]
Cell 10 on	Given by cooling tower facility [30]
Cell 11 on	Given by cooling tower facility [30]
Cell 12 on	Given by cooling tower facility [30]
Observed power (output attribute)	Given by cooling tower facility [30]

.

Neural networks can approximate large families of functions that depend on input variables. The investigation focuses on a bracket with approximately twice to half the number of inputs. The ANNs were developed, trained, and validated using the inputs presented in Table 3. JMPro software version 16.2.0 was chosen as the tool to implement the ANNs [41]. JMPro is a statistical software designed for data science, tool development, and implication for data preparation, analysis, and graphing. One of the successful ANN configurations is shown in Figure 4.

To assure that no overfitting was produced and that the model was ANN configuration independent, two approaches for model development were executed. The first two configurations of the ANN model, consisting of 10 and 47 nodes, respectively, were trained with 2/3rd of the data followed by a test with 1/3rd of the data to see if $R^2>0.95$. Afterwards, the 10-node configuration was developed using the 5 cross-fold validation method. For the first set of tests, the training and testing $R^2$ values were $0.987$ and $0.986$, respectively, for 10 nodes. For 47 nodes, the $R^2$ values were $0.998$ and $0.984$, respectively. Applying the 5 cross-fold validation resulted in $R^2$ values of $0.985$ and $0.969$, respectively. The results of the ANN using 10 nodes and a 5 cross-fold validation are presented in Figure 5.

These results indicate that using a multi-modal approach as configured in this study is successful for this case, as the individual case fitness was increased from $R^2=0.36$ using a physically based modeling approach to $R^2>0.95$ for the case study examined.

Successfully using two ANN configurations, followed by a 5 cross-fold validation, supports the generality and reusability of the technique. These results pave the way for further investigation into generalization to a variety of site cases. However, in order to be demonstrated as applicable to other sites, the multi-modal approach would need to be further tested, deployed, and validated with the different cooling tower site(s) and/or all the data would need to be combined to assess if a generalized model is obtainable. In addition, when looking at the variety of other ANN studies, some studies primarily focused on operating performances based on the internal operations of the cooling tower and the ambient conditions applied in the cooling tower performance. The ambient temperature (including wet/dry bulb temperatures) was important to study cooling towers in terms of thermal performances under deep neural network (more than one hidden layer) [28] and cross-wind conditions [29].

Another case was Blackburn et al.’s study, which closely resembles the current investigation since it focused on a 12-cell MDCT, numerical simulations based on operations, and optimization methods before applying ANNs. In their study, they simulated cooling tower operations by assessing water droplets for cooling tower performances under 12 operating fans in addition to ambient and cooling tower flow rate operations [23]. The power output from the 1D model was a steady-state case to capture the simulated and actual power output estimated from the time that the plume images were taken to calculate the volume. The steady-state case best represents the time the plume volume was recreated based on the simulated power output as well as the weather and operational data collected at that specific time. In the current study, however, this is the first application to focus on field-study validations by combining operational, weather, and plume volume (from ground-based images) data in an attempt to improve simulated operations with actual operations. With Blackburn et al., their application was the motivator and helped validate our work by providing similar levels of accuracy.

Ultimately, the neural network approach was used to assess if the information obtained from the data collection was sufficient to capture the predictability of the power estimates. The high $R^2$ indicated that the information content was sufficient to identify that the accuracy of the model could be improved through physics and weather data optimization (e.g., the dewpoint temperature). For the physics model application, an additional investigation would need to be conducted on what knowledge was gained by the ANNs. The next steps are to determine what the neural network uncovered to produce these high accuracy results, and assess the knowledge gained in the physics approach. Future studies may eventually determine that physics plus AI/ML may be the best approach.

7. Conclusions

Multi-modal ML techniques advanced the accuracy of predicting power generation levels from the previous study. Whereas the previous study produced an individual case prediction accuracy of $R^2=0.36$, this multi-modal method produced multiple $R^2$ results greater than $0.95$. Initially, the thought was that the entrainment coefficients needed improvement to increase $R^2$. After running approximately one million calculations with two global (RGA and DDS) optimization algorithms and one local (GLM) optimization algorithm, the coefficient value did not significantly improve. The insignificant change in correlation motivated further study of AI methods to increase the correlation and help to generate further studies to find the missing physics or optimize the weather data on an individual basis. Focusing on the ANN, two cases provided similar accuracy when there was several more or less than half than the number of inputs. Note that this study used the same data available during the previous study leveraging the information content more fully to create better predictive ability [30]. This multi-model analysis has a demonstrated ability to increase the accuracy, and hence, applicability, of this remote sensing approach for estimating power source generation using externally observable information. One question that remains, however, is if the accuracy could be improved with smaller or larger datasets. In this research, 289 cases were studied using an ML algorithm. Various applications could have a smaller number of examples available.

For the physics model, there were cases in which the simulated power overestimated or underestimated the actual power output because of the physical limitations of the 1D model and the sensitivity that the dewpoint temperature has on the error of the model. As previously mentioned, an average dewpoint temperature error of 1 °C increases the average error by $17\%$, where the weather data are a significant contributor to the physics model [30]. The results between observations 75 and 200 show that the data cannot always be controlled.

To further improve the ANN and 1D plume model integrated approach, the single site can be used as an experimentation test bed. For example, it can be used to determine the necessary (i.e., minimum) number of cases to be collected to generate reliable results. If a site has a large range of data, how many data points are necessary for the multi-modal study? These results could also assist with efficient equation development for physics informed ML, making the physical more complete and comprehensive. An additional case study using this dataset could be applied in design of experiments to explain the influence between the atmospheric conditions, operational parameters, and plume volume in order to identify and explain the control variables.

Further action can progress this field of study and improve the current work for applications in cooling tower monitoring:

• Determine what the ANN uncovered to produce these high accuracy results and assess the knowledge gained in the physics approach.
• Investigate datasets greater than 289 cases at the same site to anticipate how much data are sufficient and if different ANN structures are needed.
• Investigate datasets less than 289 cases at the same site and structure the ANN based on methods such as bootstrapping.
• Improve the dewpoint temperature calculation based on the weighted contribution on the individual weather stations as opposed to an equally weighted average.
• Apply white box methods, e.g., decision trees, integrated gradients, etc., to study the consequences of the actual power output.
• Assess other cooling tower scenarios in different weather environments and adapt the 1D model to see if the ANN structure is site specific.