# Development of Neural Network Prediction Models for the Energy Producibility of a Parabolic Dish: A Comparison with the Analytical Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Novelties of This Study

- The numerical models investigated are based on a collection of experimental data obtained from the real operation of a prototype dish–Stirling solar concentrator installed at the campus of the University of Palermo. The direct availability of such data, in the case of the aforementioned technology, is not common, and several previous studies were exclusively theoretical, such as [29,30,31,32]. Furthermore, in the case of the application of artificial intelligence techniques, sometimes the data used are mainly obtained from other analytical or numerical models and not from experimental measurement campaigns [24];
- One of the most important characteristics of the following research is that the tools that are used to develop the proposed models are explicitly stated and belong to the category of open-source software (Python and TensorFlow), ensuring the absolute replicability of the algorithms by the scientific community. This feature is not particularly common in the previous literature; even if tools are declared, it is still not possible to faithfully reproduce the models as they lack a multitude of details typical of proprietary software, as is the case in [27,28,33];
- Finally, a further innovative element concerning other models already available consists of the use of an input parameter representing the level of cleanliness of the mirrors. This parameter has been shown by the authors to be among the most influential for the energy production of the system [18].

## 3. Experimental Set-Up

_{e}and features a geometric concentration ratio equal to 3217 (see Table 1). The reference system has a paraboloidal collector consisting of an assembly of 54 mirrors with a high reflection coefficient; each mirror is characterised by a sandwich structure and a double curvature calibrated in order to concentrate the incident DNI on a fixed point corresponding to the small aperture of the cavity receiver. Subsequently, the Stirling engine and the electric generator convert the thermal energy into mechanical power and then electricity [34]. The power conversion unit (see zoom in Figure 1), including the receiver, the Stirling engine, and the electric generator, is placed at the focal point of the paraboloidal collector by a tripod.

#### 3.1. Description of the Experimental Dataset

_{p}) illustrated in Figure 5 shows that the net electrical power output of the dish–Stirling system is strongly correlated with the DNI.

#### 3.2. Outlier Removal Procedure

#### 3.3. Statistical Analysis of Input Datasets

## 4. Energy Modelling of the Dish–Stirling Concentrator

#### Energy and Heat Balance Equations

- ${A}_{r}$ is the aperture area of the cavity receiver (m²);
- ${h}_{r}$ is the convective heat transfer coefficient of the receiver (W/(m²∙K));
- ${T}_{r}^{ave}$ is the average value of the receiver temperature (°C);
- ${T}_{air}$ is the temperature of the external air (°C);
- ${\sigma}_{SB}$ is the Stefan–Boltzmann constant equal to 5.67∙108 W/(m²∙K4);
- ${\epsilon}_{r}$ is the emissivity of the cavity receiver (-);
- ${T}_{sky}$ is the sky’s apparent temperature calculated using the empirical formula [40] of Equation (5):$${T}_{sky}=0.0552\cdot {\left({T}_{air}+273.15\right)}^{1.5}-273.15\phantom{\rule{1em}{0ex}}\left[\xb0\mathrm{C}\right]$$

- ${a}_{1}$ (-) and ${a}_{2}$ (W) are two fitting parameters of the mechanical efficiency curve of the Stirling engine;
- ${R}_{T}$ is a dimensionless correction factor of the ambient air temperature (${T}_{air}$) for the reference temperature (${T}_{0}$ set equal to 25 °C) (both expressed in °C) defined as:$${R}_{T}=\frac{{T}_{0}+273.15}{{T}_{air}+273.15}$$

- ${I}_{b}$ is the DNI arriving on the mirrors (W/m²);
- ${T}_{air}$ is the ambient air temperature (°C);
- ${\eta}_{cle}^{ave}$ is the average level of cleanliness of the mirrors, ranging between 0 and 1 (-);
- ${\eta}_{e}$ is the mechanical-to-electric conversion efficiency of the electric generator (-);
- ${\eta}_{o}$ is the optical efficiency of the solar concentrator (-);
- ${A}_{n}$ is the net aperture area of the paraboloidal collector (m²);
- ${\dot{Q}}_{r,out}$ is the thermal power loss at the cavity receiver (W);
- ${\dot{E}}_{p}^{ave}$ is the average value of electric power consumed by parasitic equipment, such as the tracking system and dry cooler of the cooling system (W).

## 5. Artificial Neural Network Models

#### 5.1. Machine Learning Deployment Using TensorFlow and Python

#### 5.2. Artificial Neural Networks

#### 5.2.1. Multilayer Perceptron Neural Network

#### 5.2.2. Radial Basis Function

#### 5.3. Development of Neural Network Models

#### 5.4. Description of Supplementary Materials

#### 5.5. Definition of Performance Measures

_{1}), at 50% (second quartile, Q

_{2}), and 75% (third quartile, Q

_{3}). In order to graphically compare all the developed neural networks in terms of the accuracy of predicting the energy production of the dish–Stirling plant, the following graphs were produced for each of them:

## 6. Results and Discussion

#### 6.1. Performance of Neural Network Models

#### 6.2. Comparison with an Analytical Model

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a_{1} | first fitting parameter of the mechanical efficiency curve of the engine (-) |

a_{2} | second fitting parameter of the mechanical efficiency curve of the engine (W) |

a_{ik} | weight of the first layer in a MLP neural network |

A_{n} | net aperture area of the dish collector (m²) |

A_{r} | aperture area of the receiver (m²) |

b_{i} | bias value |

C | size of the validation dataset |

e | mean value of residuals |

e_{i} | i-th value of residuals |

${\dot{E}}_{n}$ | net electrical power produced by the dish–Stirling system (W) |

${\dot{E}}_{p}^{ave}$ | average value of electric power consumed by parasitic equipment (W) |

g | Gaussian function |

h_{r} | convective heat transfer coefficient of the receiver (W/(m²∙K)) |

I_{b} | DNI arriving on the mirrors (W/m²) |

m_{i} | i-th of the measured values |

N | number of records in the dataset |

Q_{1} | first quartile of the frequency distribution of residuals |

Q_{2} | second quartile of the frequency distribution of residuals |

Q_{3} | third quartile of the frequency distribution of residuals |

${\dot{Q}}_{r,\hspace{0.17em}out}$ | thermal power lost at the receiver (W) |

${\dot{Q}}_{S,in}$ | thermal input power to the Stirling engine (W) |

R² | coefficient of determination |

R_{T} | correction factor of the ambient air temperature (-) |

T_{0} | reference temperature of the external air ( °C] |

T_{air} | temperature of the external air (°C) |

T_{r} | temperature of the receiver (°C) |

T_{r}^{ave} | average value of the receiver temperature (°C) |

T_{sky} | sky apparent temperature (°C) |

w_{i} | weight of the output layer |

${\dot{W}}_{S}$ | mechanical output energy at the engine crankshaft (W) |

$\overline{x}$ | mean of the data |

x | vector of input data to the neural network |

x_{i} | vector of parameters of each neuron of a hidden layer |

x_{i} | i-th record in the dataset |

y | output signal from the neural network |

Greek letters | |

α | absorption coefficient of the cavity receiver (-) |

ε_{r} | emissivity of the cavity receiver (-) |

ϕ_{i}(x)
| output signal of each neuron of the hidden layer |

γ | interception factor of the concentrator (-) |

η_{cle} | index of cleanness of mirrors (-) |

${\eta}_{cle}^{ave}$ | average level of cleanliness of the mirrors (-) |

η_{e} | mechanical-to-electric conversion efficiency of the electric generator (-) |

η_{o} | optical efficiency of the concentrator (-) |

μ | mean of the values of residuals |

ρ | reflectivity of clean mirror (-) |

ρ_{p} | Pearson correlation coefficient (-) |

σ | standard deviation |

σ_{SB} | Stefan–Boltzmann constant (W/(m²∙K^{4})) |

ζ | non-linear function |

Acronyms | |

ANFIS | Adaptive Neuro-Fuzzy Inference System |

ANN | Artificial Neural Network |

CPV | Concentrating Photovoltaics |

CSP | Concentrating Solar Power |

DNI | Direct Normal Irradiance |

GA | Genetic Algorithm |

MAE | Mean Absolute Error |

MLP | Multiple-Layer Perceptron |

PCU | Power Conversion Unit |

PSO | Particle Swarm Optimisation |

PTSTPP | Parabolic Trough Solar Plant |

PV | Photovoltaic |

RBF | Radial Basis Function |

TMY | Typical Meteorological Year |

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**Figure 2.**Dynamic viscosity for hydrogen, helium, and air versus temperature with a pressure value of 20 MPa.

**Figure 5.**Correlation matrix of variables of the original dataset without outliers (Pearson correlation coefficient).

**Figure 8.**General architectures of (

**a**) multilayer perceptron and (

**b**) radial basis function neural networks.

**Figure 9.**Histogram of residuals showing the probability density distribution of residuals resulting from the validation process of V-MLP-2.

**Figure 11.**A predicted versus measured power output (W) resulting from the validation process of V-MLP-2.

**Figure 12.**Histogram of residuals showing the probability density distribution of residuals resulting from the validation process of V-MLP-12.

**Figure 15.**Performance comparison between an analytical and the best neural network model, V-MLP-12 [18].

Parameter | Value | Unit |
---|---|---|

Paraboloidal reflector | ||

Net aperture area of the dish collector (${A}_{n}$) | 106 | m² |

Aperture area of the receiver (${A}_{r}$) | 0.0314 | m² |

Focal length | 7.45 | m |

Geometric concentration ratio | 3217 | - |

Reflectivity of clean mirrors ($\rho $) | 0.95 | - |

Power conversion unit | ||

Peak electric output (DNI equal to 960 W/m²) | 31.5 @ 2300 rpm | kW_{e} |

Type of Stirling engine | 4 cylinders double acting | |

Displaced volume | 4 (95 × 10^{−6}) | m³ |

Max operating pressure of hydrogen | 20 | MPa |

Temperature of the receiver (${T}_{r}$) | 720 | °C |

Parameter | Description | Unit |
---|---|---|

Direct normal irradiance | Direct normal solar radiation incident per unit area by the reflector | W·m^{−2} |

Global horizontal irradiance | Global solar radiation incident per unit area by the reflector | W·m^{−2} |

Diffuse horizontal irradiance | Diffuse solar radiation incident per unit area by the reflector | W·m^{−2} |

Ambient temperature | Outdoor air temperature | °C |

Average wind speed | Average wind speed on site | m·s^{−1} |

Wind speed | Wind speed on site | m·s^{−1} |

Wind direction | Wind direction on site | degree |

Humidity | Relative humidity of external air | % |

Air pressure | Outdoor air pressure | mbar |

Solar azimuth | Instantaneous position of the sun relative to the south direction | degree |

Solar elevation | Instantaneous position of the sun relative to the horizontal plane | degree |

Total CSP net power output | Instantaneous power output of CSP less parasitic consumption | W |

Statistical Quantity | Description | Formula |
---|---|---|

Arithmetic mean | the sum of a set of values divided by the number of values in the set | Equation (2) |

Variance | measures how much a set of values quadratically deviates from its arithmetic mean | ${\sigma}^{2}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}(}{x}_{i}-\overline{x}{)}^{2}$ |

Standard deviation | a measure of how much a set of values deviates from its arithmetic mean | Equation (2) |

Standard error | a measure of how much the sample statistic (i.e., sample mean) deviates from the actual population mean | $se=\frac{\sigma}{\sqrt{N}}$ |

Skewness | a measure of the asymmetry of the probability distribution of the data | $\frac{N}{(N-1)(N-2)}\frac{1}{{\sigma}^{3}}{\displaystyle \sum _{i=1}^{N}(}{x}_{i}-\overline{x}{)}^{3}$ |

Kurtosis | a measure of the thickness of tails or the flattening of a probability distribution | $\frac{(N+1)N}{(N-1)(N-2)(N-3)}\frac{{\displaystyle \sum _{i=1}^{N}(}{x}_{i}-\overline{x}{)}^{4}}{{\sigma}^{4}}-3\hspace{0.17em}\frac{{(N-1)}^{2}}{(N-2)(N-3)}$ |

Variable | Max Value | Arithmetic Mean | Variance | Standard Deviation | Standard Error | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|

Clean day | 131 | 46.61 | 1533 | 39.15 | 0.45 | 0.74 | −0.95 |

Direct normal irradiance (W/m²) | 957.17 | 774.63 | 7399.4 | 86.02 | 0.99 | −0.14 | −0.75 |

Global horizontal irradiance (W/m²) | 1118 | 765.95 | 27869 | 166.44 | 1.94 | −0.73 | −0.49 |

Diffuse horizontal irradiance (W/m²) | 512.2 | 157.06 | 4022.8 | 63.43 | 0.73 | 1 | 1.85 |

Ambient temperature (°C) | 30.46 | 21.77 | 18.93 | 4.35 | 0.05 | −0.26 | −1.12 |

Average wind speed (m/s) | 10.29 | 2.87 | 1.76 | 1.33 | 0.01 | 1.12 | 2.94 |

Wind speed (m/s) | 11.43 | 3.13 | 2.08 | 1.44 | 0.02 | 1.23 | 3.38 |

Wind direction (deg) | 340.65 | 145.32 | 5576.3 | 74.67 | 0.87 | 0.94 | −0.59 |

Humidity (%) | 71 | 52.15 | 90.45 | 9.51 | 0.11 | −0.19 | −1.06 |

Air pressure (hPa) | 1026.1 | 1006.7 | 34.85 | 5.90 | 0.06 | 0.57 | 1.87 |

Solar azimuth (deg) | 265.05 | 167.99 | 2512.3 | 50.12 | 0.58 | 0.17 | −1.20 |

Solar elevation (deg) | 75.42 | 54.04 | 217.07 | 14.73 | 0.17 | −0.35 | −1.05 |

Total CSP net power output (W) | 25531 | 19516 | 9.65 × 10^{6} | 3107.8 | 36.08 | −0.36 | −0.63 |

**Table 5.**Main parameters used as input to the analytical model of the dish–Stirling system of Palermo.

Parameter | Value | Unit |
---|---|---|

Net aperture area of the collector (${A}_{n}$) | 106 | m² |

Aperture area of the cavity receiver (${A}_{r}$) | 0.0314 | m² |

Convective heat transfer coefficient of the receiver (${h}_{r}$) | 10 | W/(m²∙K) |

Emissivity of the cavity receiver (${\mathsf{\epsilon}}_{\mathrm{r}}$) | 0.88 | - |

${a}_{1}$ parameter | 0.475 | - |

${a}_{2}$ parameter | 3319 | W |

Average receiver temperature (${\mathrm{T}}_{\mathrm{r}}^{\mathrm{a}\mathrm{v}\mathrm{e}}$) | 720 | °C |

Average level of cleanliness of the mirrors (${\eta}_{cle}^{ave}$) | 0.85 | - |

Electric efficiency of the PCU (${\eta}_{e}$) | 0.924 | - |

Clean mirrors’ optical efficiency (${\eta}_{o}$) | 0.85 | - |

Average electric power consumption (${\dot{E}}_{p}^{ave}$) | 1600 | W |

**Table 6.**Input and output variables of datasets implemented in both MLP and RBF neural network models.

Long Dataset | Short Dataset |
---|---|

Input variables | |

Direct normal irradiance | Direct normal irradiance |

Ambient temperature | Ambient temperature |

Clean day | |

Global horizontal irradiance | |

Diffuse horizontal irradiance | |

Average wind speed | |

Wind speed | |

Wind direction | |

Humidity | |

Air pressure | |

Solar azimuth | |

Solar elevation | |

Output variables | |

Total CSP net power output | Total CSP net power output |

**Table 7.**Input and output variables of datasets implemented in both MLP and RBF neural network models.

ANN Code | Number of Layers | Neurons | Trained Parameters |
---|---|---|---|

S-MLP-2 | 4 | 2 + 20 + 5 + 1 | 181 |

S-MLP-12 | 4 | 12 + 50 + 10 + 1 | 1351 |

S-RBF-2 | 4 | 2 + 20 + 5 + 1 | 181 |

S-RBF-12 | 4 | 12 + 50 + 10 + 1 | 1351 |

M-MLP-2 | 4 | 2 + 40 + 20 + 1 | 971 |

M-MLP-12 | 4 | 12 + 150 + 30 + 1 | 6691 |

M-RBF-2 | 4 | 2 + 40 + 20 + 1 | 971 |

M-RBF-12 | 4 | 12 + 150 + 30 + 1 | 6691 |

D-MLP-2 | 5 | 2 + 140 + 300 + 80 + 1 | 66,891 |

D-MLP-12 | 5 | 12 + 140 + 300 + 80 + 1 | 68,461 |

D-RBF-2 | 5 | 2 + 140 + 300 + 80 + 1 | 66,891 |

D-RBF-12 | 5 | 12 + 140 + 300 + 80 + 1 | 68,461 |

V-MLP-2 | 8 | 2 + 130 + 200 + 400 + 700 + 100 + 50 + 1 | 462,897 |

V-MLP-12 | 8 | 12 + 130 + 200 + 400 + 700 + 100 + 50 + 1 | 464,371 |

V-RBF-2 | 8 | 2 + 130 + 200 + 400 + 700 + 100 + 50 + 1 | 462,901 |

V-RBF-12 | 8 | 12 + 130 + 200 + 400 + 700 + 100 + 50 + 1 | 464,371 |

Statistical Index | Symbol | Formula |
---|---|---|

Coefficient of determination | R² | $1-\frac{{\displaystyle {\sum}_{i}{e}_{i}^{2}}}{{\displaystyle {\sum}_{i}{\left({y}_{i}-\mu \right)}^{2}}}$ (11) |

Mean absolute error | MAE | $\frac{{\displaystyle {\sum}_{i=1}^{C}\left|{y}_{i}-{m}_{i}\right|}}{C}$ (12) |

Count | C | Size of the validation dataset |

Mean | $\mu $ | $\frac{1}{C}{\displaystyle \sum _{i=1}^{C}{e}_{i}}$ |

Standard deviation | $\sigma $ | $\sqrt{\frac{{\displaystyle \sum _{i=1}^{C}{\left({e}_{i}-\overline{e}\right)}^{2}}}{C-1}}$ |

Minimum | min | min(${e}_{i}$) |

Maximum | max | max(${e}_{i}$) |

Quartile at 25% | Q_{1} | Value for which the cumulative percentage frequency of the sample is at least 25% |

Quartile at 50% | Q_{2} | Value for which the cumulative percentage frequency of the sample is at least 50% |

Quartile at 75% | Q_{3} | Value for which the cumulative percentage frequency of the sample is at least 75% |

**Table 9.**Values of all statistical quantities calculated on residuals resulting from the validation process of all 16 neural networks tested.

ANN Code | R^{2} | MAE | m | s | Min | Max | Q_{1} | Q_{2} | Q_{3} |
---|---|---|---|---|---|---|---|---|---|

S-MLP-2 | 0.57 | 1597.8 | −191.4 | 2038.6 | −8113.1 | 5141 | −11198.2 | −22.4 | 1234 |

S-MLP-12 | 0.92 | 599.8 | 68.0 | 872.8 | −4827.7 | 6164.7 | −323.1 | 107.1 | 509.5 |

S-RBF-2 | 0.55 | 1650.5 | 7.4 | 2065.1 | −6563.9 | 4908.6 | −1320.3 | 173.2 | 1474.1 |

S-RBF-12 | 0.80 | 964.1 | 19.5 | 1341.9 | −7987.8 | 8043.8 | −630.6 | 46.7 | 705.9 |

M-MLP-2 | 0.63 | 1325.1 | −148.2 | 1891.9 | −8370.4 | 4382 | −777.2 | 9.8 | 946.7 |

M-MLP-12 | 0.94 | 465.9 | −5.5 | 720.8 | −7290 | 3536.8 | −281.3 | 59.5 | 390.9 |

M-RBF-2 | 0.62 | 1375.5 | −250.7 | 1956.2 | −8494.6 | 4800.2 | −909.3 | 35.7 | 836.6 |

M-RBF-12 | 0.85 | 795.1 | −62.9 | 1167.7 | −5386.1 | 5695.8 | −583.8 | −44 | 451.2 |

D-MLP-2 | 0.72 | 1059.4 | −99.4 | 1633.2 | −7544.2 | 6653.0 | −634.6 | −17.8 | 576 |

D-MLP-12 | 0.95 | 419.7 | −58.4 | 653 | −6596.8 | 3117.6 | −335 | −21.1 | 285.7 |

D-RBF-2 | 0.70 | 1047.2 | −39.3 | 1671.5 | −7516.9 | 6034.8 | −506.8 | 44.4 | 585.1 |

D-RBF-12 | 0.94 | 458.5 | −87.1 | 695.7 | −5627.9 | 6163.3 | −362.4 | −15.6 | 317.5 |

V-MLP-2 | 0.76 | 904.8 | −124.6 | 1546.9 | −9183.2 | 6220.4 | −518.4 | −28.1 | 385.2 |

V-MLP-12 | 0.98 | 306.9 | −50.9 | 421 | −3050.8 | 2484.5 | −275.2 | −45.0 | 205.4 |

V-RBF-2 | 0.73 | 936.2 | 91.7 | 1615.2 | −8514.5 | 7946.0 | −421.5 | 22.1 | 476.2 |

V-RBF-12 | 0.95 | 420 | −66.7 | 682 | −5950.3 | 7282.2 | −353.6 | −29.4 | 241.3 |

**Table 10.**Training time and velocity of all 16 neural networks tested with an i7 CPU with 32 GB of RAM.

ANN Code | Elapsed Time (s) | Velocity (epochs/s) |
---|---|---|

S-MLP-2 | 1662 | 0.487 |

S-MLP-12 | 2607 | 0.500 |

S-RBF-2 | 711 | 0.555 |

S-RBF-12 | 454 | 0.603 |

M-MLP-2 | 1558 | 0.217 |

M-MLP-12 | 1221 | 0.300 |

M-RBF-2 | 1613 | 0.203 |

M-RBF-12 | 1074 | 0.458 |

D-MLP-2 | 1595 | 0.333 |

D-MLP-12 | 1215 | 0.341 |

D-RBF-2 | 2505 | 0.385 |

D-RBF-12 | 1662 | 0.480 |

V-MLP-2 | 15731 | 0.507 |

V-MLP-12 | 5848 | 0.506 |

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**MDPI and ACS Style**

Lo Brano, V.; Guarino, S.; Buscemi, A.; Bonomolo, M.
Development of Neural Network Prediction Models for the Energy Producibility of a Parabolic Dish: A Comparison with the Analytical Approach. *Energies* **2022**, *15*, 9298.
https://doi.org/10.3390/en15249298

**AMA Style**

Lo Brano V, Guarino S, Buscemi A, Bonomolo M.
Development of Neural Network Prediction Models for the Energy Producibility of a Parabolic Dish: A Comparison with the Analytical Approach. *Energies*. 2022; 15(24):9298.
https://doi.org/10.3390/en15249298

**Chicago/Turabian Style**

Lo Brano, Valerio, Stefania Guarino, Alessandro Buscemi, and Marina Bonomolo.
2022. "Development of Neural Network Prediction Models for the Energy Producibility of a Parabolic Dish: A Comparison with the Analytical Approach" *Energies* 15, no. 24: 9298.
https://doi.org/10.3390/en15249298