Developing a machine learning model for fast economic optimization of solar power plants using the hybrid method of firefly and genetic algorithms, case study: optimizing solar thermal collector in Calgary, Alberta

1.   Introduction

It has recently been demonstrated that renewable energy is a viable alternative to fossil fuels in solving environmental problems and the energy crisis caused by conventional fuel use (Ashouri et al., 2015). This makes the use of renewable resources, such as solar energy, essential to promoting human sustainability and alleviating environmental concerns. One of the most effective measures to combat global warming has been the development of solar energy utilization. This type of energy, particularly in areas with higher average radiation levels, can reduce the consumption of fossil-based energy. As a result of new policies and government subsidies, the cost of utilizing solar energy has decreased, and installed capacity has increased. However, the proportion of this type of energy is still less than 3.6% despite all these worldwide initiatives (Pourasl et al., 2023). As a result of climate change, it appears that an increase in the proportion of solar energy is impossible unless government policies are changed in a paradigmatic manner.

Solar energy is a low-emission technology with a high potential for scaling up. It is therefore inevitable and necessary to increase the capacity of solar electricity to scales of terawatts in order to combat climate change. During the past few years, solar generation capacity has increased significantly, technology has improved, prices have decreased, and innovative business models have been developed to encourage the purchase of residential solar panels. However, further improvements are required in order to increase the share of solar energy at a price that is acceptable to society. Solar energy will only be able to fulfill this role if it is cost-competitive and cost-effective compared to fossil fuels, if carbon dioxide emissions are appropriately penalized, and if subsidies are likely to be significantly reduced (Sultan et al., 2020). Any new technology must be able to compete with existing commercially available technologies (Nguyen et al., 2023; Omidkar, Alagumalai, et al., 2024; Omidkar, Haddadian, et al., 2024; Omidkar et al., 2023). It has been proven that concentrated solar power (CSP) is an efficient technology for the production of clean and renewable energy. In essence, it works via focusing the sun's rays by using a reflective surface. There are two types of CSP systems: point focused and line focused. Parabolic dish collectors and solar towers are two types of point-focused collectors. There are two types of line-focused collectors: parabolic trough collectors (PTCs) and linear Fresnel reflectors (LFRs). Focus temperatures can reach over 1000 ℃ when using point-focused CSPs. At temperatures below 500 ℃, line-focused CSP systems produce thermal and electrical energy (Cuce & Cuce, 2023; Cuce et al., 2021). PTC technology is widely regarded as the preferred CSP technology. As a stand-alone system, it can also be combined with other power generation systems to create hybrid systems. The nominal power produced by solar collectors directly correlates with radiation, so the determination of the geographical location is crucial. Furthermore, the average of sunny hours, wind speed, and humidity affect the performance of solar collectors. On the other hand, the performance of solar collectors can be described by the exergy conception, which means the maximum theoretical work obtainable from a given thermodynamic state when this reaches thermo-mechanical and chemical equilibrium with a reference state on environmental conditions entering a state called "dead state" (García-García et al., 2019). Cuse et al. designed a hybrid PTC/TEG energy system created with a thermoelectric generator (TEG) energy system (Cuce et al., 2024). They concluded that the overall thermal efficiency of PTC-TEG hybrid systems can be increased by 70% compared to PTC systems.

Elfeky and Wang (Elfeky & Wang, 2023) conducted techno-environ-economic assessment of two energy generation technologies, photovoltaic and CSP, in China and Egypt. According to their findings, establishing photovoltaic power plants in China is the best option, but concentrating on solar power plants in Egypt is the best option. The results show that in optimal conditions, a parabolic trough collector (PTC), which concentrates sunlight, produces 33.34% more electricity than a photovoltaic plant. Deai et al. introduced a PTC solar system using a micro-structured polymer foil. This system generates electricity through an organic Rankine cycle and produces freshwater using a multi-effect distillation process (Desai et al., 2021). The proposed system holds significant potential for regions facing electricity and water shortages. Using cyclopentane as the working fluid for the organic Rankine cycle, the plant achieved a levelized electricity cost of 0.116 EUR/kWh_e and a levelized water cost of 1.13 EUR/m³ for Antofagasta, Chile, and 0.163 EUR/kWh_e and 1.62 EUR/m³ for Cape Town, South Africa. Gilani and Hoseinzadeh conducted a comparative techno-economic analysis of compound parabolic collectors (CPC) in which all incident radiation is focused onto a receiver in solar water heating systems in the northern hemisphere (Azad Gilani & Hoseinzadeh, 2021). As a result of the study, CPCs are found to consume less auxiliary power in all selected locations than flat plate collectors. In all research conducted so far, optimizing design parameters, operational conditions, and input variables has remained a major challenge. Energy system efficiency can be affected by factors such as operational conditions, material properties, kind of collector and operating fluid, concentrating geometry, and cycle system (Mehdipour et al., 2020; Pal & K, 2021; Shafieian et al., 2020; Tabarhoseini et al., 2022). Myriads of studies have thus been conducted to determine the thermodynamic behavior of these systems using calculational models and predictive technologies like machine learning, specifically artificial neural networks (ANN) (Alawi, Kamar, Salih, et al., 2024; Brenner et al., 2023; Kottala et al., 2023; Mustafa et al., 2022; Ruiz-Moreno et al., 2022; Vakili & Salehi, 2023). Experiments in pilot plants are ineffective due to their high operational costs and time requirements. Neural network simulators offer the advantages of simplicity, speed, and the ability to handle nonlinear and complex interactions between inputs and variables, making them a promising solution for complex data analyses. (Haykin, 1998). A common method of solving multi-objective optimization problems (MOPs) is to use evolutionary algorithms (EAs), such as genetic algorithms and particle swarm algorithms. Following the development of swarm intelligence (SI) algorithms, particle swarm algorithms expanded into multi-objective particle swarm optimization algorithms (MOSPO). A number of studies have attempted to utilize ANNs in addition to other concepts for evaluating renewable energy systems; however, the number of studies is limited. Recently, there has been a surge in interest in employing artificial intelligence techniques to optimize and predict renewable energy production systems (Elsheikh et al., 2024; Shboul et al., 2024; Shboul et al., 2024; Zayed et al., 2023a; Zayed et al., 2023b; Zayed et al., 2021). Other challenges include high CPU usage, optimization, and interpretation. The knowledge gap that motivated this study is illustrated in Table 1.

To overcome these challenges and to be useful in industrial settings, a supervised machine-learning approach has been developed in this study. The purpose of this study is to optimize energy, exergy, heat cost, and carbon dioxide emission costs for a complex renewable energy system using a machine-learning approach.

2.   Methods

2.1.   System description

The system used in this study is depicted in Figure 1. The components of the parabolic trough solar collector are a parabolic collector, pump, storage tank or thermal heat exchanger, flow transducer, DC motor, piping, working fluid, pressure and temperature sensors, CPU, and data storage. Through an absorb tube, the working fluid receives heat and enters the storage tank at a high temperature. Energy from the storage tank can be used simultaneously or whenever needed. It is also possible to use the storage tank to supply the energy of another working fluid via a heat exchanger for other applications such as the Rankine cycle, absorption refrigeration cycle, or the supply of hot water for residential use. CPUs can utilize the thermal and hydrodynamic properties of working fluids and receive atmospheric conditions from sensors. The system can also be optimized by using the aforementioned parameters, the mass flow of the working fluid, and the angle of the reflector.

2.2.   Governing equations

The difference between solar time and local time can be derived via Equation (1) (Duffie et al., 2020).

In Equation (2), the $\beta = \frac{{360\left({n - 1} \right)}}{{365}}$ and the n is the n^th day of the year.

The angle of sunset can be calculated using Equation (3), in which $\varphi$ and $\delta$ are the latitude and deviation angle, respectively (Duffie et al., 2020).

The pressure drop inside the absorber tube can be calculated by Equation (5).

In Equations (5) and (6), $\Delta P$ is pressure loss along the length of L in the pipe, D₂ is the inner diameter of the absorber pipe, g is the gravitational acceleration, V₁ is the average velocity of fluid inside the pipe, $\dot m$ is mass flow, and $\rho$ is the density of the fluid. Depending on the laminar or turbulent flow regime, $f_{Darcy}$ can be derived using Equations (7) or (8), in which f is the friction factor and $\varepsilon$ is the pipe's effective roughness height.

Based on Figure 2, the energy balance in one element of the tube in which the temperature gradient is linear can be established using the system of Equations (9–13).

In the energy conservation system of equations, ${\dot q'_{12, conv}}$ is the convection heat transfer between the absorber tube's inner surface and working fluid derived by Equations (14) and (15) (Bergman, 2011; Gnielinski, 1976; Jehring, 1992). h₁ is the convection heat transfer coefficient of fluid at T₁, D₂ is the inner diameter of the absorber tube, T₂ is the temperature of the absorber tube's inner surface, T₁ is the temperature of working fluid, and Nu_D2 is the Nusselt number based on D₂ and k₁. Depending on the flow regime (laminar, transient, and turbulent), the Nusselt number can be derived.

Between the inner and outer surfaces of the absorber pipe, the mechanism of heat transfer is based on Equation (16). k₂₃ is the conduction heat transfer coefficient of the absorber tube, T₃ is the outer surface temperature of the absorber tube, and D₃ is the outer diameter of the absorber tube.

Heat transfer mechanisms can vary depending on the pressure between the envelope cover (mostly made of glass) and the absorber tube. Heat is transferred between the envelope and the absorber tube by free molecular convection when the pressure is lower than 1 Torr. When the pressure exceeds 1 Torr, the heat transfer mechanism becomes natural convection.

D₃ is the inner diameter of the envelope, D₄ is the diameter of the outer surface of the envelope, h₃₄ is the convection heat transfer coefficient at the average temperature T₃₄, T₄ is the inner temperature of the envelope, and k_std is the conduction heat transfer of the gas. b is the interaction factor, $\lambda$ is the mean free path, a is a correlation factor, $\gamma$ is the ratio of specific heats of the gas, T₃₄ is the average temperature, P_a is the pressure of the gas inside the envelope (mmHg), and $\delta$ is the molecular diameter of the gas. For the equation related to the higher pressure, Pr₃₄ is the Prandtl number, Ra_D3 is the Rayleigh number, and $\beta$ is the thermal expansion coefficient of the gas.

The heat transfer mechanism between the absorber tube and the envelope is radiation, which can be derived via Equation (19). $\sigma$ is the Stefan- Boltzmann constant, ${\varepsilon _3}$ is the emissivity coefficient of the absorber, and ${\varepsilon _4}$ is the emissivity coefficient of the envelope.

The conduction heat transfer in the envelope is similar to the conduction heat transfer in the absorber tube [Equation (16)].

There are two mechanisms for heat transfer between the atmosphere and the envelope: convection and radiation. Depending on the wind speed, convection can be natural or forced. The radiation heat loss exists due to the temperature difference between the envelope and the sky. The convection heat transfer between the envelope and the atmosphere is the main resource for heat loss, specifically when there is wind. T₅ is the outer temperature of the envelope, T₆ is the atmospheric temperature, h₅₆ is the convection heat transfer coefficient of the air at the average temperature, k is the conduction heat transfer coefficient at the average temperature, D₅ is the outer diameter of the envelope, and Nu_D5 is the Nusselt number based on the outer diameter of the envelope.

If there is no wind (wind speed lower than 0.1 m/s), the heat transfer mechanism between the envelope and the atmosphere is natural convection.

Ra_D5 is the Rayleigh number for the air based on the outer diameter of the envelope, g is the gravitational acceleration, ${\alpha _{56}}$ is the thermal diffusivity of the air at T₅₆, $\beta$ is the volumetric expansion coefficient for an ideal gas, Pr₅₆ is the Prandtl number for the air at temperature T₅₆, ${\nu _{56}}$ is the kinematic viscosity of the air and temperature T₅₆, and T₅₆ is the average temperature of the film. If the wind speed becomes higher than 0.1 m/s, the heat transfer between the envelope and atmosphere is forced convection.

The radiation heat transfer between the envelope and the sky because of the temperature difference between the outer surface of the envelope and the sky can be derived from Equation (23).

${\varepsilon _5}$ and T₇ are the emissivity of the outer surface of the envelope and sky temperature, respectively. Generally, the sky temperature is 8 ℃ lower than the atmospheric temperature.

The absorption of sunlight can be derived by Equation (24).

${\dot q'_{si}}$ is the radiation of light along the receiver, ${\eta _{env}}$ is the light efficiency of the envelope, and ${\alpha _{env}}$ is the absorption of the envelope. ${\varepsilon '_1}$ is the collector's shadow, ${\varepsilon '_2}$ is the localization error, ${\varepsilon '_3}$ is the geometry error, ${\varepsilon '_4}$ is the fouling on the mirrors, ${\varepsilon '_5}$ is the fouling on the collectors, ${\varepsilon '_6}$ is the unconsidered error, and ${\rho _{cl}}$ is the reflection factor of the clean mirror. The factor K is the correction factor of the light angle, and $\theta$ is the deviation angle of the solar collector.

The light absorption in the absorber tube can be derived from Equation (25).

${\eta _{abs}}$ is the light efficiency in the absorber tube, ${\alpha _{abs}}$ is the absorption coefficient of the absorber tube, and ${\tau _{env}}$ is the light transmission coefficient in the envelope.

The envelope and the absorber tube at the focal length of the solar collector are maintained by the supporters. Generally, at the end of each solar element, there is one supporter every 4 m. The heat loss, assuming the support bracket behaves as an infinite fin with a temperature 10 ℃ lower than T₃, can be derived from Equation (26):

In Equation (26), ${\bar h_b}$ is the convection heat transfer of the supporter, which depends on the wind speed, P_b is the perimeter of the supporter, k_b is the conduction heat transfer coefficient, A_{cs, b} is the area of the cross-section of the bracket, T_base is the base temperature of the bracket supporter, T₆ is the atmospheric temperature, and L_HCE is the length of the solar collector.

2.3.   Exergy analysis

By applying the second law of thermodynamics to the control volume of the solar collector, Equation (27) is obtained:

$\dot E{x_{f, in}}$ is the inflow of exergy to the fluid, $\dot E{x_s}$ is the inflow of exergy from sunlight, $\dot E{x_{f, out}}$ is the outflow of exergy of the fluid, $\dot E{x_l}$ is the exergy loss, and $\dot E{x_{des}}$ is the exergy destruction. The exergy of the sunlight can be calculated using (Landsberg & Mallinson, 1976). T_sun is the surface temperature of the sun (5780 K), and l is the interaction factor. f_H is the view factor, which explains the geometry correlation between the radiation source and the receiver, ${\varepsilon _H}$ is the dilution factor, and f_r is the geometry factor of reflection by the absorber.

The useful exergy is defined as the difference between the inflow exergy and the outflow exergy of the fluid (Badescu, 2018).

The exergy loss can be calculated using Equation (30) (Bellos & Tzivanidis, 2017). $\dot E{x_{l, opt}}$ is light exergy loss and $\dot E{x_{l, thermal}}$ is the thermal exergy loss.

The exergy destruction shows irreversibility due to heat loss. Specifically, this parameter defines a probable work that disappears when there is thermal energy flow between hot and cold sources. The exergy destruction can be derived using Equation (31) (Kalogirou, 2004). In the solar collector, exergy destruction can occur mainly from two sources: between the absorber tube and the sun and between the absorber tube and the working fluid.

Exergy efficiency can be calculated using Equation (32).

2.4.   Economical modeling

The net price cost of the solar collector can be calculated as Equation (33). The CRF is the capital recovery factor, C_O&M is the operational and maintenance cost (which is considered as 2% of capital investment), C_s is the capital investment, i is the interest rate (4%), and n is the total operational years (Frangopoulos, 1987). C_H is the cost of heat production by the collector, N is the total annual operational hours, and q_u is the efficient heat transfer to the working fluid.

The purchase cost of various components of solar collectors can be found in Table 2.

2.5.   Energy-environment analysis

The environmental emission of a solar collector system can be derived using Equation (34) (Faizal et al., 2015).

${x_{C{O_2}, eq}}$ is the equivalent emitted CO₂ during the operational time of t_operational, ${y_{C{O_2}, eq}}$ is the emitted CO₂ of the reference boundary control system, which is calculated using life cycle assessment methods, and ${\dot Q_u}$ is the produced power of the reference boundary system.

2.6.   Exergy-environment analysis

Environmental assessment can also be carried out by another method. In this kind of assessment, which is similar to that of section 2.5, instead of power, the amount of exergy inside the system will be taken into account. This method is more conservative and accurate than energy-environment analysis.

2.7.   Energy-environment-economy analysis

According to (Deniz & Çınar, 2016), the economic effects of producing carbon dioxide can be derived from Equation (36). C_CO2 is the economic-environmental parameter of the system, and c_CO2 is the cost of carbon dioxide production. It is possible to prevent climate change and other phenomena such as global warming by reducing carbon dioxide emissions as much as possible. By employing a conservative approach, the carbon emission cost method aims to provide a more tangible view of the environmental impacts. By formulating incentives and punishment policies, this method seeks to reduce the environmental impact of energy systems.

2.8.   Machine learning method: Support vector regression

The support vector regression technique follows the principle of structural risk minimization and has been successfully applied to data classification, regression, and nonlinear systems modeling (Vapnik, 1999). In a regression model with a training set $G = \left\{ {\left({{x_i}, {y_i}} \right)} \right\}_i^n \subset {R^d} \times R$, in which x_i and y_i are the input and output variables, respectively, of i^th pair of datasets, and n indicates the total number of datasets, the SVR is a kernel method that performs the nonlinear regression using a kernel trick. The kernel-induced feature space F is used to perform linear regression on each input ${x_i} \in {R^d}$ via a nonlinear feature map $\phi \left(\bullet \right)$. For all training data, SVR aims to find a function that has minimum deviation $\varepsilon$ from the actual target y_i. Deviations greater than $\varepsilon$ are not accepted. Accordingly, SVR considers the following linear estimation function in order to achieve the stated objective:

In Equation (37), $\omega$ and b are coefficients and $\phi \left(x \right)$ indicates the high-dimensional feature space that is nonlinearly mapped from the input space x. The coefficients w and b can be estimated by minimizing the regularized risk function.

In Equation (38), $\frac{{\left\| {{\omega ^2}} \right\|}}{2}$ is the Euclidean norm; this parameter is used to estimate the flatness of the function. This feature is used to avoid over-fitting. The difference between measured values and values calculated by the regression function is shown by $\varepsilon$. It is possible to visualize this difference as a tube surrounding the regression function. The parameter C represents the cost function measuring empirical risk; it is used to determine the trade-off between empirical risk and model flatness. In this context, C > 0 represents the penalty degree of the sample with error exceeding $\varepsilon$. The loss function ${L_\varepsilon }\left({f\left({{x_i}} \right) - {y_i}} \right)$ is referred to as the $\varepsilon$-insensitive loss function. In the empirical analysis, C and $\varepsilon$ are parameters that are defined by the user.

Overall, SVR is a versatile and powerful tool for regression tasks, especially when dealing with nonlinear data and high-dimensional feature spaces. However, the choice of the machine learning method should always be guided by the specific characteristics of the dataset and the problem at hand.

Statistical parameters are used to evaluate the performance of the SVR model for the training and testing sets: the coefficient of determination (R2), mean absolute error (MAE), root mean square error (RMSE), and mean squared error (MSE).

2.9.   Firefly opt imization algorithm

An optimization algorithm based on the flashing behavior of fireflies is called the firefly algorithm (FA). Bioluminescence is used by fireflies to communicate and attract mates. In order to optimize solutions to various optimization problems, the algorithm simulates this behavior. As a result of this knowledge, we can idealize some of the flashing characteristics of fireflies in order to develop algorithms that are inspired by fireflies. The FA is described by the following rules:

● Since fireflies are unisex, one firefly will be attracted to another firefly regardless of their gender.

● The attractiveness of a firefly is directly proportional to its brightness. Consequently, if there are two flashing fireflies, the less bright one will move toward the brighter one. There is a direct relationship between attractiveness and brightness, both of which decrease as their distance increases. A firefly will move randomly if there is no other that is brighter.

● Fireflies' brightness is affected or determined by the objective function's landscape.

The FA is a relatively simple, yet effective, optimization algorithm that has gained significant attention due to its ease of implementation and strong global exploration capabilities. The algorithm's inherent ability to thoroughly search the solution space reduces the likelihood of becoming trapped in local optima. Its adaptability to various problem types, including continuous, discrete, and multi-objective optimization, further enhances its versatility. Furthermore, FA's parallel processing nature can significantly accelerate the optimization process. Its robustness to dynamic and noisy environments makes it well-suited for real-world applications.

Compared to other metaheuristic algorithms, FA offers several advantages. Unlike genetic algorithms (GA), FA relies on a more straightforward attraction mechanism, which can lead to faster convergence. While simulated annealing (SA) explores the solution space sequentially, FA's simultaneous exploration of multiple solutions can potentially accelerate the search process. Additionally, FA's independence from pheromone-based information reduces computational overhead and simplifies implementation. Its suitability for non-differentiable, complex, and multi-modal functions further underscores its versatility.

More information about this method and pseudo code is available in (Fister et al., 2013; Johari et al., 2013; Yang, 2010).

2.10.   Research methodology

The research methodology is illustrated in Figure 3. Initially, a dataset is produced using input variables and solving the governing equations mentioned in previous sections. The produced dataset is used for training the SVR model. Then, the SVR model is optimized by the firefly optimization method to maximize accuracy. Then, this model is used by the genetic algorithm (GA) for the optimization of design parameters.

The FA and GA are metaheuristic optimization algorithms that offer complementary strengths. FA excels in global exploration, efficiently traversing the search space, while GA is adept at local exploitation, refining solutions through crossover and mutation. By hybridizing these algorithms, it is possible to achieve a more balanced search process, combining the global exploration capabilities of FA with the local exploitation capabilities of GA. The hybrid approach can potentially improve convergence speed and solution quality. FA can guide the search toward promising regions of the solution space, while GA can refine these solutions further. This synergy can result in faster convergence to high-quality solutions compared to using either algorithm alone. Optimization problems in AI models, like tuning hyperparameters for SVR, often involve complex, multi-dimensional search spaces with multiple local optima. The hybrid method is well-suited for navigating such landscapes, as it combines the global search ability of GA with the local search refinement of FA (El-Shorbagy & El-Refaey, 2022; Mazen et al., 2016; Wahid et al., 2019; Wahid et al., 2019).

3.   Results and discussion

3.1.   Mathematical model validation

The mathematical model has been validated by (Forristall, 2003). The weather conditions and solar collector characterization in (Forristall, 2003) have been used as the input data. Table 3 depicts the performance of the mathematical model in terms of each component.

The maximum relative error is the energy efficiency, which equals 5.97%. This high relative error is mainly due to the high nonlinearity degree of the system.

3.2.   Application of the recommended SVR model

In this study, the SVR machine learning model is presented for solving nonlinear and complex problems in order to calculate economic metrics and assess environmental impacts. The model will be implemented in detail, and it can be applied to other complex projects as well. An example of the complexity of this study is dealing with a system of nonlinear equations and numerous variables. Time, energy, and money will be consumed in solving five nonlinear equations in order to derive the temperature profile. According to the solution, the CPU time for the solution of the main function is approximately 10 s. The optimization of a problem with 250 generations and a population of 200 takes approximately 138 h. The optimization time for this problem is high, as can be seen from the example. Minitab V 22.1 was used to generate the standard input data. The discrete and continuous factors were selected as 3 and 7, respectively (Table 4). By using Minitab software, 1500 input datasets were created. Then, by using computational code, all datasets were used as input, and a system of equations using these input data was solved. For the training of the SVR model, the sets of inputs and outputs were selected. The outputs include energy efficiency, exergy efficiency, heat cost, CO₂ emission cost based on energy, and CO₂ emission cost based on the exergy. The heat cost function includes cost function ($) and heat gain (W), which relate to energy efficiency. The input and output of the SVR model are depicted in Table 4 and Table 5. The input design is also depicted in Table 6.

For the regression of the dataset, a Gaussian kernel was selected. $\varepsilon$ was selected for optimization using the firefly algorithm with the characterization as in Table 7.

Figure 4 (a) illustrates the process of optimizing the SVR machine learning model. The MSE becomes constant after the 25th iteration of the optimization using the Firefly algorithm. Figure 4 (b) shows the regression results for all training, validation, and test data. According to the results, an ideal regression is obtained, and the regression number for the test data is 0.994. As well as using different methods to enhance accuracy, accuracy is primarily achieved by scaling input and output data to an appropriate range and applying nonlinear constraints to ensure meaningfulness. As a result of scaling data, large weights are not produced from the training model. An unstable model often exhibits high coefficient values, which may result in poor learning performance and sensitivity to input values, which may result in larger errors. Based on the analysis of the results, it is evident that there is a lack of data for regression for areas with nonlinear limitations on data generation. As a result, there are no data in this area, since these data are contradictory to reality and the fundamental physics of the problem. By following this approach, it is possible to avoid misdirection, incorrect training of machine learning algorithms, and the creation of unrealistic results. The optimization results are depicted in Figure 4 (c) by comparison between SVR and optimized SVR.

3.3.   Optimization

An effective and efficient way to provide accurate and appropriate machine learning models is to optimize and analyze the desired goals using different decision variables in order to solve the previously complex and time-consuming problem. Two object functions have been defined for the optimization of the design parameters of the system derived from the SVR model: exergy efficiency (%) and heat cost ($\$$/kWh). As previously mentioned, the main reason for selecting the heat cost function was that it includes both the cost function ($\$$) and the increase in useful heat gain (W), which are related to overall energy efficiency. In general, the optimization process aims to maximize exergy efficiency and minimize heat costs. Table 8 illustrates the main parameters of the genetic algorithm used in this study to optimize design parameters. The SVR model has the advantage of reducing the optimization process.

Figure 5 illustrates the Pareto front, which indicates the two objective functions derived from the trained model. In accordance with the Pareto front, exergy efficiency increases as the heat cost function increases. Since energy efficiency decreases with an increase in exergy efficiency, which increases the cost function, there is an ideal point in the Pareto front that cannot be reached, but the optimal point is the closest to the ideal point.

As a result, the optimum reachable point based on the optimized design is the exergy efficiency of 14.05% with a heat cost of 0.0137 $/kWh. The optimized design parameters are illustrated in Table 9.

3.4.   Relation between design parameters and economic metrics

Figure 6 shows the energy efficiency, exergy efficiency, heat cost, and CO₂ emission cost based on energy and exergy according to the inner diameter. The heat cost will increase with an increase in diameter from 0.01 to 0.1 m. There is a descending trend in energy efficiency and exergy efficiency except for small diameters. Consequently, as the diameter decreases, both efficiencies will decrease. In small diameters, the pressure drop and the required power for pumping is high. As the diameter increases, the efficiency decreases due to a decrease in the Nusselt number and convection coefficient, which results in less heat being generated and more exergy being destroyed. So, there is a maximum point for the energy and exergy efficiencies when the inner diameter varies depending on whether the pressure drop or Nusselt number effect is dominant.

It is necessary to provide a gap between the tubular absorber and the metallic reflector in order to prevent conduction heat loss from the absorber to the reflector and to provide a glass envelope around the absorber, which will increase the thermal efficiency of the compound parabolic concentrator (CPC) module at high temperatures. As a result of the gap between the absorber and the envelope, light incident on the absorber is lost, which is referred to as "gap losses". It is therefore necessary a compromise between optical and thermal performance. Figure 7 depicts energy efficiency, exergy efficiency, heat cost, and CO₂ emission cost based on the energy and exergy efficiency vs. distance between absorber and envelope. With an increase in the distance from 0.01 to 0.05 m, the energy and exergy efficiencies will decrease due to the increase of convection heat transfer in annulus space.

Figure 8 illustrates the effect of working fluid mass flow on efficiency and economic parameters. The increase in the mass flow will result in an exergy efficiency decrease. There is a maximum point for energy efficiency when mass flow rises. By increasing the mass flow by more than 0.2 kg/s, energy efficiency will decrease because of the lower heat transfer. The higher speed of the working fluid decreases the residence time of volume control of fluid inside the collector; therefore, the heat transfer and output temperature will decrease. In this analysis, energy efficiency is influenced by two main parameters: pressure drop and heat transfer coefficient. With the increase of mass flow rate, both increase; at a low mass flow rate, the convective heat transfer coefficient is dominant. However, energy efficiency is first upward and then downward. The cost of carbon dioxide emission based on energy and the cost of carbon dioxide emission based on exergy is shown as a function of the previously presented parameters.

Based on Figure 9, with an increase in the inlet temperature, the energy efficiency decreases because the temperature difference between the fluid and the ambient environment decreases, resulting in a rise in the cost of heat. A temperature gap between hot and cold sources contributes to exergy destruction. By reducing the temperature difference, exergy destruction is reduced, which increases exergy efficiency.

3.5.   Sensitivity analysis

Based on the trained model, a sensitivity analysis of all continuous and discrete variables was performed using the proposed machine learning approach. The sensitivity analysis was carried out based on the one factor at a time (OFAT) methodology. Based on the ±10% change in the design parameters, the percentage of change in energy efficiency, exergy efficiency, and heat cost was calculated. As can be seen from Figure 10, the results indicate that the most important parameter is the initial temperature of the working fluid.

3.6.   Case study: Economic optimization of a solar collector in Calgary

In order to demonstrate the applicability of the proposed platform, a case study was developed to optimize a parabolic trough solar collector in Calgary (51.0447° N, 114.0719° W). Table 10 depicts the average monthly weather conditions with average solar irradiation.

The platform was used to optimize the design of a parabolic trough solar collector and the optimum parameters for this system were determined (Table 11). As can be seen from Table 11, due to a higher average wind speed in Calgary, the average heat loss will be higher than the design parameters in Table 9. This resulted in a decrease in the distance between the envelope and the absorber tube. Therefore, the envelope cross-sectional area would be smaller, thereby decreasing forced convection heat transfer. Because the average number of sunny hours per day in (Forristall, 2003) is higher than in Calgary, the mass flow in the optimized design in Calgary was reduced, and the width of the collector was increased.

It is clear that the proposed platform can be used for optimizing the solar collector in each location based on the inputs and weather conditions. At optimum design parameters and operational conditions, the heat cost would be 0.0141 $\$$/kWh, and the levelized cost of gas generation would be 0.15 $\$$/kWh. However, the expected levelized cost of renewable energy in Alberta is 0.05 $/kWh by 2035 (Ali, 2018; Barrington-Leigh & Ouliaris, 2017; Patel & Parkins, 2023).

4.   Conclusions

In this comprehensive study, a support vector regression (SVR) algorithm was synergistically integrated with the firefly metaheuristic and genetic algorithms to forecast the economic profitability of solar power plants. Initially, mathematical modeling was employed to construct a robust dataset, which was subsequently utilized as the foundation for training the predictive model. Following the optimization of the SVR algorithm, the model was adeptly applied to evaluate the economic feasibility of the solar power plant. It was found that an increase in the inner diameter of the absorber tube was correlated with an escalation in heat costs, attributable to diminished energy and exergy efficiencies. Furthermore, it was observed that an increase in the distance between the envelope and the absorber tube resulted in a decline in heat cost, energy, and exergy efficiency, while paradoxically elevating CO₂ emission costs. An optimal heat transfer fluid flow rate of 0.4 kg/s was identified. Through sensitivity analysis, the paramount importance of the initial fluid temperature as a critical design parameter was underscored. Additionally, it was determined that the width of the collector and the pressure within the envelope space significantly influenced the leveled cost of energy.

In conclusion, the proposed hybrid optimization strategy was found to proficiently generate highly precise models, while concurrently fulfilling the objective of cost minimization. This methodology is considered to hold substantial promise for feasibility studies and conceptual design phases, offering a robust framework to evaluate the economic and environmental viability of energy projects prior to their construction.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare no conflict of interest.