Analysis of the Performance of a Hybrid Thermal Power Plant Using Adaptive Neuro-Fuzzy Inference System (ANFIS)-Based Approaches

Abstract

The hybridization of conventional thermal power plants by the incorporation of renewable energy systems has witnessed widespread adoption in recent years. This trend aims not only to mitigate carbon emissions but also to enhance the overall efficiency and performance of these power generation facilities. However, calculating the performance of such intricate systems using fundamental thermodynamic equations proves to be both laborious and time-intensive. Nevertheless, possessing accurate and real-time insights into their performance is of utmost significance to ensure optimal plant operation, facilitate decision making, and streamline power production planning. This paper explores the novel application of machine learning techniques to predict the performance of hybrid thermal power plants, specifically the integrated solar combined cycle power plant (ISCCPP). These plants combine conventional thermal power generation with renewable energy sources, making them crucial in the context of carbon reduction and enhanced efficiency. We employ three machine learning approaches: the adaptive neuro-fuzzy inference system (ANFIS), ANFIS optimized via particle swarm optimization (ANFIS-PSO), and ANFIS optimized through a genetic algorithm (ANFIS-GA). These methods are applied to the complex ISCCPP, comprising steam and gas turbine sections and a concentrated solar power system. The results highlight the accuracy of ANFIS-based models in evaluating and predicting plant performance, with an exceptional overall correlation coefficient of 0.9991. Importantly, integrating evolutionary algorithms (PSO and GA) into ANFIS significantly enhances performance, yielding correlation coefficients of 0.9994 for ANFIS-PSO and 0.9997 for ANFIS-GA, with ANFIS-GA outperforming the others. This research provides a robust tool for designers, energy managers, and decision makers, offering valuable support in assessing the performance of hybrid thermal power plants. As the world transitions to cleaner energy sources, the insights gained here are poised to have a significant impact on the growing number of these thermal power plants globally.

1. Introduction

As part of a transition step to zero carbon emission, some classic thermal power plants have been hybridized with renewable energy, especially concentrated solar power systems. On the other hand, the hybridization of renewable energy is mainly driven by the challenge of the non-availability of renewable energy sources (sun, wind, geothermal, ocean, biomass, etc.) at times throughout the year [1]. Not only does this affect the overall performance or efficiency of the system but reduces fossil fuel consumption and brings up the challenge of the computation of the said efficiency, since the level of complexity of the system increases. Hybridizing thermal power plants is the way forward if governments take energy transition and climate change seriously. Moreover, doing so can improve the overall efficiency of the power plant [2]. For the system studied in this research, the gas system and the solar system interact through the heating unit, where low-grade energy in the exhaust gas from a gas turbine is used to produce steam by heating water, in addition to the solar heating through a heat-transfer fluid that flows in a concentrated solar power system which is made of parabolic troughs, in this case. Engineers are very concerned and sensitive to the use of energy or the performance of systems or plants that they design or operate. With the increasing level of complexity of hybrid thermal power plants, it becomes much harder to assess or evaluate the performance of such systems from a thermodynamics standpoint or governing equations [3]. Therefore, there is a need for alternative approaches such as those developed as part of machine learning.

Machine learning (ML) techniques have been adopted in recent years to model and analyze hybrid energy systems. ML models have been hybridized and refined with optimization strategies in a quest for better solutions. As a result, an outstanding rise in the accuracy, precision, robustness, and generalization ability of ML models in energy systems using hybrid models has been noted. Hybridization of ML models has even been reported to be effective in the advancement of prediction models [4,5], particularly for renewable energy systems. However, recent trends suggest that the research direction is moving toward customized ML models or models that are designed for a particular application. In other words, the highest degree of accuracy can be achieved through the development of a case-based ML model [6]. It was found that the enhancement of the predictability of renewable energy systems and demand enables the replacement of expensive standby power generation assets with advanced control and optimization systems [7]. In this study, ANFIS, which is a combination of an artificial neural network (ANN) and fuzzy logic, is used to evaluate the performance of an integrated solar combined cycle gas/steam power plant. Thereafter, it was hybridized for the training phase with evolutionary algorithms, namely particle swarm optimization (PSO) and genetic algorithm (GA), which are metaheuristic algorithms. Some studies have successfully applied ANFIS in the prediction of power generation in solar power plants, control and modeling of interconnected combined cycle gas turbine plants, and diagnosis of these systems. However, until recently, no work applying the ANFIS model in the design or analysis of the efficiency or the performance of a combined cycle gas turbine was found in the literature [8]. To date, this gap is further widened by the integration of renewable energy systems, namely solar systems, in conventional combined cycle gas turbine plants in an effort to reduce their carbon footprint and increase the overall efficiency of the plants. Furthermore, there is a necessity to explore advanced optimization techniques to improve the prediction accuracy of the ANFIS model for these hybrid power systems. The present investigation is motivated by and undertaken to fill this gap.

Some researchers have conducted related work; for example, Khosravi et al. [9] tried to determine the optimal design parameters of a solar-only power tower system using molten salt for storage. They used machine learning, specifically a hybrid adaptive neuro-fuzzy inference system with a combination of a genetic algorithm and a teaching-based optimization algorithm. They used four parameters as inputs, namely latitude, longitude, design point direct normal irradiance (DNI), and solar multiple (SM), which is the ratio of the solar field size to the power block, all expressed as nominal thermal power. Three parameters (annual energy produced, levelized cost of energy, and capacity factor) were used as output parameters in the analysis. An extremely high correlation coefficient, close to 1, for the hybrid ANFIS-GATLBO was reported in this study.

Yaici and Entchev [10] investigated the suitability of the adaptive neuro-fuzzy inference system (ANFIS) method for predicting the performance parameters of a solar thermal energy system (STES) used for household hot water and space heating applications. They found that the predicted values were in agreement with the experimental data, with mean relative errors less than 1%. The ANFIS results were compared to the ANN results, and it was found that the ANFIS approach performed slightly better than the ANN one because of higher accuracy and reliability in the prediction of the performance of the energy system. However, the ANN model was more flexible in terms of implementation and reduced computation time.

Zaaoumi et al. [11] presented a comparison between ANN and ANFIS to predict the daily power output of a solar power plant in eastern Morocco. The plant itself is an integrated solar combined cycle (ISCC) plant, which is made of a CSP plant and a natural gas-fired combined cycle (NGCC) power plant. The whole system has two gas turbines fueled by natural gas, a steam turbine, two recovery boilers, a solar field (made of parabolic troughs), and a heat exchanger. The total installed capacity is 472 MW, of which 20 MW is of solar source. For modeling purposes, six variables (daily direct normal irradiance, day of the month, mean wind speed, daily mean ambient temperature, relative humidity, and previous daily electric production) were used as inputs, while the daily electricity generation of the plant was used as the output. They concluded that both the ANN and ANFIS models had similar performance with regard to prediction accuracy. The coefficient of correlation, R², was in the range of 0.94 for the training and testing phases, while the RMSE was in the range of 0.072 for training and 0.089 for testing.

In an effort to identify the operating variables that can improve the efficiency of a combined cycle gas turbine (CCGT), Rodriguez et al. [8] modeled the cycle using an adaptive neuro-fuzzy inference system (ANFIS). Three input variables, namely, the compression ratio in the gas cycle, the pressure of the bled steam for water heating, and the heat lost to the steam turbine exhaust, were considered. They found that the pressure ratio in the gas turbine had the most significant effect on the efficiency of the combined system. The ANFIS results were compared to the analytical results and found to be similar.

Azfal et al. [12] conducted a critical review of optimization techniques for the thermal performance of solar energy devices using metaheuristic algorithms. Many power arrangements integrating solar PV, CSP (dish collectors, heliostats, parabolic troughs), geothermal wells, etc. were covered. It was highlighted in this study that more research is needed in hybrid optimization strategies to solve complex obstacles and obtain high efficacy in solar energy systems. Evolutionary algorithms for multi-objective optimization of hybrid renewable energy systems were recommended for future research.

Reyes-Belmonte et al. [13] studied the optimization of an integrated solar combined cycle. The system was made of an open-air Brayton cycle, which was thermodynamically connected to a base steam Rankine cycle and a CSP hybrid plant. The CSP plant was based on pressurized air receiver technology assisted by a natural gas burner. For analysis, the Thermoflex software tool was used. An exclusive contribution of thermal energy through the solar thermal receiver was first considered, and then a mixed thermal contribution of solar energy and natural gas was examined. Scenarios of different configurations of the combined system were considered. It was found, among other things, that the overall system efficiency was far abovethat of modern conventional combined cycle systems, whose conversion efficiencies are around 60% because of pressure limitations for pressurized air receivers.

An investigation the performance of an integrated solar combined cycle (ISCC) plant situated in the tropical climate of southern Algeria was carried out by Achour et al. [14]. The plant was a combination of a parabolic trough solar field with a fossil fuel combined cycle, which was made of two gas turbines and a steam turbine. The authors developed, from first principles, a model for each component of the plant and concluded that an overall thermal efficiency of about 60% could be reached. It is worth noting that developing a thermodynamic model for such a complex system or plant is a very tedious process, and some assumptions need to be properly made.

Temraz et al. [15] developed and validated a dynamic simulation model for an integrated solar combined cycle (ISCC) power plant in Karaymat in Egypt using Apros (Advanced PROcess Simulation), a design software. The power plant (135 MW total electrical power) consisted mainly of a parabolic trough collector solar field, a gas turbine (70 MW), a steam turbine (65 MW), and a heat recovery steam generator (HRSG). The boiler used hot water from the heat exchanger with the heat-transfer fluid of the CSP field, and also flue gas from the gas turbine. The model was initialized and tuned using operational data measured from the plant. The authors concluded that the model represented reality with high accuracy and showed a good predictive capability. Once again, it can be seen that the approach followed by the authors was tedious and did not make use of machine learning.

Benabdellah et al. [16] analyzed, from a thermodynamics point of view, the energy, exergy, and economics of an integrated solar combined cycle (ISCC) power plant situated in Algeria. It was made of two gas turbines of 40 MW each, one steam turbine of 80 MW, which was fed by two HRSGs (heat recovery steam generators), one solar steam generator (SSG), and a solar field of a total area of 183.120 m² and comprising 224 parabolic trough collectors (PTC). The plant operates as ISCC-PTC during sunny times and as a conventional combined cycle plant at other times. Hence, the SSG works as a boiler in parallel to the HRSG to increase the steam quantity. They found that energy and exergy efficiencies were, respectively, 56.0% and 53.29%. The levelized cost of energy (LCOE) was promising but still higher than a simple combined cycle. The ISCC-PTC power plant allowed some savings in natural gas consumption and CO₂ emission taxes. Many other researchers [17,18,19,20,21] have adopted similar thermodynamic analysis approaches to assess the performance of the integrated solar combined cycle power plant, even in recent years.

Some ANFIS-based approaches have recently been investigated in other applications. For example, Pradeep et al. [22] used a deep neural network (DNN) and hybrid models of ANFIS and metaheuristic optimization algorithms for the prediction of rock strain. For ANFIS training, they used four optimization algorithms, namely grey wolf optimizer (GWO), fireflies algorithm (FF), particle swarm optimization (PSO), and genetic algorithm (GA). They used a rank index to determine the most robust model, and for the data analyzed, concluded that DNN was the best model. Abba et al. [23] employed ANFIS-PSO, ANFIS-GA, and ANFIS-BBO (biogeography-based optimization) to identify groundwater salinization in the coastal region of eastern Saudi Arabia. Their simulated results showed that the ANFIS-PSO algorithm had the highest accuracy (99%), followed by the ANFIS-GA and ANFIS-BBO. Yomar et al. [24] conducted a study on modeling air pollution by integrating ANFIS and three metaheuristic algorithms, which were GA, PSO, and DE (differential evolution). They compared the results of these methods to those of classical ANFIS and found that these methods were more successful than classical ANFIS for modeling and predicting air pollution. To find an efficient and reliable streamflow forecasting model, Aghelpour et al. [25] developed an ANFIS model coupled with an ant colony optimization (ACO) algorithm to predict the streamflow of a river for 1 day, 2 days, and 3 days ahead. They found that the accuracy of the simple ANFIS model obtained was good. However, the hybridization of ANFIS with the ACO algorithm significantly improved the streamflow prediction accuracies by 12.1, 12.91, and 13.66% in comparison with a simple ANFIS model.

From the existing literature, and to our best knowledge, it appears that some authors have tried to model the performance of hybrid thermal power plants by means of ANFIS, but the hybridization of ANFIS with evolutionary algorithms (PSO and GA) has not been applied or investigated on the performance of hybrid thermal power plant (gas/steam/solar) systems yet.

The aim of this study is to investigate and demonstrate the capability of metaheuristic methods (PSO and GA) combined with ANFIS to accurately predict the performance of a hybrid solar/gas/steam power plant or an integrated solar combined cycle power plant. It promotes the hybridization of thermal power plants with renewable energy systems like solar systems, where possible, to mitigate the effects of climate change and shows that the challenge of prediction or real-time knowledge of the performance of such a complex plant for energy planning and sustainability can be overcome by making use of suitable machine learning approaches. Hence, this paper contributes to the body of knowledge in the application of metaheuristic approaches in the modeling of hybrid thermal power plants.

The structure of this paper is as follows: Section 2 provides the methodology followed in this investigation for the implementation and deployment of ANFIS-based approaches. Section 3 shows and discusses the results obtained, and Section 4 is a summary of the major findings of this research.

2. Materials and Methods

The methodology used in this work is shown in Figure 1. It takes into account data preparation, the procedure, modeling, and results analysis.

It should be noted that the coefficient of correlation, R², is usually less than 1. However, the higher it is (closer to 1), the more accurate is the model.

2.1. Data Collection

The data used in this paper were obtained from the Mendeley repository [26]. They were measured from an integrated solar combined cycle power plant comprising a gas turbine rated at 87.7 MW, a steam turbine rated at 37.1 to 42.4 MW, and a concentrated solar power system in the form of parabolic troughs rated at 16 MW thermal. Steam was produced in the steam generator from heat in the exhaust gas of the gas turbine and the heat-transfer fluid from the CSP field, as shown in Figure 2. The details of the hybrid power plant can be found in [27], and it is a good step towards carbon emission reduction or a transition from a fully fossil fuel-based energy generation to green or renewable energy.

Six parameters, namely ambient temperature (tamb), direct normal irradiance (DNI), mass flow rate of air (ma), mass flow rate of gas or exhaust gas from the gas turbine (mg), mass flow rate of fuel or natural gas (mf), and mass flow rate of heat-transfer fluid or thermal oil (mHTF) were used as input variables to the model, while the power output of the plant was used as the output parameter of the model. These input parameters were measured factors that influence the efficiency, or the power plant production, from a thermodynamic point of view. In total, 108 datasets were used in this investigation, and

Table 1. A sample of datasets used in this investigation.

tamb (°C)	DNI (w/m2)	ma (kg/s)	mg (kg/s)	mf (kg/s)	mHTF (kg/s)	Power Output (kW)
11.0767	162.547	214	219	5	85.34	128,037.2
20.9667	749.527	201.47	205.76	4.3	85.34	116,162.6
24.9233	447.82	196.51	200.81	4.3	85.33	112,374.3
25.42	434.645	196.95	200.82	4.3	85.33	112,389.7
22.9978	551.628	198.95	203.26	4.32	85.34	114,308.4
0.61	88.15	223.85	229.14	5.29	85.34	136,802.8
36.97	332.64	183.25	187.23	3.98	85.33	101,213
−0.61	64.63	225.16	230.5	5.34	85.34	138,351.3
33.025	137.475	187.24	191.48	4.24	85.34	104,705.4
44	0	174.84	178.72	3.88	85.34	93,186.5

shows a sample of these. For a full range of the data used in this study, tamb varied from −0.01 to 44 °C, DNI from 0 to 909.669 W/m², ma from 174.84 to 225.16 kg/s, mg from 178.72 to 230.5 kg/s, mf from 3.61 to 5.34 kg/s, mHTF from 85.33 to 85.34 kg/s, and the power output of the plant (W_tot) from 93,186.5 to 138,940.8 kW.

2.2. Model Development and Implementation

2.2.1. Adaptive Neuro-Fuzzy Inference System Modeling

The analysis carried out in this paper uses adaptive neuro-fuzzy system- or ANFIS-based approaches to analyze the ISCCPP’s performance. Neuro-fuzzy is actually a hybrid artificial intelligence technique that incorporates artificial neural networks and fuzzy logic. The development of this technique over the years has produced different types of fuzzy logic, including the outstanding adaptive neuro-fuzzy inference system (ANFIS), which was discovered in 1993 by Jang. An ANFIS model borrows the structure and learning capability of the artificial neural network and the decision-making protocol of fuzzy logic [28]. In other words, an ANFIS model is a hybrid method that combines the architecture of a fuzzy inference system with an artificial neural network and provides the solution to a complex non-linear problem. The nodes in the feedforward network are flexible and change with specific parameters of the membership functions associated with fuzzy rules. Usually, an ANFIS model comprises 5 layers, namely the fuzzy layer, product layer, normalized layer, de-fuzzy layer, and the total output layer, as shown in Figure 3. The de-fuzzy layer transforms a fuzzy set into a classical or crisp value. For training, the input and output data are obtained from the parameters of the problem being analyzed, and the ANFIS model presents a fuzzy inference system (FIS) for which the parameters of the membership function are tuned or refined by means of a classic optimization method or a metaheuristic optimization method. The optimization is done in the training step in order to minimize the error function or the difference between the targets or desired values and the output values [8]. A number of training algorithms have been presented in the literature, e.g., backpropagation as part of heuristic methods, evolutionary algorithms like particle swarm optimization (PSO), and genetic algorithm (GA), etc. In the present study, classic methods PSO and GA were tried for training and compared for model performance analysis and assessment.

The first layer starts with the incoming signals or inputs x and y, which are transferred to the neurons in Layer 2. The output can then be written using the membership function as:

$${\mathrm{O}}_{\mathrm{i}}^1={\mathsf{\mu }}_{{\mathrm{A}}_{\mathrm{i}}}(\mathrm{x}) \mathrm{or} {\mathrm{O}}_{\mathrm{i}}^1={\mathsf{\mu }}_{{\mathrm{B}}_{\mathrm{i}}}\left(\mathrm{y}\right),\mathrm{i}=1,2$$

(1)

where i is the node’s label, and A_i or B_i is the linguistic label (small, very small, large, very large, etc.) associated with the membership function. In other words, ${\mathrm{O}}_{\mathrm{i}}^1$ is the membership function of A_i and B_i, and it indicates the degree to which the given input, x or y, satisfies the quantifier, A_i or B_i. There are a number of membership function types that can be used: trapezoidal, triangular, bell-shaped, Gaussian, etc. It must be noted that the Gaussian membership function is determined by:

$${\mathsf{\mu }}_{{\mathrm{A}}_{\mathrm{i}}}(\mathrm{x})=\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left(\frac{\mathrm{x}-{\mathsf{\gamma }}_{\mathrm{i}}}{{\mathsf{\alpha }}_{\mathrm{i}}}\right)}^2\right]$$

(2)

The same can be written for input y.

For the bell-shaped membership function,

$${\mathsf{\mu }}_{{\mathrm{A}}_{\mathrm{i}}}(\mathrm{x})=\frac{1}{1+{\left[{\left(\frac{\mathrm{x}-{\mathsf{\gamma }}_{\mathrm{i}}}{{\mathsf{\alpha }}_{\mathrm{i}}}\right)}^2\right]}^{{\mathsf{\beta }}_{\mathrm{i}}}}$$

(3)

$$\mathrm{Layer} 2: {\mathrm{O}}_{\mathrm{i}}^2={\mathsf{\omega }}_{\mathrm{i}}={\mathsf{\mu }}_{{\mathrm{A}}_{\mathrm{i}}}(\mathrm{x})\ast {\mathsf{\mu }}_{\mathrm{B}\mathrm{i}}(\mathrm{y}), \mathrm{i}=1, 2$$

(4)

$$\mathrm{Layer} 3: {\mathrm{O}}_{\mathrm{i}}^3={\overline{\mathsf{\omega }}}_{\mathrm{i}}=\frac{{\mathsf{\omega }}_{\mathrm{i}}}{{\sum }_{\mathrm{i}=1}^2{\mathsf{\omega }}_{\mathrm{i}}}$$

(5)

$$\mathrm{Layer} 4: {\mathrm{O}}_{\mathrm{i}}^4={\overline{\mathsf{\omega }}}_{\mathrm{i}}{\mathrm{f}}_{\mathrm{i}}={\overline{\mathsf{\omega }}}_{\mathrm{i}}\left({\mathrm{p}}_{\mathrm{i}}\mathrm{x}+{\mathrm{q}}_{\mathrm{i}}\mathrm{y}+{\mathrm{r}}_{\mathrm{i}}\right)$$

(6)

where p, q, and r are coefficients of the i^th neuron or node.

Layer 5: This is the last layer, which has an elementary neuron.

It computes the entire output as:

$${\mathrm{O}}_{\mathrm{i}}^5={\sum }_{\mathrm{i}}{\overline{\mathsf{\omega }}}_{\mathrm{i}}{\mathrm{f}}_{\mathrm{i}}=\frac{{\sum }_{\mathrm{i}}{\mathsf{\omega }}_{\mathrm{i}}{\mathrm{f}}_{\mathrm{i}}}{{\sum }_{\mathrm{i}}{\mathsf{\omega }}_{\mathrm{i}}}$$

(7)

2.2.2. Particle Swarm Optimization

Particle swarm optimization (PSO) is a computational technique that optimizes a stochastic or non-linear problem by iteratively improving a candidate solution relative to a defined fitness function. It was first presented by Kennedy and Eberhart in 1995 and simulates social behavior like the movement of organisms in a bird flock or fish school [30]. The approach helps solve a numerical problem by generating candidate solutions called particles and moving them around in the search space until the optimal solution is attained. The movement of each particle is guided by its knowledge of its personal best or local best position and the global best position of the swarm or population. In this way, the swarm moves towards the best solution. The flowchart of the method is shown in Figure 4.

The velocity and the position of a particle are updated as follows [31]:

$${\mathrm{v}}_{\mathrm{i}}\left(\mathrm{t}+1\right)={\mathrm{v}}_{\mathrm{i}}\left(\mathrm{t}\right)+{\mathrm{C}}_1{\mathrm{r}}_1\left(\mathrm{t}\right)\left[{\mathrm{x}}_{\mathrm{P}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{i}}\left(\mathrm{t}\right)-{\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right)\right]+{\mathrm{C}}_2{\mathrm{r}}_2\left(\mathrm{t}\right)\left[{\mathrm{x}}_{\mathrm{G}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{i}}\left(\mathrm{t}\right)-{\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right)\right]$$

(8)

$${\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}+1\right)={\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right)+{\mathrm{v}}_{\mathrm{i}}(\mathrm{t}+1)$$

(9)

where ${\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right)$ and ${\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}+1\right)$ are the particles’ positions in the search space at time step t, and t + 1. ${\mathrm{v}}_{\mathrm{i}}\left(\mathrm{t}\right)$ and ${\mathrm{v}}_{\mathrm{i}}\left(\mathrm{t}+1\right)$ are the velocity vectors of particle i at time step t and t + 1. ${\mathrm{x}}_{\mathrm{P}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{i}}\left(\mathrm{t}\right)$ and ${\mathrm{x}}_{\mathrm{G}\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t},\mathrm{i}}\left(\mathrm{t}\right)$ are the personal (local) and global best positions of particle i. ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ are the acceleration factors. ${\mathrm{r}}_1\left(\mathrm{t}\right)$ and ${\mathrm{r}}_2\left(\mathrm{t}\right)$ are random numbers between 0 and 1.

2.2.3. Genetic Algorithm

A genetic algorithm (GA) is one of the algorithms that can explore a wide search space. The idea was introduced by John Holland in the 1960s and 1970s and emulates natural biological genetics and evolution. An initial and unlimited population of solutions is randomly generated. Each solution is evaluated by calculating the values of the objective or fitness function. Solutions with fitter chromosomes stand a higher chance of being selected to participate in the next generation. Through this probabilistic approach, an intermediate population with a higher representation of the strong species is created. The intermediate population is allowed to undergo crossover through mutation, and the next population is randomly generated. This process continues until the termination condition is satisfied [32]. In the genetic analogy, which is a competition-based process, individuals or candidate solutions represent chromosomes, and the variables represent genes. Each solution is given an eligibility score that presents an individual’s “competition” abilities [33,34]. Figure 5 illustrates the flowchart of a genetic algorithm.

An offspring population is created through the crossover operation, which, for example, interchanges a subsequence of two of the selected chromosomes to generate two offspring. It may be noted that GA makes use of a selection operator to implement the principle of natural selection or survival of the fittest. Therefore, unlike the PSO, where all the population members are kept in the process, not all the candidate solutions are kept as part of the population in the genetic algorithm [35].

2.2.4. Hybrid ANFIS-PSO Modeling

For the hybrid ANFIS-PSO model, the data were divided into 69 datasets for training and 39 datasets for testing, and both were loaded. A basic fuzzy inference system (FIS) was then generated. Its parameters were set and tuned in the training step by means of particle swarm optimization (PSO). In the optimization phase, the error function between the model outputs and the expected values is minimized until the required number of iterations (which was set at 1000) is reached. From this point, the results were plotted, the testing phase could start, and its corresponding results were equally plotted. The main steps of the model are illustrated in the flowchart in Figure 5. Once training and testing plots were obtained, the results data were exported to Microsoft Excel to generate regression plots and determine the coefficient of correlation, R².

2.2.5. Hybrid ANFIS-GA Modelling

The approach followed for the hybrid ANFIS-GA model was similar to that of the ANFIS-PSO up to the setting of the FIS parameters. From this point, an option could be chosen to conduct the training phase either by the particle swarm optimization or by the genetic algorithm approach, as depicted in Figure 6.

2.3. Procedure

The model was implemented in the Matlab 2022a environment. The data were divided into 64 to 70% datasets for training, while the remaining 36 to 30% were used for testing. The ANFIS model was first run with a classic training algorithm to get basic results. After importing data into the neuro-fuzzy designer, a fuzzy inference system (FIS) was generated using a Sugeno fuzzy inference system, and a subtractive clustering approach (with default parameters) was chosen, as the number of input parameters was relatively high. The set FIS was trained using a hybrid learning strategy that combines gradient descent and linear and least squares methods. The maximum number of epochs was set to 100 (refer to Section 3.1 for further details). After this stage, the FIS was tested for generalization capability. Second, the ANFIS model was optimized with particle swarm optimization (refer to Section 3.2 for details), and finally with a genetic algorithm (refer to Section 3.3 for details) for comparison purposes.

3. Results and Discussion

3.1. ANFIS

For the purpose of this investigation, the data (training data and testing data) were loaded into the model, a FIS was generated and trained, and then its reliability was tested with the training data and the testing data, as shown in Figure 7. The Takagi–Sugeno fuzzy inference system (FIS) was used, and eight Gaussian membership functions were used for each input variable, while a linear membership function was used for the output. A total of eight rules were applied to define the FIS, whose structure is shown in Figure 8.

It can be observed from Figure 7 that the set FIS is reliable for training and testing because the errors are negligible compared to the range of total power output.

Figure 9 displays a rule viewer of the model, while Figure 10 shows the surface viewer for two input parameters, namely ambient temperature (tamb) and direct normal irradiation (DNI). The output power (W_output) in Figure 9 is to be multiplied by 10⁵. The rule viewer also shows reasonable results.

As expected, it can be observed from the surface viewer that the power output increases as the ambient temperature and the direct normal irradiation increase.

The ANFIS model appraisal was also conducted by means of the coefficient of correlation, R², obtained from the regression plots shown in Figure 11 for training, testing, and overall results.

It emerges from Figure 11 that the ANFIS model performed well, with an overall correlation coefficient of 0.9991.

3.2. ANFIS-PSO

The Matlab codes for running ANFIS-PSO were obtained from [28]. Particle swarm optimization is one of the widely known evolutionary or metaheuristic algorithms that are used to solve complex non-linear problems. It mimics the swarming behavior of creatures (insects, animals, or birds). A swarm of particles is initially randomly generated. In the iterative search for the optimum solution, each particle updates its position based on its prior experience and the experience of its neighbors. The velocity with which the particle flies the search space is also consequently updated until the optimal or global best solution is found. The details of PSO can be found in [31,35]. For the purpose of this study, the following parameters were used: inertia weight, w = 1, inertia weight damping ratio, wdamp = 0.99, personal learning coefficient, c₁ = 1, global learning coefficient, c₂ = 2, number of rules = 10, maximum number of iterations = 1000, and the population size or swarm size, nPop = 25. The corresponding ANFIS-PSO results are shown in Figure 12 and Figure 13.

It is clear from Figure 12 and Figure 13 that the ANFIS-PSO model outputs track very closely the expected or target data for both training and testing. For more insight, the regression plots are displayed in Figure 14.

It can be observed from Figure 14 that the ANFIS-PSO model performed well for training and testing, with an overall coefficient of correlation, R² = 0.9994. This performance metric is slightly higher than the one obtained with the ANFIS model, but the computation time was longer.

3.3. ANFIS-GA

The genetic algorithm is one of the most powerful optimization algorithms. It emulates the so-called biological evolution process in nature, or the natural selection of the fittest. The basic elements of a GA comprise the fitness function, which is to become optimized, the population of chromosomes, the selection, by means of an operator, of chromosomes that will reproduce, and the production through mutation of the next generation in a random fashion. The crossover operation is employed to generate offspring. It swaps a sequence or subsequence of the two selected chromosomes to create one or two offspring, depending on the strategy followed. The uniqueness of GA compared to PSO resides in the fact that, unlike PSO, GA has the selection operator for easy optimization, and not all individuals are retained as members of the population. This last factor somewhat reduces the computation time. The main steps in the deployment of a GA are the initialization of the population, calculation of the fitness function, crossover, mutation, selection of survivors, and finally termination of the process, keeping the best if the criteria is met. The details of GA can be found in [31].

In this investigation, the parameters of the ANFIS section were kept as in the case of the PSO (number of rules = 10, nPop = 25, and maximum iterations = 1000), and the following parameters were used for the GA: crossover percentage, pc = 0.4; number of offsprings nc = 2*round(pc*nPop) = 10; mutation percentage, pm = 0.7; number of mutants, nm = round(pm*nPop) = 18, gamma = 0.7; mutation rate, mu = 0.15; and selection pressure, beta = 8. The corresponding ANFIS-GA results are shown in Figure 15 and Figure 16.

It is obvious from Figure 15 and Figure 16 that as in the case of ANFIS-PSO, the ANFIS-GA model outputs track very closely the expected or target data for both training and testing. The corresponding regression plots are depicted in Figure 17.

It should be noted in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 that Output is the predicted power or the value of the power output predicted by the model, while Target is the measured value of the output power of the plant.

It can be observed from Figure 17 that the ANFIS-GA model performed very well for both training and testing, with an overall correlation coefficient, R², equal to 0.9997, which was slightly higher than the one obtained in the case of ANFIS-PSO, and the corresponding MSE or RMSE was relatively lower than in the case of ANFIS-PSO. It was also noted that the computation time was relatively shorter than that required for ANFIS-PSO, which may be attributed to the reduction in the population size due to the selection of the fittest. This finding confirms the theory about GA. The ANFIS model is susceptible to the problem of local minima during the training process, owing to the use of heuristic methods. To mitigate this issue and achieve the global optimal solution, researchers have increasingly turned to metaheuristic or evolutionary techniques such as particle swarm optimization (PSO) and genetic algorithms (GA) [24]. This study reaffirms that optimizing ANFIS with PSO and GA yields significantly improved accuracy, resulting in minimal disparities between predicted output values and the actual target output values.

4. Conclusions

In this investigation, the ANFIS model and the hybrid ANFIS model combined with evolutionary algorithms (PSO and GA) were alternatively employed to model and analyze the performance of a combined cycle gas turbine power plant integrated with a concentrated solar power system utilizing parabolic troughs. The results demonstrated remarkable accuracy and efficacy across all the models, with coefficient of correlation (R²) values reaching an impressive 0.9991 for ANFIS, 0.9994 for ANFIS-PSO, and 0.9997 for ANFIS-GA. Additionally, the root mean square errors were consistently minimal, substantiating the precision of these ANFIS-based approaches. Notably, the accuracy exhibited an upward trajectory as the foundational ANFIS model was enriched through integration with metaheuristic optimization techniques. The application of evolutionary algorithms (PSO or GA) to hybridize ANFIS showcased its robustness and reliability in analyzing and predicting the integrated solar combined cycle power plant’s performance. However, it is essential to acknowledge that this hybridization, while enhancing accuracy, also led to an increase in computation time. Remarkably, among the ANFIS-based methodologies explored, the ANFIS-GA model emerged as a standout performer for the scenarios investigated in this study. The significance of this work is underscored by its revelation of the potential inherent in ANFIS-based methodologies for accurate performance prediction within hybrid thermal power plants. These methodologies present themselves as practical alternatives to more phenomenological approaches. As we look towards the future, several avenues for further exploration come to light:

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Delving into the influence of clustering techniques, varying parameters, and the intrinsic model parameters on the ANFIS approach’s performance.

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Thoroughly investigating the pivotal parameters of the hybrid ANFIS-PSO and ANFIS-GA models and discerning their impact on accurately forecasting the integrated solar combined cycle power plant’s performance.

In conclusion, this study highlights the efficacy of ANFIS-based methodologies in precisely predicting hybrid thermal power plant performance. By leveraging the capabilities of evolutionary algorithms, these methodologies can serve as invaluable tools, offering a level of accuracy that is both robust and practical when compared to traditional phenomenological methodologies. The performance of power plants is related to the output power, but the direct impact on the environment can’t be ignored, noting that pollution, especially for thermal power plants is reduced when high efficiency or performance is maintained. This pollution reduction is due to, among other things, lower fossil fuel consumption for the same output power. The major limitation in this work was the non-availability of data on the performance of hybrid thermal power plants or integrated solar combined cycle power plants.