Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000

Metaheuristic algorithms are extensively utilized to find solutions and optimize complex industrial systems' performance. In this paper, metaheuristic algorithms are utilized to predict the optimum value of the operational availability of a cooling tower in a steam turbine power plant. These techniques have some flaws like poor convergence speed, being stuck in local optima, and premature convergence. For this purpose, a novel efficient stochastic model is proposed for a cooling tower that is configured with six subsystems. The Markovian birth-death process is utilized to develop the Chapman-Kolmogorov differentialdifference equations. All the random variables are statically independent, and repairs are perfect. Failure rates are exponentially distributed, while repair rates follow the arbitrary distribution. Steady-state availability (SSA) of the system is derived concerning various failure and repair rates. The sensitivity analysis of SSA is also performed to identify the most critical component. Further, system availability is optimized using genetic algorithm (GA) and particle swarm optimization (PSO) because they are found to be more suitable for such types of problems. It is revealed that the PSO outperforms GA in predicting the availability of cooling towers used in steam turbine power plants. INDEX TERMS Particle Swarm Optimization; Genetic Algorithm; Cooling Tower; Availability; Markov Modeling


I. INTRODUCTION
In physics, energy is termed as the capability to do work and occurs in the form of nuclear, thermal, renewable, electrical, chemical, kinetic, and potential. Energy remains in practice to transfer from one body to another. It is classified according to its nature. The heat converts itself as thermal while work done by becomes mechanical energy. But all these kinds are associated with motion. Electrical energy is termed electricity generated through the transformation of other energy sources. This transformation/conversion is done in power plants. These industrial entities generate electricity from coal, steam, thermal, wind, tidal, and nuclear-using generators and transform mechanical energy into electrical energy and transfer it to the grid to use it into society and industries. Nuclear, thermal, solar, wind, and coal feed power plants are situated in various countries and dominated according to the availability of primary energy sources as fuel. Thermal power plants most established plants in various countries. In thermal power plants, water is heated up to generate steam, and then at high pressure, it passes through the turbine. The turbine spin operates the generator, and motion started between the coil of wire and magnet available in the generator. This whole process resulted in electricity flow inception. Very high heat is generated during this process, and waste heat is rejected to the atmosphere with the help of heat rejection devices, namely cooling towers. It uses the process of evaporation of water to remove heat. Cooling towers are vast entities divided into several zones like rain zone, fill packing zone and spray zone. These systems are very complex, and the operation of such systems is very crucial. The failure causes complete power plant failure that causes a severe effect on societal and industrial production. Hence, it becomes necessary to handle cooling towers with high reliability. The cooling tower and steam generators of power plants have been simulated using an artificial neural network [1]. A modified and improved version of the butterfly optimization algorithm utilized in combined cooling, heat, and power (CCHP) system operated by proton exchange membrane fuel cells (PEMFC) [2]. A systematic review of the optimization algorithms applying sustainability energy is presented in the literature [3]. A study on the performance management of solar thermal power plants has been carried out using multiobjective dynamic programming for optimization [4]. An optimal design using the Cuckoo search algorithm (CSA) was proposed for a solar-hybrid cogeneration system [5]. The results of the cuckoo search have been compared with a genetic algorithm using MATLAB toolbox and an effective time-saving procedure using simple parallel computing is the key finding of this study. A ground-breaking technology developed for geothermal and cooling cogeneration systems [6]. To improve the efficiency zeotropic mixtures have been used in subsystems. The thermodynamical and optimization characteristics of this system are also analyzed. Thermo-flow measures of the dry cooling system designed with only one tower in power plants have been investigated [7]. An experimental study has been carried out to investigate the efficiency of the wet cooling tower having diverse packing compaction using the artificial neural network and particle swarm optimization algorithm [8]. An analysis to reduce water intake for cooling towers used in thermal power plants as a pilot study [9]. Membrane capacitive deionization (MCDI) has been done in this study. A response surface methodology is developed through which optima process conditions of MCDI cooling tower can be determined given cost and efficiency. Supercritical combined gas-steam cycle systems analyzed technically as well as economic aspects [10]. Here, it is recommended that investments in adopting components of the steam part may be balanced from higher profit. The impact of the integration of hybrid thermal plants into energy complexes has been investigated [11]. A generalized methodology for the selection and calculation of technology schemes for mini plants has been utilized. The concept of Thermoeconomic has been chosen for deciding the criteria for the best option of placing a mini thermal plant. Failure evaluation of power industry instruments done using probabilistic arguments [12]. It is concluded through experimental results that the proposed framework gain superiority over other typical data-based approaches. Several techniques like failure mode and effect analysis, reliability block diagram, semi-Markov process, minimal cut set approach, and Markov birth-death process, exist in the literature for reliability evaluation of industrial systems with a certain set of assumptions [67]. But when failure and repair rate of components follow memoryless property and exponential distribution then the Markov birth-death process approach is recommended [79][80][81]. Nature-inspired algorithms (NIAs) are very efficient algorithms used to find solutions and optimize complex industrial systems' performance. NIA is a group of efficient methodologies derived from natural activities [13]. In the present study, NIAs have been used to predict the optimum value of the operational availability of a cooling tower in a steam turbine power plant. These techniques have some lacks like being stuck in local optima and slow convergence rate. For this purpose, an efficient stochastic model has been proposed for cooling towers configured with six subsystems. Markovian birth-death process has been utilized to develop the Chapman-Kolmogorov differential-difference equations. All the random variables are statically independent, and repairs are perfect. Failure rates are exponentially distributed, while repair rates follow the arbitrary distribution. Steady-state availability of the system has been derived concerning various failure and repair rates. Further, system availability has been optimized using the Genetic Algorithm (GA) [50][51] and Particle Swarm Optimization (PSO) [52]. The results will be shared with plant personnel. It is revealed the PSO outperforms GA in predicting the availability. In short, the major contribution of this work is highlighted as follows:  A novel efficient stochastic model is developed using the concept of cold standby redundancy for a colling tower of Steam Turbine Power Plants.  Availability and profit analysis of cooling towers are achieved by considering all failure rates as exponentially distributed while repair rates as arbitrary.  Sensitivity analysis of availability function is carried out to identify the most critical component.  Metaheuristic techniques namely Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) are applied to obtain the optimum availability of the proposed model.  Validation of obtained results is statistically investigated using Mann-Whitney U Test.
The remainder of this paper is organized as follows: Section 2 presents some work related to the work presented in this manuscript. Nomenclature, system description, and assumptions are provided in Section 3. Section 4 presents the stochastic modeling and sensitivity analysis of the cooling tower subsystem. Profit analysis is depicted in Section 5. Section 6 depicts materials and methods viz. reliability measures, Markov process, simulation environment, and optimization strategies. Numeric results and discussions are presented in Section 7. Finally, Section 8 presents the concluding remarks with some future directions.

II. RELATED WORK
Many studies have been done by researchers related to steam turbine power plants and their subsystems. Reliability aspects of combined cycle power plants (CCPP) were studied through mathematical modeling and availability analysis because generators play a critical role in the operation of CCPP and steam turbine power plants [15]. Operational availability investigation of these systems has been done using the Markovian approach [14]. Advanced data mining techniques supported by data like support vector regression and data reconciliation have been used in air-cooling condensers o thermal power plants [16]. The applicability of dry cooling towers as condensers has been observed in geothermal power plants [17]. A reliability-based approach has been proposed for maintenance in combined cycle power plants based on identifying the most critical component of it [18]. Some amendments in the VGB guidelines suggested improving the design for manufacturing cooling towers in power plants [19]. The effect of climate change on cooling towers' performance was investigated to optimize technical, economic, and design perspectives [20]. Performance evaluation of thermal power plants to improve accuracy and reliability model-based data reconciliation techniques has been used [21]. A systematic review was done to observe the performance of natural draft dry cooling towers by using inlet air spray [22]. The use of cooling towers in chimneys and solar power plants is discussed in [23]. The technical aspects of cooling towers associated with river basins have been appended in [24]. A new design of the hybrid cooling system for large-scale steam turbine power plant generators was developed to assess the performance [25]. The water flow in the power plant's cooling towers also shows a significant effect, and a numerical investigation has been done in this direction [26][27]. An investigation was made to assess thermal power plants' sustainability in low water areas using cooling towers [28]. Fault tree analysis and system reliability evaluation have been done for combined cycle power plants [29]. The performance assessment and optimization of spray cooling system design in solar power plants is studied in [30]. Machine learning techniques are frequently used to optimize evaporation-based cooling towers, and performance is optimized using particle swarm optimization [31]. A novel model exists in the literature to understand water use in power plants [32][33][34][35][36]. A prediction model for performance and cost analysis in hybrid cooling towers in power plants has been developed [33]. The stochastic Petri nets technique is utilized to develop efficient and optimized maintenance strategies for a coal-fired power plant [34]. The mathematical model for efficiency prediction of thermal plants' cooling towers was discussed and extended it up to three towers [35]. A new methodology for the reliability evaluation of thermal power plants is proposed under different assumptions [37]. Energy-saving benefits and economic evaluation of cooling towers using fuel gas are analyzed [38]. Variable ambient conditions that impact the cooling system of power plants are investigated using mathematical modeling [39].
During the last few years, reliability analysis and optimization of thermal plants have attracted researchers. A Multicriteria decision-making model for optimization of operational routes in thermal power plants has been developed [40]. A study for reliability improvement of power systems using the idea of transmission line switching has been conducted with ac power flows [41]. The reliability evaluation of power systems using power outage modeling is carried out by the researcher [42]. The combination of Markov and matrix methods is extensively used in reliability assessment [43]. The reliability evaluation of wind power plants also attracted researchers [44,45]. Recently, several advanced metaheuristics approaches were developed for the performance optimization of industrial systems [46][47][48]67]. Some popular optimization algorithms namely Chimp, dynamic Levy flight chimp, and weighted Chimp optimization algorithms are proposed for the optimization of industrial systems [68][69]71]. For sonar dataset classification some improved migration models based on biogeography-based optimization has been proposed using neural network [70]. Recently, a new binary meta-heuristic algorithm Binary Chimp Optimization Algorithm (BChOA) is developed for optimization problem solving [72]. Various algorithms like chaotic fractal walk trainer, whale trainer, modified grey wolf optimizer, and fuzzy grasshopper optimizer are proposed for sonar data analysis [73][74][75][76]. Sensitivity analysis of the algorithms and parameter initializations in metaheuristics is also important and has wide applicability in optimization algorithms [77][78]. The literature review observed that several studies were conducted on performance analysis of steam turbine power plants. But reliability aspects of steam turbine power plants and their components like cooling towers are still not explored extensively. The optimization of reliability measures of cooling towers and steam turbine power plants were not discussed yet. Therefore, in this paper, to optimize the availability of cooling towers, nature-inspired algorithms like genetic algorithms and particle swarm optimization are used.

A. NOMENCLATURE
Following notations are utilized to develop the mathematical model of the cooling tower:

B. SYSTEM DESCRIPTION
A concise depiction of a cooling tower in a steam turbine power plant has been given in this part. Cooling tower mainly comprises seven parts: hydro turbine, pressure-driven valves, water splash framework, programmed deaerator valves, cooling water siphon, engine valves, and standpipe. All segments are organized in a series arrangement. The concept of cold standby redundancy at the component level is adopted. All the time-dependent random variables associated with failure and repair rates of components are statistically independent. The failed component after the repair was performed as a new one. A sufficient repair facility is available with the system. The visual portrayal of segments is attached in Figure 1. The working of different components of the cooling tower subsystem are briefed as under: 1) Hydro turbine: It comprises one unit of a hydro turbine. This current unit's disappointment causes total framework disappointment as it is associated with different units in series. 2) Hydraulic valves: It comprises one bunch of Hydraulic valves. This present unit's failure causes total framework failure as it is associated with different units in series.
3) Water supply framework: It comprises one unit of water splash framework. This current unit's failure causes total framework failure as it is associated with different units in series. 4) Programmed deaerator valves: It comprises two arrangements of programmed de-aerator valves; one is employable, and the other is on standby. The disappointing pace of both units is the same, and the disappointment of the two units keeps an eye on framework disappointment. 5) Cooling water siphon: It comprises one unit of Cooling Water Pump. This present unit's failure causes total framework failure as it is associated with different units in series. 6) Engine valves: It comprises one bunch of engine valves.
This current unit's failure causes total framework failure as it is associated with different units in series. 7) Standpipe: It comprises one bunch of engine valves. This present unit's failure causes total framework failure as it is associated with different units in series.

C. ASSUMPTIONS
For the development of the model, several assumptions are incorporated as follows: 1) The failure rate of subsystems follows an exponential distribution, whereas repair rates are arbitrarily distributed (as per Table 1). 2) Random variables are independent and identical to each other. 3) Case of concurrent failures not considered in model development. 4) Perfect repairs and switch-over devices.

IV. STOCHASTIC MODELING AND SENSITIVITY ANALYSIS OF COOLING TOWER
By using simple probabilistic arguments and the Markov birth-death process, a novel stochastic model is developed as shown in Figure 2. Analytical solution of the proposed stochastic model is obtained using the supplementary variable technique. The differential-difference equations are as follows: Dividing both the sides of Eq. (1) by , we get VOLUME -, -

V. Profit analysis
Let K1 be the total revenue per unit uptime of the system and K2 be the total repair cost then profit incurred to the system model in steady-state is obtained as: The value of profit function after considering the arbitrary values for K1 = 3500 and K2 = 500 is defined in Eq. (33) as follows:

RELIABILITY
It is the probability that the system performs its intended function under stated operating conditions without any failures. It ranges from 1 to 0.

AVAILABILITY
It is a kind of probability states that a system is available for utilization as and when required. It ranges from 0 to 1. At time t=0, availability is 1, and at infinity, its value is 0. It is defined as the ratio of uptime and total life duration [67].

B. Markov Process
It is a well-known stochastic process used in reliability and queuing theory. Russian mathematician A.A. Markov develops it. It states that the behavior of future states depends on the present conditions and is independent of the past [79][80][81]. A process of continuous time discrete space is utilized in the reliability estimation of mechanical and electrical systems.

C. Optimization Strategies
Computational intelligence-based optimization strategies are widely used to obtain optimal solutions to several problems of humankind. Metaheuristics are one of the prominent computational intelligence-based optimization strategies that are problem independent in nature. Meta-heuristics are generally classified into three main classes viz. evolutionary, physics-based, and swarm intelligence algorithms. Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) are some of the most popular algorithms [53]. These algorithms have achieved more efficient results for various problems such as availability optimization of process industries. Also, GA and PSO are not influenced by the size and non-linearity of problems [49]. There are many other algorithms such as Chimp optimization, dynamic Levy flight chimp, and weighted Chimp optimization [54][55][56][57][58][59][60][62][63][64][65][66] that can be applied in the future for estimating their performances in availability computations of STPP.

SIMULATION ENVIRONMENT
For simulating the experiments, we have used MATLAB R2019a on the Windows 10 64-bit operating system with 8 GB of RAM and Intel Core i5 8th generation CPU.

GENETIC ALGORITHM
GA is one of the well-known Evolutionary Computation (EC) approaches that is inspired by the biological evolution process [50][51]. The process of EC is mainly characterized by the fitness computation of individuals who participate in achieving global optimization. Individuals with fitness greater than the specified threshold are retained for further evolution.
Once the problem is encoded using decision parameters, the optimum solution can be obtained. GA starts by generating an initial random population based on initialization of values of decision parameters followed by computation of fitness of individuals as depicted in Figure 3.
The overall search process comprises of the following steps: 1. Problem encoding and generation of an initial random population (chromosomes) based on threshold values (lower bound and upper bound) of decision parameters. In our case, failure rates and repair rates of various subsystems of the cooling tower of steam turbine power plant are considered as decision parameters and their threshold values according to distributions (exponential in case of failure rates, whereas arbitrary for repair rates). 2. Computation of fitness of individuals according to fitness function specified. In our case, the availability function (Av) is considered as a fitness function that needs to be optimized. 3. Identification of terminating conditions (i.e., stopping criteria for convergence of algorithm after meeting specified constraints). If terminating conditions are satisfied, the algorithm converges, otherwise, it moves further.

Select of parents (based on threshold values) and
generation of new offspring by applying genetic operator (crossover and mutation). 5. Go to step 2.

Figure 3: Flowchart of Genetic Algorithm
Numerous variations of genetic operators (i.e., crossover and mutation) are presented in the literature. The main aim of these operators is to exchange genes so that a new population with high fitness values can be obtained. In a crossover, the parts of chromosomes are shuffled in different ways whereas in mutation new genetic material is introduced as a replacement of older one. One-point crossover and uniform mutation are adopted in this work for a generation of new offspring.

PARTICLE SWARM OPTIMIZATION (PSO)
PSO developed by Eberhart and Kennedy [52] is one of the efficient swarm intelligence-based metaheuristic techniques that are inspired by birds' social behavior. The principal mechanism behind Swarm intelligence techniques is the coordination of individuals for finding out optimal solutions to a problem. The term 'particles' refers to the individuals in a problem space, whereas swarm refers to their population. To obtain an optimal solution, particles move themselves to an appropriate position at a specified speed referred to as 'velocity'. For achieving a 'global best' solution, at every instance, each particle identifies its best position known as 'personal best'. This process is generally referred to as learning from experience. The algorithm converges upon either satisfying the stopping criteria or reaching maximum iterations as specified.

Figure 4: Flowchart of Particle Swarm Optimization algorithm
The position of the particles and their velocity is updated on every iteration. The working mechanism of PSO is depicted with the help of a flowchart in Figure 4. The basic Eqs. of PSO are given in Eq. 33 and 34. The position of a particle i at time t in the search space is denoted by ( ). The new position of particle i is achieved by adding the velocity ( ) to its current position, as stated in (33).
Here, ( + 1) indicates the combined value of inertia coefficient, cognitive component, and the social component as given in eq. 34 In eq. (34), indicates the inertia coefficient, ( ) the initial velocity, ( ) to be personal best, ( ) indicates the global best, and 1 and 2 are acceleration coefficients. On every iteration, the position of individuals is getting updated based on various parameters viz. personal best, global best, and the velocity of movement from one position to another (towards global optimization).

Problem encoding of cooling tower subsystem using GA and PSO
To obtain the maximum availability of the cooling tower subsystem, the optimum values of failure and repair rates have been computed using GA and PSO. The availability function (Eq. 24) consisting of failure and repair rates of several cooling towers' subsystems (as given in Table 1) is considered as a fitness function. Optimizing the availability of cooling towers is measured as single-objective optimization subject to certain constraints of failure and repair rates. The optimization of the availability function has been checked against varying certain parameters of GA viz. population size, evolutions, crossover probability, and mutation probabilities, as shown in Tables 6-9.
The optimization of the availability function has been checked against varying certain parameters of PSO (i.e., iteration, population size, and damping ratio) by keeping the values of inertia weight, p-best, and g-best to be 1, 1.5, and 2, respectively, as shown in Tables 10-12.

D. Mann-Whitney U Test
The Mann-Whitney U test is the non-parametric analogous of the independent t-test. It is used for testing the significant difference between two populations or to identify both samples are drawn from the same distribution. If the null hypothesis is accepted, then both the samples have more or less the same values. The test statistic can be defined as: where n1 and n2 are sample sizes, R1 and R2 are rank sums of corresponding samples.
By using (35), the test statistics is as follows: In the present analysis, a statistical comparison of GA and PSO results has been done using Eq. (36), as both the samples are independent and assumptions of parametric tests like normality, heteroscedasticity does not achieve [61].

VII. Numerical Results and Discussion
Here, the influence of variation in the failure rate of various subsystems of the cooling tower is estimated on availability (Eq. 24) and profit function (Eq. 32) for an arbitrary set of parametric values. The initial parametric values are as follows: The variation is observed concerning hydraulic valves failure rate (ϑ2). It is revealed that hydro turbine, cooling tower pump, and standpipe are the most sensitive components in the cooling tower. By changing the value of ϑ1=0.006 to ϑ1=0.06 and keeping rest values as constant, it is revealed from Table 2, that the availability cooling tower reduces up to 0.534392 while in a subsystem where the provision of standby unit is given availability does not show high variation. From Table 3, the same behavior is depicted for the profit function. The availability and profit function values decline by increasing the failure rates of all the components. Availability of the system decreases by 5.35% approximately with the increase in failure rate 1 = 0.006 1 = 0.06. Similarly, the availability of the system decreases 6.53%, 7.49%, 5.82 %, 7.32%, and 4.21% approximately with the increase in the failure rate of 3 from 0.0009 to 0.009, 4 from 0.00075 to 0.0055, 5 from 0.0018 to 0.035, 6 from 0.0054 to 0.15 and 7 from 0.0008 to 0.052 respectively and in Table 3, it is revealed that profit of the system decreases 6.99% approximately with the increase in failure rate 1 from 0.006 to 0.06. Similarly, the profit of the system decreases 8.05%, 8.95%, 7.40%, 8.79%, and 6.02% approximately with the increase in the failure rate of 3 from 0.0009 to 0.009, 4 from 0.00075 to 0.0055, 5 from 0.0018 to 0.035, 6  from 0.0054 to 0.15 and 7 from 0.0008 to 0.052 respectively. From Table 4, it is found that the availability of the system shows a 2.31% enhancement along with the increase in repair rate 1 from 0.09 to 2.1. Also, the availability of the system is improved by 2.23%, 2.19%, 2.23%, 2.33%, and 2.22%, respectively with the increase in repair rate 3 from 0.033 to 0.95, 4 from 0.26 to 0.2, 5 from 0.075 to 1.9, 6 from 0.066 to 0.95 and 7 from 0.045 to 0.81 respectively and Table 5 reflected that profit of the system increases 2.72% approximately with the increase in repair rate 1 from 0.09 to 2.1. Similarly, profit of the system increases 2.65%, 2.60%, 2.64%, 2.74%, and 2.63% approximately with the increase in repair rate 3 from 0.033 to 0.95, 4 from 0.26 to 0.2, 5 from 0.075 to 1.9, 6 from 0.066 to 0.95 and 7 from 0.045 to 0.81 respectively. From Tables (4)(5), it is observed that water spray pumps and motor valves are highly influential components. Any increment in their repair rate significantly contributes to increasing system availability and profit. Availability showed a highly inclined trend when the variation was made in repair rates as given in Table 4.     As depicted in Table 6, the availability of the cooling tower subsystem has been computed by varying the population size. The values of evolution, mutation, and crossover are kept fixed at 400, 0.65, and 0.85, respectively. It is observed that the maximum availability has been obtained when the population size is 15. An increase in population results in decreasing overall availability. The optimum values of failure and repair rates of different subsystems can also be seen from Table 6 against maximum availability when the size of the population is 15. The relationship of availability vs. population in GA as computed for the cooling tower subsystem is also depicted in Figure 5. The impact of evolution has also been observed towards optimizing availability computations of cooling tower subsystems. As shown in Table 7, the availability of the cooling tower subsystem has been computed by varying evolutions. The values of population, mutation, and crossover are kept fixed at 60, 0.65, and 0.85, respectively. It is observed that the maximum availability has been obtained on 360 evaluations. An increase in evolution results in an increase in overall availability. The optimum values of failure and repair rates of different subsystems can also be seen from Table 7 against maximum availability when the value of evolution is 360. The relationship of availability vs. evolutions in GA as computed for cooling tower subsystem is also depicted in Figure6.
The impact of crossover rate has also been observed towards optimizing availability computations of cooling tower subsystems. As shown in Table 8, the availability of the cooling tower subsystem has been computed by varying the crossover rate. The population, evolution, and mutation values are kept fixed at 60, 400, and 0.65. It is observed that the maximum availability has been obtained when the value of crossover is 0.4. The optimum values of failure and repair rates of different subsystems can also be seen from Table 8 against maximum availability when the value of crossover is 0.4. The relationship of availability vs. crossover probability in GA as computed for the cooling tower subsystem is also depicted in Figure 7. The impact of mutation rate has also been observed towards optimizing availability computations of cooling tower subsystems. As shown in Table 9, the availability of the cooling tower subsystem has been computed by varying the crossover rate. The values of population, evolution, and crossover are kept fixed at 60, 400, and 0.65, respectively. It is observed that the maximum availability has been obtained when the value of mutation is 0.48. The optimum values of failure and repair rates of different subsystems can also be seen from Table 9 against maximum availability when the value of mutation is 0.48. The relationship of availability vs. mutation probability in GA as computed for the cooling tower subsystem is also depicted in Figure 8.  Cross Over VOLUME -, -

Figure 8. Availability vs. Mutation Probability in GA
In PSO, the impact of iterations over-optimizing availability computations of cooling tower subsystems has been observed, as shown in Table 10. The values of other parameters such as population size, inertia weight, damping ratio, p-best, and gbest are kept at 60, 1, 0.9, 1.5, and 2, respectively. It is observed that the maximum availability has been obtained when the value of iteration is 28.

Figure 9. Availability vs. Number of Iterations in PSO
The optimum values of failure and repair rates of different subsystems can also be seen from table 10 against maximum availability when the value of iteration is 28. The relationship of availability vs. iteration in PSO as computed for the cooling tower subsystem is also depicted in Figure 9.
In PSO, the impact of population size over-optimizing availability computations of cooling tower subsystems has also been observed, as shown in Table 11. The values of other parameters such as iteration, inertia weight, damping ratio, pbest, and g-best are kept at 20, 1, 0.9, 1.5, and 2, respectively. It is observed that the maximum availability has been obtained when the value of the population is 80. The optimum values of failure and repair rates of different subsystems can also be seen from Table 11 against maximum availability when the value of the population is 80. The relationship of availability vs. iteration in PSO as computed for the cooling tower subsystem is also depicted in Figure 10.

Figure 10. Availability vs. Population Size in PSO
In PSO, the impact of damping ratio over-optimizing availability computations of cooling tower subsystems have also been observed, as shown in Table 12. The values of other parameters such as iterations, inertia weight, population, pbest, and g-best are kept at 20, 1, 60, 1.5, and 2, respectively. It is observed that the maximum availability has been obtained when the value of the damping ratio is 0.36. The optimum values of failure and repair rates of different subsystems can also be seen from Table 12 against maximum availability when the value of damping ratio is 0.36. The relationship of availability vs. iteration in PSO as computed for the cooling tower subsystem is also depicted in Figure 11.      Figure 12 and 13 depicts the availability of the cooling tower subsystem concerning the number of iterations performed by GA and PSO respectively. As shown in figure 13, the maximum availability of 99.8% is achieved by PSO in just 18 iterations, whereas GA took 60 iterations in achieving a maximum availability of only 98.8%. PSO is efficient in providing optimum values of failure and repair rates of cooling tower subsystems in achieving maximum availability of the whole system in a few iterations. Table 13 shows that the failure rate of automatic deaerator valves is most sensitive, and the operation of this component needs high attention. The comparison of GA and PSO availability results is statistically validated using Mann-Whitney U-test. The cooling tower availability corresponding to population size has been statistically analyzed at a 5% level of significance, under null hypothesis: both algorithms are equally effective and alternative hypothesis: PSO performs better than GA. Using SPSS (Version 21) software the absolute calculated value of test statistics z = 3.576 while absolute critical z value is 1.645 at a 5% level of significance. Here, the calculated value is greater than the tabulated value so cannot accept the null hypothesis. So, it is statistically significant that PSO outperforms GA.

VIII. CONCLUSION
The failure mechanism and repair policies have a significant impact on the system's operational availability and profit function. A mathematical model for the cooling tower in a steam turbine power plant (STPP) has been developed using the Markov birth-death process and supplementary variable technique. A critical evaluation of the model was carried out, and the impact of various failure and repair rates has been investigated. It was revealed that the standpipe, hydro turbine, and cooling water pump failure rates are very sensitive. While repair rates of motor valves and water supply systems show steep increments concerning their repair rate. The availability and profit decline concerning the failure rate of all subsystems. The sensitivity analysis shows that the failure rate of Automatic Deaerator Valves (ADV) is highly influential that can reduce the overall operational availability of the system. The overall availability was optimized using two well-known metaheuristic approaches viz. Genetic Algorithm (GA) and Particle Swarm Optimization (PSO). GA and PSO proved to be more efficient for such types of problems. It was revealed from the experimental evaluations that the PSO outperforms GA in optimizing the availability computations. Optimum availability of cooling tower achieved through GA was found to be 0.9884, whereas through PSO the same was recorded to be 0.9980. It has been observed that the convergence rate of PSO was very fast in achieving the optimum availability of cooling towers as compared to GA. Therefore, the present model and derived results are very helpful to STPP designers to develop highly reliable and available systems. Further, GA, PSO, and other metaheuristic techniques can be utilized to obtain the optimum availability of various process industries,