Performability evaluation, validation and optimization for the steam generation system of a coal-fired thermal power plant

The present paper talks over performability evaluation for a steam generation system of a Coal Fired Thermal Power Plant (CFTPP) using the concept of the Markov method. A steam generation system provides a suitable amount of steam for the sound functioning of the plant. The system comprises five subsystems, i.e., High-Pressure Heater, Economizer, Boiler Drum, Water Tubes, and Super Heater. First, the transition diagram of the concerned system is designed based on the state probabilities of various subsystems. The differential equations are derived based on the mnemonic rule. After that, the performability model is developed by using the normalizing condition. The performability levels for various subsystems are obtained by placing the appropriate value of failure and repair rates in the developed model. The performability of each subsystem is evaluated based on performability matrices. It is observed that the economizer subsystem is most critical in which the availability increased from 0.7640 to 0.8827, i.e. (11.87 %). In contrast, boiler drum is the least crucial subsystem with availability enhanced from 0.8627 to 0.8657 (i.e., 0.3 %). The results show that the economizer subsystem must be given top priority, and the boiler drum be given the least priority from the maintenance outlook. The performability levels obtained through the Markov method are compared with those obtained through the Artificial Neural Network to validate. Moreover, machine learning (artificial neural network) and optimization technique (particle swarm optimization) is also employed to check the adequacy of the results and optimized process parameters.• The aim of the present study is evaluate the performance of steam generation system of a coal fired thermal power plant.• The probabilistic approach (i.e. Makov Method) is used to formulate the transition diagram of the steam generation system. Then, the first-order differential equations are obtained using the mnemonic rule and further solved recursively.• The results show that the economizer system must be given top priority, and the boiler drum subsystem must be given the least priority from the maintenance outlook.

• The aim of the present study is evaluate the performance of steam generation system of a coal fired thermal power plant. • The probabilistic approach (i.e. Makov Method) is used to formulate the transition diagram of the steam generation system. Then, the first-order differential equations are obtained using the mnemonic rule and further solved recursively. • The results show that the economizer system must be given top priority, and the boiler drum subsystem must be given the least priority from the maintenance outlook. Aggarwal et al. [2] 2015 Markov modeling and reliability analysis of urea synthesis system Urea synthesis system of a fertilizer plant Arora and Kumar [3] 1997 Availability analysis of steam and power generation systems thermal power plant Ansari et al. [4] Particle swarm optimization technique PV System Bahl et al. [5] 2018 Availability analysis of distillery plant using Petri Nets

Distillery plant
Braglia et al. [6] 2012 multivariate statistical approach an oil refinery plant Chokshi et al. [7] 2018 Artificial Neural Network thermal power plants Gupta et al. [11] 2020 Markov Method steam turbine powerplants The ordinary differential equations have been solved by using the fourth-order Runga-Kutta method to obtain the reliability of the butter manufacturing plant. Kaur et al. [13] discussed a numerical method to obtain the transient state for systems having inconsistent failure and repair rates. Kumar et al. [14] analyzed the availability of a system in a thermal power plant with the help of the Markov approach and suggested the maintenance schedule for various subsystems of the system concerned. Marinakis et al. [15] proposed the Ant Colony (ACO and the PSO algorithms to solve the financial classification model. They tested the proposed methods through two different financial classification problems. Yazdi [16] discussed a heuristic optimization model using fuzzy numbers and triangular intuitionistic to find the reliability of the model and used Spearman Correlation. Liu et al. [17] discussed the maintenance strategies of the product under warranty period having limited repair time and repair number. Further, they discussed the repair cost of the product is based on the repair time. Rajpal et al. [18] described the use of artificial neural networking (ANN) to determine a repairable system's performance. The neural network was trained with the help of past plant data. The simulation results were used to formulate a strategy for the optimum working of the system. Chen [19] discussed the preemptive scheduling with a single machine to minimize the total weighted late work and weighted number of tardy jobs. Further, they also discussed the Pareto-scheduling problem. Kumari et al. [20] discussed the solution of constrained problems using particle swarm optimization (PSO). Li and Zhang [21] used dynamic programming to solve the redundancy allocation optimization for the multi-state series and parallel systems. Zeng and Sun [22] analyzed the competing failure in the system based on stochastic Petri nets in their study and also discussed the effect of common cause failure of the system. Moreover, Table 1 shows the comparison of previous published work.
In the current scenarios, automation in industries is a vital problem owing to massive capital investment. For example, a power plant industry requires a multifaceted system with tremendous capital investment and planning. Besides this, the failure of components is also another problem related to this industry. To overcome all these problems, a proper arrangement for preventive maintenance must be necessary. Thus, a mannered maintained schedule is required for all the components of the power plant industry. In addition, the power plant industry equipment's are always working in harsh conditions. Because of this, it requires to repair and replacement of components from time to time. It means the condition of the machine depends upon the operating conditions. These conditions may be different for different systems. The priority for the power plant industry is to retain the availability of all the systems during the process. The availability of the plant can be measured on the basis of operational time without failure. In current research work, the main focus is to preserve the plant in working condition without any failure by maintaining the different systems (i.e., steam generation) of a plant in a failure-free State. It is a vital part of the power plant industry and consists of five subsystems organized in sequences. The outcome of the concerned system is interconnected with the reliability and maintainability of the equipment. It depends upon the number of failures. For this, an optimal maintenance strategy is required by taking the highest maintenance priority to the most critical subsystem of the system. In this current research, a birth-death Markov method is utilized for solving the above-mentioned problems related to power plant industries. The transition diagram for different power plant industry systems is drawn as per the working condition of the system. Moreover, the differential equations are generated for each system by utilizing the transition diagram. In addition, the performance model is designed as per the transition diagram. Finally, performance analysis of various subsystems is measured as per the decision matrix obtained from the developed performance model.

Steam generation system description
A steam generation system ensures a regular supply of steam for the efficient execution of a thermal power plant. The concerned system comprises mainly five subsystems (as shown in Figure 1 ) in a chain configuration with the following description: (i) High-Pressure Heater (HP): It has three high-pressure heaters in series. The function of a highpressure heater is to increase the temperature of feed water. If one of them fails, the system goes into a complete failure state. (ii) Economizer (EC): It is used to capture the heat from the flue gases and transfer it to the boiler feed water. It consists of a single economizer; its failure causes the complete failure of the concerned operating system. (iii) Boiler Drum (BD): It comprises one boiler drum to separate the saturated steam from the steam-water mixture. Its failure causes the whole failure of the system. (iv) Boiler Tubes (BT): Hot gasses come from the furnace and contact with water tubes where the heat of these hot gases transfers to the water, and consequently, steam is produced in the boiler. Failure of the water tube causes complete failure of the system. (v) Super Heater (SH): It is a device that converts saturated steam into superheated steam.
Superheater failure leads to the whole shutdown of the system.

Notations
H, E, D, T and S : Show the operating state of the subsystems of HP, EC, BD BT and SH respectively. h, e, d, t and s : Indicate that failed state of the subsystems HP,EC,BD, BT and SH P S0 ( ˆ t ) : Probability of the system working with full capacity at time t. P S1 ( ˆ t )-P S5 ( ˆ t ) : Probabilities of the system in a failed state. F i, i = 1-5 : Mean failure rates of HP, EC, BD, BT and SH respectively. R i, i = 1-5 : Mean repair rates of HP, EC, BD, BT and SH respectively. K i : It is ration of failure rate (F i ) to repair rate (R i ) : Denotes the working of the system without failure.
: Exhibits the reduced capacity of the system. (iv) The exponential distribution will be followed by failure and repair rates.
(v) The subsystems cannot fail simultaneously.
The transition diagram for the steam generation system consists of 11 states as depicted in Figure 2 . The initial state (S0) is working with full capacity. State S1 is a reduced capacity state, and S2 to S10 shows that the system is in a failed state due to the complete failure of one or the other subsystem of the steam generation system.

Mathematical formulation using Chapman-Kolmogrove differential equations
The differential equations associated with the transition diagram derived by using the mnemonic rule are as follows: P S0 ˆ P S1 ˆ t + R 4 P S1 ˆ t + F 1 P S1 ˆ t + F 2 P S1 ˆ t + F 3 P S1 ˆ t + F 4 P S1 ˆ t + F 5 P S1 ˆ t

Solution of equations by steady state behavior
The steady-state or long-run behavior of the system can be analyzed by setting P ' = 0 as t → ∞ . The equations from (3.1) to (3.11) are written as: R 4 P S1 ˆ t + F 1 P S1 ˆ t + F 2 P S1 ˆ t + F 3 P S1 ˆ t + F 4 P S1 ˆ t + F 5 P S1 ˆ t Solving these Eqs. (12)-(22) recursively, P S1 = K 4 P S0 , P S2 = K 1 P S0 , P S3 = K 2 P S0 P S4 = K 3 P S0 , P S5 = K 5 P S0 P S6 = K 1 K 4 P S0 P S7 = K 2 K 4 P S0 , P S8 = K 4 K 4 P S0 P S9 = K 4 K 5 P S0 P S10 = K 3 K 4 P S0 Use of normalizing condition i.e., the sum of all the state probabilities is equal to one [ 10 i=0 P Si = 1 ], gives the solution as follows: P S0 + P S1 + P S2 + P S3 + P S4 + P S5 + P S6 + P S7 + P S8 + P S9 + P S10 = 1 By putting the value of different stages ( P S1 to P S10 ) in the form of P S0 The performability model of the steam generation system is obtained by the addition of reduced and full working state probabilities.

Performability evaluation
The appropriate values of FRR (Failure and Repair Rate) are taken from maintenance records available in the history cards, maintenance sheets, etc., and also on the basis of discussion with concerned plant employees. The performability of subsystems is obtained by placing the suitable values of FRR in the developed model and solved in MATLAB. The numerous performability levels of different subsystems have been presented in Tables 2-6 , as well as in Figs. 3-7 , respectively. The impact of FRR on various subsystems of steam generation system is shown below: Table 2 presents the performability analysis of the steam generation system. The maximum value of performability, i.e., 0.8705, is obtained at a failure rate of 0.015 and repair rate of 0.27. Fig. 3 depicts the 3D surface plot between repair rate and failure rate for HPH. It is observed that the performability is enhanced up to 3.60 % when the failure and repair rate varies between 0.007 to 0.015 and 0.23 to 0.27, respectively. Table 3 Performability matrix for economizer of steam generation system.  Table 4 Performability matrix for boiler drum of steam generation system.  Table 5 Performability matrix for boiler tubes of steam generation system. The variation in performability of the Economizer subsystem is shown in Table 3 . It is concluded that the maximum performability is observed at a failure rate of 0.0 0 022 and a repair rate of 0.005. Fig. 4 illustrates the graphical analysis of the performability of for Economizer subsystem. The failure rate (F 2 ) of the Economizer is increased from 0.0 0 018 to 0.0 0 022, thus resulting in a decrement in the performability from 0.7881 to 0.7640. Conversely, results are obtained for the repair rate.
The performability analysis for the boiler drum subsystem is depicted in Table 4 and Fig. 5 . The system performability is enhanced from 0.8627 to 0.8657 with a failure rate, and the repair rate varies between 0.0 0 08 to 0.0 012 and 0.2 to 0.6, respectively. Table 5 and Fig. 6 reveal the performability analysis of the boiler tube subsystem. The failure rate (F 4 ) of Boiler Tubes increases from 0.006 to 0.010, and the performability of the system decreases merely from 0.8640 to 0.8558 i.e., 0.82%. In the same way, as the repair rate (R 4 ) increases from 0.09 to 0.13, the performability of the system increases just from 0.8558 to 0.8659 i.e., 1.01%. Table 6 and Fig. 7 indicate the effect of FRR of the superheater subsystem on the performability of the steam generation system. When the failure rate (F 5 ) of the superheater increases from 0.0 0 01 to 0.0 0 05, then the performability of the system decreases noticeably from 0.8812 to 0.8323 i.e., Table 6 Performability matrix for superheater of steam generation system.    It uses a two-layer feed-forward network for solving the data fitting problems. In addition, it also helps in dividing the data set into training and testing data sets. MATLAB software is utilized for employing the ANN technique on the given data set. The outcomes of the present study are mentioned below: The predicated output performability levels obtained through ANN for various systems of RGTPP are compared with the output of performability levels through the Markov method. The error is attained by the subtraction of the performability levels (obtained through the Markov approach) and predicated output performability levels (obtained through ANN). The predicated output performability levels obtained through ANN are much closer to the output of performability levels obtained through

Performability optimization
In the recent past, Particle Swarm Optimization has been found to be the most effective technique applied in many engineering and management applications for the optimization of the processes. This learning algorithm is based on the flying birds as the birds change their direction and control their speed as per their past experience to locate their destination by minimizing the distance gradually. The searching algorithm considers each solution as a particle (bird), and the fitness value of each particle has been estimated with the help of the fitness function. By tuning the cumulative speed and the position of the particle, the group performance (g-best) and the best performance of the individual (p-best) have been estimated. Multiple iterations have been performed to control the cumulative speed and the position of the particle. The speed and the position of i th particle for the population size of ' n' are estimated by the following relations: The particles are continuously allowed to move in the multidimensional search space, and with the help of successive iterations, the best optimum solution has been obtained.
In the present work, an attempt has been made to optimize the performability level for a thermal power plant i.e., by estimating the various groupings of FRR and multiple iterations on different population sizes. This problem of the thermal power plant has been considered 10-dimensional space of different failure rates and repair rates, as shown in Table 7 . The constrained range of different FRR parameters to optimize the performability of the thermal power plant are described below: The position and the speed of the particles is reconfigured with the help of the following algorithms.
Where ' w ' is the Inertia Weight, ' c 1 ' is the Cognitive Parameter, ' c 2 ' is the Social Parameter, and ' r 1 & r 2 ' are the Random Numbers arbitrarily selected.  In this part of the optimization technique, the algorithm has been terminated either at the maximum count of generations or at a minimum value of fitness function. The different parameters considered for the PSO algorithm are shown in Table 8 .
The algorithm has been explained with the help of the following flow diagram as depicted in Figure 8 .
By fine-tuning PSO parameters such as population size, step size of both the algorithms, and the number of iterations, the performability of the steam generation system has been optimized.

Results and discussion
The optimum performability for steam generation system has observed 91.74% by using Particle swarm optimization approach at a PS of 45 and by taking GS constant i.e. 100. Table 9 indicates the best arrangements of FRR as F 1 = 0.009, F 2 = 0.0002, F 3 = 0.0009, F 4 = 0.00 6 6, F 5 = 0.0001, R 1 = 0.3442, R 2 = 0.0047, R 3 = 0.5378, R 4 = 0.0898 and R 5 = 0.0085. Further, the performability of the system has been represented in Fig. 9 by taking the different parameters like PS and GS constant. The performability levels for steam generation system at PS varied from 5 to 50 in a step of 5 taking constant GS are specified as follows: Table 9 Impact of constant GS (100) and PS on system performability.   The optimum performability of steam generation system obtained is 91.74% by using Particle Swarm Optimization approach at GS (70) and PS constant i.e. 40. Table 10 Fig. 10 . The performability levels for steam generation system at GS varied from 10 to 100 in a step of 10 taking constant PS are given as follows:

Conclusions
In current research work, the Markov Birth death technique is utilized to find out the best economical maintenance schedule for thermal power plant steam generation system. The following conclusion is drawn listed below: 1. The most critical unit is found to be the economizer unit in which the availability increased from 0.7640 to 0.8827, i.e., (11.87 %), whereas boiler drum is observed as the least critical unit with an availability enhanced from 0.8627 to 0.8657 (i.e., 0.3 %). 2. A drastic fall of 2.41 % in the level of unit availability occurs with the rise in the failure rate of the economizer (F 2 ) from 0.0 0 018 to 0.0 0 022. Also, a significant boost of 11.87% in unit availability can be observed with the rise in repair rate economizer (R 2 ) from 0.001 to 0.005. 3. A fall of 1.9 % in the level of highe pressure heater availability occurs with the rise in the failure rate (F 1 ) from 0.007 to 0.015. Also, a noticeable increase of 3.6% in unit availability can be observed with the rise in repair rate (R 1 ) from 0.232 to 0.27. 4. A radical fall of 4.89 % in the level of unit availability occurs with the rise in failure rate of super heater subsystem (F 5 ) from 0.0 0 01 to 0.0 0 05. Also, an appreciable enhancement of 2.38 % in unit availability can be observed with the rise of repair rate of super heater subsystem (R 5 ) from 0.002 to 0.014. 5. A fall of 0.82% in the level of unit availability occurs with the rise in failure rate of centrifuge subsystem (F 4 ) from 0.006 to 0.010. Also, a substantial improvement of 1.01 % in unit availability can be observed with the rise of repair rate of centrifuge (R 4 ) from 0.09 to 0.13. 6. A marginal fall of .15% in the level of unit availability occurs with the rise in failure rate of sugar grader (F 3 ) from 0.0 0 08 to 0.0012. Also, a mere step up of 0.03 % can be observed in unit availability with the rise in repair rate of sugar grader (R 3 ) from 0.2 to 0.6. 7. For the complete analysis of thermal power plant, it can be arranging the different units of thermal power plant according to maintenance priorities as follows: economizer, high pressure heater, super heater, boiler tubes, and boiler drum. 8. The difference/error between the results obtained through the Markov method and ANN is very less, up to 5%. So, this small variation validates the output of the Markov method with the help of ANN. Furthermore, the performability of the concerned system has been optimized by using the PSO algorithm to improve the total performance of the concerned system. It is observed that the highest performability level, i.e., 91.74 at a population size of 40 and at a generation size of 70.

Declaration of Competing Interest
There is no conflict of interest on this article.

Data availability
No data was used for the research described in the article. Table 11   Table 11 Comparison of performability levels obtained through markov method and ANN for steam generation system.