Performance evaluation and optimization for urea crystallization system in a fertilizer plant using genetic algorithm technique.

Kumar, Sanjeev ; Tewari, P.C. ; Kumar, Sunand 等

Introduction

Over the years as engineering systems have become more complex and sophisticated, the performance evaluation and optimization of engineering systems is becoming increasingly important because of factors such as cost, risk of hazard, competition, public demand, usage of new technology. High reliability level is desirable to reduce overall costs of production and risk of hazards for larger, more complex and sophisticated systems such as fertilizer plant. During the last three decades reliability technology has been developed for use in various technological fields. The technology is mainly used in the development of electrical and electronics equipments. The technology has also been used in a number of industrial and transportation problems.

A fertilizer plant is a complex engineering system comprising of various systems: urea synthesis, urea crystallization, urea decomposition and urea prilling etc. [3, 4]. The optimization of each system in relation to one another is imperative to make the plant profitable and viable for operation. Effectiveness of fertilizer plant is mainly influenced by the availability, reliability and maintainability of the plant, and its capability to perform as expected. Reliability analysis techniques have been gradually accepted as standard tools for the planning and operation of automatic and complex fertilizer plants.

The maintenance of repairable systems has been widely studied by many authors, considering different focus of interest, such as the repair/replacement policy, periodic inspections, degrading, optimization problems, among other topics. Reliability analysis is one of the main tools to ensure agreed delivery deadlines which in turn maintain certain intangible factors such as customer goodwill and company reputation [1]. Downtime often leads to both tangible and intangible losses. These losses may be due to some unreliable subsystems/components, thus an effective strategy for maintenance, replacement and design changes related to those subsystems and component needs to be framed out [2,4].

The equipment that fails very frequently can be said to have very low reliability and the equipment that takes very long time to be replaced or repaired, can be said to have very low maintainability. Both are critical and result in low availability. Availability predictive modeling can provide insight in these cases by identifying exactly those equipments that are critical for availability. With a predictive model experiments can be made with different maintenance strategies and their influence on reliability and maintainability [5]. From an economic point of view, high reliability is desirable to reduce the maintenance costs of systems [6]. Since failure cannot be prevented entirely, it is important to minimize both its probability of occurrence and the impact of failures when they do occur. To maintain the designed reliability, availability and maintainability characteristics and to achieve expected performance, an effective maintenance program is a must and the effective maintenance is characterized by low maintenance cost [7]. The type of component failure and its frequency has a direct effect on the system's reliability. Thus it becomes very important to locate the critical components and analyze their reliability. Furthermore, in many situations it is easier and less expensive to test components/subsystems rather than entire system [8]. Availability and reliability are good evaluations of a system's performance [9].

For the prediction of availability or performance of systems, several mathematical models have been discussed in literature, which handle wide degree of complexities [10]. Most of these models [11, 12, 13] are based on the Markovian approach, wherein the failure and the repair rates are assumed to be constant. In other words, the times to failure and the times to repair follow exponential distribution. The behaviour of complex systems can be studied in terms of their reliability, availability and maintainability (RAM). For example, Kurien [14] developed a predictive model for analyzing the reliability and availability of an aircraft training facility. The model was useful for evaluating various maintenance alternatives.

Factors that affect RAM of a repairable system include machinery operating conditions, maintenance and infra-structural facilities. Reliability analysis has helped in identifying the critical and sensitive subsystems in the electricity production system, which has a major effect on system failure. Therefore, a focus on reliability is critical for the improvement of equipment performance and ensuring that equipment is available for production as per production schedules [15].

Nowadays, reliability analysis has become an integral part of system design. This is especially true for systems performing critical applications. System designers rely on commercially available dependability tools in order to assess the reliability of their systems. The usage of such tools becomes therefore crucial. Systems being built are increasingly complex and large; their components are exhibiting behaviors and interactions that are becoming more and more difficult to model and analyze using existing conventional tools. Markov Chains (MCs) and their extensions have proven to be a versatile tool for modeling complex dynamic component behavior. They have been extensively used for dependability analysis of dynamic systems and many tools have adopted, directly or indirectly, MCs as their formalism [16]. During the past decade a lot of study [17, 18, 19] has been done by on analysis tools for reliability, availability, performance modeling.

In the present article, performance of a critical urea crystallization system in the fertilizer plant has been worked out using a Markovian model. The need and the relevance for carrying out such a study have been described in the script. The actual failure and repair data on the identified urea crystallization system has been used in the analysis.

For efficient functioning, it is essential that various systems of the plant remain in upstate as far as possible. However, during operation they are liable to fail in a random fashion. The failed elements can however be inducted back into service after repairs/replacements. The rate of failure of the components in the system depends upon the operating conditions and repair policy used [20].This paper discusses the performance evaluation and optimization of urea crystallization system in a medium sized fertilizer plant.

System Description

The Urea Crystallization system comprises of four subsystems arranged in series:

(i) Subsystem (A1) consists of two-stage ejector, barometric condenser (used to generate the pressure of 175 mm of Hg), concentrator (lower part) and crystallizer (upper part), all are arranged in series. Failure of any one unit causes the complete failure of the system.

(ii) Subsystem (B) consists of five centrifugal pumps arranged in series. Failure of any unit causes the complete failure of the system.

(iii) Subsystem (D) consists of two crystallizer pumps one in operative and other in cold standby. Complete failure of the system will occur only when both pumps fail at a time.

(iv) Subsystem (E) consists of two slurry feed pumps arranged in parallel (one operative and other in cold standby). Complete failure of the system will occur only when both pumps fail at a time.

Notations

The notations used to represent the various states of the subsystems in the transition diagram of Urea Crystallization system (figure 1) and system modeling, are as follows:

A, B, D, E : denotes that the subsystems are full operating state.

Ds, Es : denotes that the subsystems D and E are working on standby unit.

A, b, c, d, e : denotes that the subsystems are in failed state.

[P.sub.0] (t) : Probability that at time t all subsystems are in original working state (without standby unit).

[P.sub.i] (t) : Probability that at time t all subsystems are in full load condition (standby mode) for i = 1-3.

[P.sub.j] (t) : Probability that at time t all subsystems are in breakdown state for j = 4-15 [[alpha].sub.i], i = 1-5 : mean failure rates in A, B, D, E.

[[beta].sub.i], i =1-5 : mean rate of repairs in A, B, D, E.

[DELTA]t : Time increment

d/dt : represents derivative w.r.t. 't'.

[circle] : System working at full load condition.

[rectangle] : System breakdown.

Assumptions

Modeling is carried out on the basis of following assumptions:

(1) Failure/repair rates are constant over time and statistically independent.

(2) A repaired unit is good as new, performance wise for a specified duration.

(3) Sufficient repair facilities are provided as and when required.

(4) Standby units are of the same nature as that of active units [4, 5].

(5) System failure/repair follows exponential distribution.

(6) Service includes repair and/or replacement [9].

(7) System may work at reduced capacity.

(8) There are no simultaneous failures.

[FIGURE 1 OMITTED]

Mathematical Modeling

Mathematical modeling has been developed for the prediction of steady state availability of the individual components as well as entire system. The failure and repair rates of the different subsystems, available from the maintenance sheets of the fertilizer plant, are used as standard input information for the analysis. The state of the system defines the condition at any instant of time and the information is useful in analyzing the current state and in the prediction of the failure state of the system. If the state of the system is probability based, then the model is a Markov probability model. Markov model is defined by a set of probabilities [P.sub.ij], where [P.sub.ij] is the probability of transition from any state i to any state j. One of the most important features of the Markov process is that the transition probability [P.sub.ij] ; depends only on states i and j and is completely independent of all past states except the last one, state i.

Let the probability of n occurrences in time t be denoted by [P.sub.n](t), i.e.,

Probability(X = n, t) = [P.sub.n] (t) (n = 0, 1, 2 ...).

Then, [P.sub.0] (t) represent the probability of zero occurrences in time t. The probability of zero occurrences in time (t + [DELTA]t) is given by equation (1); i.e.

[P.sub.0] (t + [DELTA]t) = (1 - [beta]t), [P.sub.0](t) (1)

Similarly [P.sub.1] (t + [DELTA]t)) = ([alpha] + [DELTA]t). [P.sub.0] (t) + (1 + [beta]. [DELTA]t). [P.sub.1] (t) (2)

The equation 2 shows the probability of one occurrence in time (t + [DELTA]t) and is composed of two parts, namely, (a) probability of zero occurrences in time t multiplied by the probability of one occurrence in the interval [DELTA]t and (b) the probability of one occurrence in time t multiplied by the probability of no occurrences in the interval [DELTA]t [21]. Then simplifying and putting [DELTA]t [right arrow] 0, one gets

(d/dt + [alpha]) [P.sub.1](t) = [beta] [P.sub.0](t) (3)

Using the concept used in equation 3 and various probability considerations, the following differential equations associated with the transition diagram of urea crystallization system are formed [26, 27, 28].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

Where in equation (8), for,

m=1, i = 4, 6, 9, 12, j = 0,1, 2, 3. m=2, i = 5, 7, 10, 13, j = 0,1,2,3 m=3, i = 8, 14, j = 1, 3 m=4, i = 11, 15, j = 2, 3

With the initial condition [P.sub.0] (0) =1, otherwise zero, since any urea plant is a process industry where raw material is processed through various subsystems continuously till the final product is obtained [29, 30]. Thus, the long run availability of the urea crystallization system of a fertilizer plant is attained by putting derivatives of all probability functions equal to zero as:

t [right arrow] [infinity] d/dt = 0

into differential equations, one gets,

[P.sub.4] = ([[alpha].sub.1]/[[beta].sub.1])[P.sub.0] = (9)

[P.sub.5] = ([[alpha].sub.2]/[[beta].sub.2])[P.sub.0] = (10)

[P.sub.6] = ([[alpha].sub.1]/[[beta].sub.1])[P.sub.0] = (11)

[P.sub.7] = ([[alpha].sub.2]/[[beta].sub.2])[P.sub.1] = (12)

[P.sub.8] = ([[alpha].sub.3]/[[beta].sub.3])[P.sub.1] = (13)

[P.sub.9] = ([[alpha].sub.1]/[[beta].sub.1])[P.sub.1] = (14)

[P.sub.10] = ([[alpha].sub.1]/[[beta].sub.1])[P.sub.2] = (15)

[P.sub.11] = ([[alpha].sub.4]/[[beta].sub.4])[P.sub.2] = (16)

[P.sub.12] = ([[alpha].sub.1]/[[beta].sub.1])[P.sub.2] = (17)

[P.sub.13] = ([[alpha].sub.2]/[[beta].sub.2])[P.sub.3] = (18)

[P.sub.14] = ([[alpha].sub.3]/[[beta].sub.3])[P.sub.3] = (19)

[P.sub.15] = ([[alpha].sub.4]/[[beta].sub.4])[P.sub.3] = (20)

([[alpha].sub.3] + [[alpha].sub.4])[P.sub.0] = [[beta].sub.3][P.sub.1] + [[beta].sub.4][P.sub.2] (21)

([[alpha].sub.4] + [[beta].sub.3])[P.sub.1] = [[beta].sub.4][P.sub.3] + [[beta].sub.3][P.sub.0] (22)

([[alpha].sub.3] + [[beta].sub.4])[P.sub.2] = [[beta].sub.3][P.sub.3] + [[beta].sub.4][P.sub.0] (23)

([[beta].sub.4] + [[beta].sub.4])[P.sub.3] = [[beta].sub.4][P.sub.1] + [[alpha].sub.3][P.sub.2] (24)

Mathematical task is performed to solve the set of equations (21)-(24) simultaneously in terms of 0 P as given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The probability of full working capacity, namely, [P.sub.0] is determined by using normalizing condition: (i.e. sum of the probabilities of all working states and failed states is equal to 1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Now, the steady state availability of Urea Crystallization system may be obtained as summation of all working state probabilities as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore, Availability of the system (Av.) represents the performance evaluating system of Urea Crystallization system. This performance-evaluating model includes all possible states of nature, that is, future events ([[alpha].sub.i]) and the identification of all the courses of action, that is, repair priorities ([[beta].sub.i]). This model is used to implement the maintenance policies for a Urea Crystallization system of a fertilizer plant. The various availability levels may be computed for different combinations of failure and repair rates / priorities. It can be used for performance optimization of this operating system of a fertilizer plant by using Genetic Algorithm Technique.

Genetic Algorithm Technique (G.A.T)

Genetic Algorithms (G.A.) are computerized search and optimization algorithms based on the mechanics of natural genetics and natural selection [32]. G. A.'s have become important because they are found to be potential search and optimization techniques for complex engineering optimization problems. The action of G.A.T. for parameter optimization in the present problem can be stated as follows:

(1) Initialize the parameters of the G.A.T.

(2) Randomly generate the initial population and prepare the coded strings.

(3) Compute the fitness of each individual in the old population.

(4) Form the mating pool from the old population.

(5) Divide the population repeatedly into random tournaments, consisting of four population members per tournament.

(6) Copy the most fit population member in each tournament to the mating pool.

(7) The process is repeated until the mating pool has the same size as the population.

(8) Determine the new generation pool.

(9) Select two parents from the mating pool randomly.

(10) Perform the crossover of the parents to produce to produce two off springs.

(11) Mutate, if required.

(12) Place the child strings to new population.

(13) Compute the fitness of each individual in new population.

(14) Replace the old population with the new population.

(15) Repeat the steps 3 to 8 until the best individuals in new population represent the optimum value of the performance function (system availability).

Performance Optimization using G.A.T.

The performance behavior of Urea Crystallization system is highly influenced by the failure and repair parameters of each subsystem. These parameters ensure high performance of the Urea Crystallization system. G.A.T. is hereby proposed to coordinate the failure and repair parameters of each subsystem for stable system performance i.e. high availability. Here, number of parameters is eight (four failure parameters and four repair parameters). The design procedure is described as follows [32]:

To use G.A.T. for solving the given problem, the chromosomes are to be coded in real structures. Here, concatenated, multi-parameter, mapped, fixed point coding is used. Unlike, unsigned fixed point integer coding parameters are mapped to a specified interval [[X.sub.min], [X.sub.max]], where [X.sub.min] and [X.sub.max] are the maximum and minimum values of system parameters. The maximum value of the availability function corresponds to optimum values of system parameters. These parameters are optimized according to the performance index i.e. desired availability level. To test the proposed method, failure and repair rates are determined simultaneously for optimal value of system availability. Effects of population size and number of generations on the availability of Urea Crystallization system are shown in Table 1 and 2. To specify the computed simulation more precisely, trial sets are also chosen for G.A. and system parameters. The performance [availability] of Urea Crystallization system is determined by using the designed values of the system parameters [32].

Failure and repair rate parameter constraints

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Here, real-coded structures are used.

Maximum number of Population size is varied from 20 to 160.

Number of Generations--100

Crossover probability--0.8

Mutation Probability--0.2

Total number of run--01

The effect of population size on availability of Urea Crystallization system is shown in figure 2. The optimum value of system's performance is 98.43 %, for which the best possible combination of failure and repair rates is [[alpha].sub.1] = 0.0103, [[beta].sub.1] = 0.9912, [[alpha].sub.2] = 0.0041, [[beta].sub.2] = 0.8811, [[alpha].sub.3] = 0.0206, [[beta].sub.3] = 0.6711, [[alpha].sub.4] = 0.0013, [[beta].sub.4] = 0.3972, at population size 140, as given in table 1.

Maximum number of Generations is varied from 50 to 300.

Population size--100

Crossover probability--0.8

Mutation Probability--0.2

Total number of run--01

The effect of number of generations on availability of the Urea Crystallization system is shown in figure 3. The optimum value of system's performance is 98.45 %, for which the best possible combination of failure and repair rates is [[alpha].sub.1] = 0.0103, [[beta].sub.1] = 0.9983, [[alpha].sub.2] = 0.0040, [[beta].sub.2] = 0.8998, [[alpha].sub.3] = 0.0215, [[beta].sub.3] = 0.6858, [[alpha].sub.4] = 0.0010, [[beta].sub.4] = 0.3709 at generation size 200, as given in table 2.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Conclusions

The performance optimization of the Urea Crystallization system of a fertilizer plant is discussed in this paper. Genetic Algorithm Technique (G.A.T) is hereby proposed to select the various feasible values of the system failure and repair parameters. Then, G.A.T is successfully applied to coordinate simultaneously all these parameters for an optimum level of system performance. Besides, the effect of G.A. parameters such as population size and number of generations on system performance i.e. availability has also been discussed. The findings of this paper are discussed with the concerned fertilizer plant management. Such results are found highly beneficial for the purpose of performance optimization of the Urea Crystallization system of a fertilizer plant concerned.

Acknowledgement

The author is grateful to Sh. Sudesh Chohan, National Fertilizer Limited Panipat (India), for providing every possible help during the work.

References

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(a) Sanjeev Kumar, (b) P.C. Tewari, (c) Sunand Kumar and (d) Meenu

(a) Department of Mechanical Engineering, AKG Engineering College, Ghaziabad--201009 (Uttar Pradesh),INDIA, E-mail : [email protected]

(b,d) Department of Mechanical Engineering, National Institute of Technology, Kurukshetra--136119 (Haryana) INDIA, E-mail: [email protected], [email protected]

(c) Department of Mechanical Engineering, National Institute of Technology, Hamirpur--177005 (Himachal Pradesh) INDIA, E-mail: [email protected]

Declaration

The research material in the paper title "Performance Evaluation and Optimization for Urea Crystallization System in a Fertilizer Plant using Genetic Algorithm Technique" submitted to the International Journal of Applied Engineering and Research (IJAER) has neither been published elsewhere nor is being considered elsewhere for publication.

Sanjeev Kumar

Assistant Professor,

Department of Mechanical Engineering

AKG Engineering College,

Ghaziabad--201009 (Uttar Pradesh)

INDIA

Email: [email protected]

Table 1: Effect of Population Size on Availability of Urea
Crystallization System.

Pop.           [[alpha].   [[alpha].   [[alpha].   [[alpha].
Size    Av.    sub.1]      sub.2]      sub.3]      sub.4]

 20    0.9775  0.0154      0.0041      0.0234      0.0013
 40    0.9824  0.0108      0.0045      0.0261      0.0012
 60    0.9830  0.0110      0.0042      0.0227      0.0012
 80    0.9837  0.0103      0.0043      0.0200      0.0010
100    0.9835  0.0110      0.0041      0.0216      0.0010
120    0.9838  0.0101      0.0043      0.0244      0.0014
140    0.9843  0.0103      0.0041      0.0206      0.0013
160    0.9843  0.0101      0.0040      0.0201      0.0011

Pop.   [[beta].   [[beta].   [[beta].   [[beta].
Size   sub.1]     sub.2]     sub.3]     sub.4]

 20    0.9250     0.7772     0.6879     0.1915
 40    0.9670     0.8518     0.6727     0.3839
 60    0.9690     0.8773     0.6766     0.3580
 80    0.9974     0.7824     0.6908     0.3941
100    0.9715     0.8889     0.6915     0.3932
120    0.9682     0.7920     0.6986     0.3992
140    0.9912     0.8811     0.6711     0.3972
160    0.9457     0.8915     0.6914     0.3951

Table 2: Effect of Number of Generations on Availability of Urea
Crystallization System.

Gen.            [[alpha].   [[alpha].   [[alpha].   [[alpha].
Size   Av.      sub.1]      sub.2]      sub.3]      sub.4]

 50    0.9819   0.0126      0.0042      0.0213      0.0012
100    0.9835   0.0110      0.0041      0.0216      0.0010
150    0.9844   0.0103      0.0040      0.0206      0.0011
200    0.9845   0.0103      0.0040      0.0215      0.0010
250    0.9834   0.0111      0.0041      0.0222      0.0011
300    0.9836   0.0109      0.0040      0.0242      0.0010

Gen.   [[beta].   [[beta].   [[beta].   [[beta].
Size   sub.1]     sub.2]     sub.3]     sub.4]

 50    0.9821     0.8999     0.6994     0.3948
100    0.9715     0.8889     0.6915     0.3932
150    0.9946     0.8722     0.6730     0.3891
200    0.9933     0.8998     0.6858     0.3709
250    0.9984     0.8670     0.6768     0.3893
300    0.9966     0.8929     0.6872     0.3955