# Studies concerning the maintenance optimization of power plant steam condenser.

1. INTRODUCTION

The steam condenser is a major component of a power plant. The condenser plays two important roles (Badea et al., 2003) : it converts the used steam back into water for return to the steam generator as feedwater and it increases the cycle's efficiency.

The performance of steam condenser decreases in time and can be restored to its initial state by performing cleaning operations. If the time period between two cleanings is small, the probability of the appearance of faults due to tube fouling is very small. In the same time, a big number of maintenance actions mean supplementary costs and less delivered energy in National Power System. Under the circumstances, it needs the assurance of a correct maintenance policy and an optimization of time period between two condenser cleanings (Putman, 2004).

2. THE PROPOSED MATHEMATICAL MODEL

The objective function of the optimization method is an economic function and it is represented by total condenser costs due to fouling (Grigore & Hazi, 2008):

* Preventive maintenance cost;

* Corrective maintenance cost;

* Cost due to less energy produced pursuant to the appearance of tube fouling;

* Cost due to production losses during planned and unplanned shutdowns due to fouling.

The variable is represented by the time between two actions of scheduling maintenance.

2.1. Preventive maintenance cost

The relation for one preventive maintenance activity is:

[C.sub.m] = [t.sub.mp] x [r.sub.m] x n + [C.sub.mat] [m.u./interv.], (1)

Where: [t.sub.mp]--the duration of preventive maintenance activity in hours, [r.sub.m]--the schedule average tariff for an worker in m.u./h, [C.sub.mat]--materials costs which appear for cleaning the condenser in m.u./intervention. The variable of objective function is [T.sub.i] in hours.

The number of action of scheduling maintenance for steam condenser in one year will be:

[N.sub.m]([T.sub.i]) = T/[T.sub.i] [interv./year], (2)

Where: T--number of operation hours in one year. For a power plant, T=8760h. The total cost of preventive maintenance actions became:

[C.sub.MP]([T.sub.i]) = T/[T.sub.i]([t.sub.mp] x [r.sub.m] x n + [c.sub.mat]) [m.u./year] (3)

2.2. Corrective maintenance cost

The number of corrective maintenance actions is determinate by the average number of faults [N.sub.d] ([T.sub.i]). The corrective maintenance cost will be:

[C.sub.MC]([T.sub.i]) = [N.sub.d]([T.sub.i]) x [C.sub.d] [m.u./year], (4)

[C.sub.MC]([T.sub.i]) = [N.sub.d]([T.sub.i]) x ([t.sub.mc] x [r.sub.m] x n + [C.sub.matc] + [C.sub.sch]), (5)

Where: [C.sub.sch]--the average cost for a replacement of an damaged tube or equipment in m.u./intervention, [C.sub.matc]--material costs in m.u./intervention, [t.sub.mc]--average time for one corrective maintenance action in hours.

2.3. Cost due to less energy produced pursuant to the appearance of tube fouling

This cost can be quantified thus:

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

Where: [p.sub.i]--delivered price of electric energy in system in m.u./kWh, [delta]P--the percentage of power produced delivered in system, T/[T.sub.i]--number of maintenance actions in one year in maintenance actions/year, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]--loss energy pursuant to the appearance of tube fouling, in MWh/maintenance action.

2.4. Cost due to production losses during planned and unplanned shutdowns due to fouling

The planned and unplanned shutdowns are the periods of preventive maintenance and corrective maintenance. The cost due to production losses during planned and unplanned shutdowns due to fouling is the cost due to energy not produced during these periods. The computation relation is:

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

Where: [P.sub.b]--output power in MW.

2.5. The optimization problem

Optimization problem is presented in the next relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The problem is a non-linear programming problem with inequality restrictions, 4000 [less than or equal to] [T.sub.i] [less than or equal to] 40000 [h].

2.6. The consideration of maintenance activity to failure rate of steam condenser

For considering the influence of time between two maintenance activities, it is propose next relation:

[[lambda].sub.i]([T.sub.i]) = m + n x [T.sub.i][[h.sup.-1]], (9)

Where: [[lambda].sub.i]([T.sub.i])--failure rate. For T=8760 h, it is consider [lambda](T)=[lambda] and for T=8760/2 h, [lambda]=[lambda]/2, where [lambda]--failure rate from reliability standard or calculated from exploitation data. In these conditions, it can determine the coefficients m and n from the next equations system:

[lambda] = m + n x 8760

[lambda]/2 = m + n x 8760/2. (10)

2.7. The resolution of optimization problem

The resolution is realized using golden section method. The optimization problem, which is a one-dimensional minimum problem, has form:

Z([T.sub.i) = min!

a < [T.sub.i] < b, (11)

Where: [a,b] is the restriction interval. It is a combination of several step of accelerated search to optimum point and bordering optimum point with Fibonacci method (Hazi, 2004).

It is obtained [T.sub.iopt]--the value for optimum period between two scheduling cleanings in the interval [[a.sub.n], [b.sub.n]], where n--number of iterations and [absolute value of [a.sub.n] - [b.sub.n]] < [epsilon]. The golden section method requires no information about the derivative of the function. If such information is available it can be used to predict where best to choose the new point T. in the above algorithm, leading to faster convergence.

2.8. A case study: optimizing the cleaning frequency of a steam condenser from a DSL 50-1 turbine.

The proposed mathematical model is applied to a concrete case, considering the exploitation data. In the analyzed case, the time between two cleanings for the steam condenser is, conform actual scheduling maintenance, 8760 h. The costs due to fouling at T=8760 h/year are presented in table 1:
```Tab. 1. Costs due to fouling for steam condenser analyzed,
T=8760 h.

Cost Monetary unit Value

[C.sub.MP] \$/year 3144
[C.sub.MC] \$/year 1876
[C.sub.P] \$/year 349365
[C.sub.E] \$/year 680032
[Z.sub.opt] \$/year 1034517
```

After the optimization method is applied, the optimized time between two action of scheduling maintenance became [T.sub.iopt] = 9843 h and the costs associated to fouling are presented in table 2.

The variation of these costs in time are presented in figure 1:

[FIGURE 1 OMITTED]

3. CONCLUSIONS

The optimization algorithm determines:

* The period between two scheduling cleanings;

* The number of cleaning actions required along with their corresponding timings.

The proposed method is a general method, available for all heat exchangers. For application it is necessary to have a base knowledge about exploitation data. It is important to generate a mathematical fouling model for calculate power losses due to appearance of fouling. Comparable with existing strategy to mitigate fouling by managing cleaning, the proposed approach leads to higher energy savings and lower total costs at the level of the entire power plant.

4. REFERENCES

Badea, A.; Necula, H.; Stan, M.; Ionescu, ,L.; Blaga, P.; Dane, G.(2003). Echipamente si instalafii termice, Editura Tehnica, ISBN 973-31-2183-5, Bucuresti,

Grigore, R.; Badea, A. (2007). Aspects regarding fouling of steam condenser--a case study, UPB, Scientific Bulletin, Series C: Electrical Engineering, vol. 69, nro4 pg. 181-189, ISSN 1454-234x

Grigore,R.; Hazi, G. (2008). Contributions regarding the maintenance optimization of power plant steam condenser, Proceedings of World Energy System Conference, ISSN 1198-0729, Iasi , june 2008

Hazi, G. (2004) Tehnici de optimizare in energetica: Teorie, programe, aplicapi, Ed. Tehnica--Info, ISBN 9975-63-242-4, Chisinau

Putman R.,(2004). Return on Investment Anaysis: The Economics of Regular Condenser Maintenance, CONCO SYSTEMS, INC. Available from: www.concosystems.com
```Tab. 2. Costs due to fouling for steam condenser analyzed,
[T.sub.iopt] = 9843 h.

Cost Monetary unit Value

[C.sub.MP] \$/year 2798
[C.sub.MC] \$/year 2219
[C.sub.P] \$/year 359208
[C.sub.E] \$/year 666216
[Z.sub.opt] \$/year 1030441
```