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### Abstract:

Unexpected fouling in condensers has always been one of the main operational concerns in

thermal power plants. This paper describes an approach to predict fouling deposits in thermal power plant

condensers by means of support vector machines (SVMs). The periodic fouling formation process and

residual fouling phenomenon are analyzed. To improve the generalization performance of SVMs, an

improved differential evolution algorithm is introduced to optimize the SVMs parameters. The prediction

model based on optimized SVMs is used in a case study of 300MW thermal power station. The experiment

result shows that the proposed approach has more accurate prediction results and better dynamic

self-adaptive ability to the condenser operating conditions change than asymptotic model and T-S fuzzy

model.

Keywords: Fouling prediction; Condensers; Support vector machines; Differential evolution

1. Introduction

Condenser is one of key equipments in thermal power plant thermodynamic cycle, and its thermal

performance directly impacts the economic and safe operation of the overall plant [1]. Fouling of steam

condenser tubes is one of the most important factors affecting their thermal performance, which reduces

effectiveness and heat transfer capability with time [2, 3]. It is found that the maximum decrease in

effectiveness due to fouling is about 55 and 78% for the evaporative coolers and condensers, respectively

[2]. As a consequence, the formation of fouling in condenser of thermal power plants has special economic

significance [4-6]. Furthermore, it represents the concerns of modem society in respect of conservation of

limited resources, for the environment and the natural world, and for the improvement of industrial working

conditions [6, 7].

The fouling of heat exchangers is a wide ranging topic coveting many aspects of technology, the

designing and operating of condenser must contemplate and estimate the fouling resistance to the heat

transfer. The knowledge of the progression and mechanisms of formation of fouling will allow a design of

* Manuscript

an appropriate fouling mitigation strategy such as optimal cleaning schedule to be made. The most common

used models for fouling estimation are the thermal resistance method and heat transfer coefficient method

[6-10]. However, the residual fouling of periodic fouling deposition process and the dynamic changes of

heat exchanger operating condition are not considered in these models. Consequently, the estimation error

of those methods is very large.

Artificial Neural Networks (ANNs) are capable of efficiently dealing with many industrial problems

that cannot be handled with the same accuracy by other techniques. To eliminate most of the difficulties of

traditional methods, ANNs are used to estimate and control the fouling of heat exchanger in recent years.

Prieto et al [11] presented a model that uses non-fully connected feedforward artificial neural networks for

the forecasting of a seawater-refrigerated power plant condenser performance. Radhakrishnan et al [12]

developed a neural network based fouling model using historical plant operating data. Teruela et al [13]

described a systematic approach to predict ash deposits in coal-fired boilers by means of artificial neural

networks. To minimize the boiler energy and efficiency losses, Romeo and Gareta illustrated a hybrid

system that combines neural networks and fuzzy logic expert systems to control boiler fouling and optimize

boiler performance in [14]. Fan and Wang proposed diagonal recurrent neural network [15] and multiple

RBF neural network [16] based models for measuring fouling in thermal power plant condenser.

Although the technique of ANNs is able to estimate the fouling evolution of heat exchanger with

satisfaction, there are some problems. The selection of structures and types of ANNs dependents on

experience greatly, and the training of ANNs are based on empirical risk minimization (ERM) principle

[18], which aims at minimizing the training errors. ANNs therefore face some disadvantages such as

over-fitting, local optimal and bad generalization ability. Support vector machines (SVMs) are a new

machine learning method deriving from statistical learning theory [18, 19]. Since later 1990s, SVMs are

becoming more and more popular and have been successfully applied to many areas such as handwritten

digit recognition, speaker identification, function approximation, chaotic time series forecasting, nonlinear

control and so on [20-24]. Established on the theory of structural risk minimization (SRM) [19] principle,

compared with ANNs, SVMs have some distinct advantages such as globally optimal, small sample-size,

good generalization ability and resistant to the over-fitting problem [18-20]. In this paper, the use of SVMs

model is developed for the predicting of a thermal power plant condenser. The prediction model was used

in a case study of 300MW thermal power station. The experiment result shows that the prediction model

based on SVMs is more precise than thermal resistance model and other methods, such as T-S fuzzy model

[17]. Moreover, to improve the generalization performance of SVMs, an improved differential evolution

algorithm is introduced to optimize the parameters of SVMs.

2. Periodic fouling process in condenser

The accumulation of unwanted deposits on the surfaces of heat exchangers is usually referred to as

fouling. In thermal power station condensers, fouling is mainly formed inside the condenser tubes, reducing

heat transfer between the hot fluid (steam that condenses in the external surface of the tubes) and the cold

water flowing through the tubes. The presence of the fouling represents a resistance to the transfer of heat

and therefore reduces the efficiency of the condenser. In order to maintain or restore efficiency it is often

necessary to clean condensers. The Taprogge system has found wide application in the power industry for

the maintenance of condenser efficiency, which is one of on-line cleaning systems [6]. When the fouling

accumulation in condensers reached a threshold, the sponge rubber balls cleaning system is activated,

slightly oversized sponge rubber balls continuously passed through the tubes of the condenser by the water

flow, and the fouling in the condenser is reduced or eliminated. The progresses of fouling accumulating and

cleaning continue alternatively with time. Therefore, the fouling evolution in power plant condensers is

periodic.

However, the sponge rubber ball system is only effective of preventing the accumulation of

waterborne mud, biofilm formation, scale and corrosion product deposition [6]. As for some of inorganic

materials strongly attached on the inside surface of tubes, e.g. calcium and magnesium salts, can not be

effectively reduced by this technique. As a result, at the end of every sponge rubber ball cleaning period,

there still exist a lot of residual fouling in the condensers, and the residual fouling will be accumulated

continuously with the time. Where, the fouling can be cleaned by the Taprogge system is called soft fouling,

and those can not be cleaned residual fouling is called hard fouling. When the residual fouling accumulated

to some degree, the cleaning techniques that can eliminate them, such as chemistry cleaning method,

should be used.

Generally, the foul degree of heat exchanger is expressed as fouling thermal resistance, defined as the

difference between rates of deposition and removal [6]. In this paper, the corresponding fouling thermal

resistance of soft fouling and hard fouling expressed as Rfs and Rfh, respectively. Then, the condenser

fouling thermal resistance Rf in any time is the sum of soft fouling thermal resistance and hard fouling

thermal resistance, expressed as Eq. (1).

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 R t R t R t R t R t t R t t f fs fh f fs fh ? ? ? ? ? ? ? (1)

where ( ) 0 R t f is the initial fouling.

Fig. 1 periodic fouling evolution in power plant condensers

Fig. 1 demonstrates the periodic evolution process of fouling in power plant condensers. In fact, the

evolution process of fouling in a condenser is very complex, which is related to a great number of variables,

such as condenser pressure, cooling water hardness, the velocity of the circulating water and the

corresponding inlet and outlet temperatures, the non-condensing gases present in the condenser, and so on.

The Rfs(t) and Rfh(t) expressed a very complex physical and chemical process, their accurate mathematic

models are very hard to be obtained. Hence, measurement and prediction of fouling development is a very

difficult task. Since the fouling evolution process is a very complex nonlinear dynamic system, the

traditional techniques based on mathematic analysis, i.e. asymptotic fouling model, are not efficient to

describe it [11]. SVMs, as a small sample method to deal with the highly nonlinear classification and

regression problems based on statistic learning theory, is expected to be able to reproduce the nonlinear

behavior of the system.

3. SVMs regression and parameters

3.1 SVMs regression

SVMs are a group of supervised learning methods that can be applied to classification or regression.

SVMs represent an extension to nonlinear models of the generalized portrait algorithm developed by

Vladimir Vapnik [18]. The SVMs algorithm is based on the statistical learning theory and the

Vapnik-Chervonenkis (VC) dimension introduced by Vladimir Vapnik and Alexey Chervonenkis [19]. Here,

the SVMs regression is applied to forecast the fouling in power plant condensers.

Let the given training data sets represented as ?( , ), ( , ), , ( , )? 1 1 2 2 n n D ? x y x y ? ? ? x y , where d

i x ? R is

an input vector, y R i ? is its corresponding desired output, and n is the number of training data. In SVMs,

the original input space is mapped into a high dimensional space called feature space by a nonlinear

mapping x ? g(x) . Let f (x) be the SVM outputs corresponding to input vector x. In the feature space,

a linear function is constructed:

f (x) ? wT g(x) ? b (2)

where w is a coefficient vector, b is a threshold.

The learning of SVMs can be obtained by minimization of the empirical risk on the training data.

Where, ? -intensive loss function is used to minimize the empirical risk. The loss function is defined as

L? (x, y, f ) ? y ? f (x) ? max(0, y ? f (x) ?? ) e

(3)

where ? is a positive parameter to allow approximation errors smaller than ? , the empirical risk is

??

?

n

i

emp i i L x y f

n

R w

1

( , , )

1

( ) ? (4)

Besides using ? -intensive loss, SVMs tries to reduce model complexity by minimizing 2 w , which

can be described by slack variables. Introduce variables i

? and i

?? , then SVMs regression is obtained as

the following optimization problem:

min ?

?

? ?

n

i

i i w C

1

2 ( ? )

2

1 ? ? (5)

s.t. i i i y ? f (x ) ? ? ?? , i i i f (x ) ? y ? ? ??? , i

? , i

?? ? 0

where C is a positive constant to be regulated. By using the Lagrange multiplier method [18], the

minimization of (5) becomes the problem of maximizing the following dual optimization problem

max ( ? )( ? ) ( , )

2

1

( ? ) ( ? )

1 1 , 1

j j i j

n

i j

i i

n

i

i i i i

n

i

i ? y ? ?? ? ? ? ? ?? ? ? ? ?? ? ?? K x x

? ? ?

(6)

s.t. ( ? ) 0

1

? ? ?

?

n

i

i i ? ? ,C = i

?? , i

? =0

where i

?? and i

? are Lagrange multipliers, and kernel ( , ) i j K x x is a symmetric function which is

equivalent to the dot product in the feature space. The kernel ( , ) i j K x x is defined as the following.

( , ) ( ) ( ) j

T

i j i K x x ? g x g x (7)

There are some kernels, i.e. polynomial kernel K(x, y) ? (x ? y ? 1) d and hyperbolic tangent

kernel ( , ) tanh( ( ) ) 1 2 K x y ? c x ? y ? c can be used. Where the Gaussian function is used as the kernel.

)

2

( , ) exp( 2

2

?

x y

K x y

?

? ? (8)

Replacing i i i ? ? ? ? ? ? and relation 0 ? ? ? ?

i , then the optimization of (6) is rewritten as

max ( , )

2

1

1 1 , 1

j i j

n

i j

i

n

i

i i

n

i

i ? y ? ? ? ? ?? ? K X X

? ? ?

? ? (9)

s.t. 0

1

? ?

?

n

i

i ?

,C ? i

? ? ? C

The learning results for training data set D can be derived from equation (9). Note that only some of

coefficients i

? are not zeros and the corresponding vectors x are called support vectors (SV). That is, only

those vectors whose corresponding coefficients i i ?? ?? are not zero are SV. Then the regression function is

expressed as equation (10).

f x K x x b i j

p

i

i i ? ? ??

?

( ) ( ? ) ( , )

1

? ? (10)

It should be noted that p is the numbers of SV, and the constant b is expressed as

? ? ? ?

?

?

? ??

?

? ? ? ? ?

?

?

? ??

?

? ? ? ? ?

? ?

p

i

i i i i

p

i

i i i i b y K x x y K x x

1 1

min ( ? ) ( , ) max ( ? ) ( , )

2

1 ? ? ? ? (11)

3.2 SVM parameters

The quality of SVMs models strongly depends on a proper setting of parameters and SVMs

approximation performance is sensitive to parameters [25, 26]. Parameters to be regulated include

hyper-parameters C, ? and kernel parameter? , if the Gaussian kernel is used [25]. The values of C,

? and ? are relate to the actual object model and there are not fixed for different data set. So the problem

of parameter selection is complicated.

The values of parameter C, ? and ? affect model complexity in a different way. The parameter C

determines the trade-off between model complexity and the tolerance degree of deviations larger than ? .

The parameter? controls the width of the ? -insensitive zone and can affect the numbers of SV in

optimization problem. The kernel parameter? determines the kernel width and relates to the input range of

the training data set. Here, parameters selection is regarded as compound optimization problem and an

improved differential evolution algorithm is proposed to select suitable parameters value.

4. Improved Differential Evolution

Differential evolution (DE) algorithm is a simple but powerful population-based stochastic search

technique for solving global optimization problems [27]. DE has three operations: mutation, crossover and

selection. The crucial idea behind DE is a scheme for generating trial vectors. Mutation and crossover are

used to generate trial vectors, and selection then determines which of the vectors will survive into the next

generation. The original DE algorithm is described in the following briefly.

4.1 Basic differential evolution

Let S ? Rn be the search space of the problem under consideration. Then, the DE algorithm utilizes

NP, n-dimensional vectors X x x xt S i NP

in

t

i

t

i

t

i ( , , , ) , 1,2, , 1 2 ? ? ? ? ? as a population for each

generation of the algorithm. t denotes one generation. The initial population is generated randomly and

should cover the whole parameter space. In each population, two operators, namely mutation and crossover,

are applied on each individual to yield a trial vector for each target vector. Then, a selection phase takes

place to determine the trial vector enters the population of the next generation or not.

For each target individual t

i X , a mutant vector { 1 , , 1}

1

?1 ? ? t ?

n

t t

i V v ? v is determined by the

following equation.

( ) 1 2 3

1 t

r

t

r

t

r

t

i V ? ? X ? F ? X ? X (12)

Where F ? 0 is a real parameter, called mutation constant, which controls the amplification of the

difference vector ( ) 2 3

t

r

t

r X ? X to avoid search stagnation. According to Storn and Price [27], the F is set

in (0, 2]. 1 r , 2 r , 3 r are indexes, randomly selected from the set {1,2,..., NP} . Note that indexes must

be different from each other and from the running index i so that NP must be a least four.

Following the mutation phase, the crossover (recombination) operator is applied on the population.

For each mutant vector t ?1

i V , a trial vector { 1 , , 1}

1

?1 ? ? t ?

n

t t

i U u ? u is generated, using the following

scheme.

??

?? ?

? ?

? ?

?

?

?

, ( ) ( )

1 , ( ) ( )

1

x rand j CR and j randn i

v rand j CR or j randn i

u

t

ij

t

t ij

ij (13)

Where j=1, 2, ?, n. rand( j) is the jth evaluation of a uniform random number generator within [0, 1].

CR is a crossover probability constant in the range [0, 1], which has to be determined previously by the user.

randn(i) ? (1,2,...,n) is a randomly chosen index which ensures that t ?1

i U gets at least one element

from t ?1

i V . Otherwise, no new parent vector would be produced and the population would not alter.

To decide whether the trial vector t ?1

i U should be a member of the population comprising the next

generation, it is compared to the corresponding target vector t

i X , and the greedy selection strategy is

adopted in DE. The selection operator is as following.

??

?? ?

?

?

? ?

?

, otherwise

1 , ( 1 ) ( )

1

t

i

t

i

t

i

t

t i

i X

U f U f X

X . (14)

4.2 Modification of Mutation

From the mutation Eq. (12) we can see that in the original DE three vectors are chosen at random for

mutation and the base vector is then chosen at random within the three, which has an exploratory effect but

it slows down the convergence of DE. In order to accelerate the convergence speed, a modified mutation

scheme is adopted.

The randomly selected three vectors for mutation are sorted by ascending in terms of the fitness

function value. The tournament best vector is t

tb x , the better vector is t

tm x and the worst vector is t

tw x .

For speeding up convergence, the base vector in the mutation equation should select t

tb x , and the direction

of difference vector should direct to t

tm x , that is to choose ( t )

tw

t

tm x ? x as the difference vector. Then the

new modified mutation strategy is as following Eq. (15).

1 ( t )

tw

t

tm

t

tb

t

i v ? ? x ? F ? x ? x . (15)

After such modification, this process explores the region around each t

tb x in the direction of ( t )

tw

t

tm x ? x

for each mutated point. The mutation operator is not random search any more, but a determinate search.

However, the vectors for mutation are selected randomly in the population space, so in the whole

evolutionary process it is still a random search, which can ensure the global optimization performance of

the algorithm [28].

5 Optimization procedures of IDE for SVMs

5.1 Objective function

The objective of SVMs parameters optimization is to minimize deviations between the outputs of

training data and the outputs of SVMs. Where, the mean square error (MSE) is used as the performance

criterion.

2

1

1

( ( , ))2

1 ?

?

?

??

?

? ? ?

?

K

k

k k y f x w

K

Obj (16)

Where K is the number of training data, k y is the output of the kth training data, and f (x ,w) k is the

output of SVMs correspond to input k x . Then the objective of the IDE is to search optimal parameter C,

? and ? to minimize Obj:

min F(C,? ,? ) ? minObj (17)

Generally, the search range of these parameters is C? [1, 1000], ? ? (0, 1], ? ? (0, 0.5]. For

special problem, the search range is changeable.

5.2 Optimization procedures

The searching procedures of the improved differential evolution (IDE) for optimization of SVMs

parameters are shown as below.

Step1: Input the training data and test data, select the Gaussian kernel function.

Step2: Specify the number of population NP, the difference vector scale factor F, the crossover

probability constant CR, and the maximum number of generations T. Initialize randomly the individuals, i.e.

C, ? and ? , of the population and the trial vector in the given searching space. Set the current

generation t=0.

Step3: Use each individual as the control parameters of SVMs, train the SVMs using training data.

Step4: Calculate the fitness value of each individual in the population using the objective function given

by equation (17).

Step5: Compare each individual?s fitness value and get the best fitness and best individual.

Step6: Generate a mutant vector according to equation (15) for each individual.

Step7: According to equation (13), do the crossover operation and yield a trial vector.

Step8: Execute the selection operation in terms of equation (14) and generate a new population.

Step9: t=t+1, return to Step3 until to the maximum number of generations.

6 Case study

6.1 Fouling prediction scheme

The formation and development of fouling in condensers is influenced not only by cooling water

hardness and turbidity but also by working conditions of condensers, such as velocity of the cooling water

and the corresponding inlet and outlet temperatures, the saturation temperature of steam under entrance

pressure of condenser, the non-condensing gases present in the condenser, and so on. According to the

previous analysis of periodic fouling process of power plant condensers, the fouling can be classified as

soft fouling and hard fouling. Therefore, two SVMs models are developed to forecast thermal resistance of

soft and hard fouling, respectively. Then, the whole prediction fouling thermal resistance ( f R? ) in

condenser is the sum of output of soft fouling prediction model ( fs R? ) and output of hard fouling prediction

model ( fh R? ).

Generally, the evolution of soft fouling is determined by the velocity (v), turbidity (d), inlet (Ti) and

outlet temperatures (To) of cooling water, saturation temperature of steam under entrance pressure of

condenser (Ts), and prediction time range (Tp) (the running time in a sponge rubber ball cleaning period).

Therefore, these variables are chosen as inputs of the soft fouling thermal resistance predictive model. As

for hard fouling of the class of calcium and magnesium salts, it is related to the residual fouling at the

beginning and the end of previous sponge rubber ball cleaning period (corresponding thermal resistance is

Rfb,n-1, Rfe,n-1, respectively), hardness of cooling water (s), saturation temperature of steam under entrance

pressure of condenser (Ts), and the accumulating running time of condenser (Ta). Hence, those variables are

chosen as the inputs of hard fouling thermal resistance prediction model. The soft and hard fouling

prediction model based on SVMs illustrated in Fig. 2 and Fig. 3, respectively.

( , ) 1 K x x

( , ) 2 K x x

( , ) p?1 K x x

( , ) p K x x

1 1 ?? ??

2 2 ?? ??

1 1 ? ? ? ? p p ? ?

p p ?? ??

( , ) 1 K x x

( , ) 2 K x x

( , ) p?1 K x x

( , ) p K x x

S

b

Ts

1 1 ?? ??

2 2 ?? ??

1 1 ? ? ? ? p p ? ?

p p Ta ?? ??

Rfh

Rfb,n-1

R fe,n-1

?

^

Fig. 2 Soft fouling prediction model Fig. 3 Hard fouling prediction model

The parameters of the two prediction models are optimized by the IDE algorithm. Fig. 4 illustrates the

fouling prediction model using SVMs optimized by IDE.

?

Fig. 4 fouling prediction model based on SVMs optimized by IDE

6.2 Experiment results

In this section, experiments on N-3500-2 condenser (300MW) in Xiangtan thermal power plant are

carried out to prove the effectiveness of the proposed approach. The cooling water of this plant is river

water that pumped from the Xiangjiang river. The Taprogge systems are installed in the plant to on-line

clean the condensers. At present, the condenser is cleaned every two days using the Taprogge system, and

every cleaning time is about 6 hours. Obviously, the fitted cleaning period is not optimal, because the

fouling accumulating process is dynamic changing with the operating conditions changing.

The experiment system consists of sensors for operating condition parameters measuring, data

acquisition system, PC-type computer, etc. A set of 1362 real-time running data in different operating

conditions in 84 cleaning periods is collected to train and optimize the SVMs model for fouling prediction,

another set of 300 data is chosen for model verification. The proposed IDE is used to optimize the SVMs

parameters. The control parameters of IDE are the following. The number of population is 30, the crossover

probability constant CR is 0.5, the mutation factor F is 0.5, and the maximum number of generations is 100.

The selection of above parameters is based on the literature [27] and [28]. After application of IDE, the

optimal SVMs parameters of soft fouling prediction model are C=848, ? =0.513, ? =0.0117, the optimal

SVMs parameters of hard fouling prediction model are C=509, ? =0.732, ? =0.0075.

The velocity, turbidity, and inlet temperature of cooling water is different in summer and winter, the

evolution of fouling in condensers is also different in the two seasons. In the experiments, four sponge

rubber ball cleaning periods in different seasons are investigated. Among them, three periods, i.e. the first,

18th and 40th period, are in summer, and the other period is in winter. The hardness and turbidity of cooling

water is 56mg/L and 17mg/L in summer, and is 56mg/L and 29mg/L in winter.

To demonstrate the effectiveness of the proposed approach, the comparison between the SVMs model,

T-S fuzzy logic model [17] and asymptotic model is considered. The asymptotic model is obtained by

probability analysis method, and the corresponding expression is the following [17].

( ) ? 41.3?[1? ?(t ?1.204) /14.57 ]

f R t e (17)

Table 1 and Table 2 show the fouling thermal resistance prediction results of the above three models in

the first and the 18th cleaning periods, respectively. From the Table 1 and Table 2, we can see that compared

with tradition asymptotic model and T-S fuzzy logic model, the SVMs based prediction model has higher

prediction precision. Fig. 5 and Fig. 6 show the predicted fouling thermal resistance evolution based on the

optimized SVMs model and asymptotic model. Fig.6 clearly shows that the asymptotic model is not able to

forecast the fouling evolution process at the beginning stage of the 18th cleaning period, the reason is that

the residual fouling in the periodic fouling formation process is not considered in the asymptotic models.

Table 1 fouling thermal resistance prediction results in the first cleaning period

Running

time Tpa

(hour)

Operating conditions Measuring

values Rf

(K.m2/kW)

Prediction values (K.m2/kW) Relative error

v(m/s) Ti(?) Ts(?)

SVMs

model

T-S

model

Asymptotic

model

SVMs

model

T-S

model

Asymptotic

model

0 2.0 19.1 33.2 0.0258 0.0260 0.0258 -- 0.62 0 --

5 2.0 18.5 33.3 0.0995 0.0992 0.1018 0.0947 0.26 2.31 4.82

10 2.0 15.6 31.9 0.2028 0.2037 0.2007 0.1872 0.45 1.04 7.69

15 2.0 14.3 31.6 0.2501 0.2494 0.2411 0.2528 0.27 3.6 1.08

20 2.0 15.5 33.5 0.2865 0.2864 0.2830 0.2993 0.03 1.22 4.48

25 2.0 15.5 34.0 0.3174 0.3172 0.3123 0.3323 0.06 1.61 4.69

30 2.0 16.1 34.8 0.3420 0.3393 0.3321 0.3558 0.79 2.89 4.04

35 2.0 14.4 34.6 0.3567 0.3562 0.3497 0.3724 0.14 1.96 4.40

40 2.0 14.2 34.9 0.3722 0.3736 0.3600 0.3842 0.37 3.28 3.22

Table 2 fouling thermal resistance prediction results of the 18th cleaning period

Running

time Ta

(hour)

Operating conditions Measuring

values Rf

(K.m2/kW)

Prediction values (K.m2/kW) Relative error

v(m/s) Ti(?) Ts(?)

SVMs

model

T-S

model

Asymptotic

model

SVMs

model

T-S

model

Asymptotic

model

632 2.0 14.0 29.8 0.0774 0.0791 0.074 -- 2.26 0 --

637 2.0 14.2 30.9 0.1772 0.1773 0.1850 0.0947 0.06 4.40 46.56

642 2.0 12.5 30.4 0.2474 0.2479 0.2438 0.1872 0.21 1.46 24.33

647 2.0 11.9 30.4 0.2898 0.2908 0.2955 0.2528 0.36 1.97 12.77

652 2.0 10.6 30.1 0.3230 0.3222 0.3354 0.2993 0.25 3.84 7.34

657 2.0 11.4 31.5 0.3447 0.3437 0.3525 0.3323 0.28 2.26 3.60

662 2.0 10.2 31.2 0.3655 0.3652 0.3648 0.3558 0.08 0.19 2.65

667 2.0 10.7 32.0 0.3831 0.3815 0.3767 0.3724 0.42 1.67 2.79

672 2.0 11.8 33.5 0.3985 0.3978 0.3912 0.3842 0.18 1.83 3.59

To eliminate the influence of residual fouling and improve the prediction precision, an improved

asymptotic models are introduced in [17] according to the probability analysis method. The improved

asymptotic model of the 40th cleaning period is the following [17].

( ) ? 41.3?[1? ?(t ?4.31) /14.57 ]

f R t e (18)

Fig. 5 predicted fouling thermal resistance evolution Fig. 6 predicted fouling thermal resistance evolution

in the first cleaning period in the 18th cleaning period

At the 27th hour of the cleaning period, the velocity of cooling water increased from 2.0m/s to 2.5m/s

to adapt the steam load change, and the experiment result shows in Table 3. From the Table 3, we can see

that from 0 to the 27th hour in this cleaning period, the improved asymptotic model is able to well forecast

the fouling evolution. At the time of 27th hour, the velocity of cooling water increased suddenly, and the

fouling thermal resistance will slowly increase after decreasing a period of time. However, the improved

asymptotic model is not able to forecast the tendency. The models of SVMs and T-S fuzzy logic are able to

predict the tendency, but the SVMs model has more accurate results than T-S model. Fig. 7 demonstrates

the predicted fouling thermal resistance evolution based on the optimized SVMs model and the improved

asymptotic model.

Table 3 fouling thermal resistance prediction results of the 40th cleaning period

Running

time Ta

(hour)

Operating conditions Measuring

values Rf

(K.m2/kW)

Prediction values (K.m2/kW) Relative error

v(m/s) Ti(?) Ts(?)

SVMs

model

T-S

model

Asymptotic

model

SVMs

model

T-S

model

Asymptotic

model

1289 2.0 10.5 28.4 0.1065 0.1070 0.1066 0.1058 0.47 0.01 0.66

1294 2.0 10.1 28.6 0.2031 0.2011 0.2026 0.1950 0.98 0.25 3.99

1299 2.0 10.8 30.1 0.2660 0.2676 0.2574 0.2583 0.60 3.23 2.89

1304 2.0 12.0 32.0 0.3130 0.3161 0.3053 0.3032 0.99 2.46 3.13

1309 2.0 12.5 32.9 0.3422 0.3408 0.3398 0.3351 0.41 0.70 2.07

1314 2.5 11.1 32.3 0.3672 0.3687 0.3625 0.3578 0.41 1.28 2.56

1316 2.5 10.1 32.0 0.3734 0.3728 0.3712 0.3648 0.16 0.59 2.30

1319 2.5 10.0 32.2 0.3173 0.3152 0.3327 0.3738 0.66 4.85 17.81

1324 2.5 11.7 33.5 0.3048 0.3054 0.3135 0.3852 0.20 2.85 26.38

1329 2.5 13.2 34.8 0.3079 0.3097 0.3202 0.3933 0.58 3.99 27.73

Fig. 7 predicted fouling thermal resistance evolution Fig. 8 predicted fouling thermal resistance evolution

in the 40th cleaning period in the 85th cleaning period

The 85th cleaning period in winter is also considered in the experiment to further evaluate the

effectiveness of the proposed approach. Table 4 shows the fouling thermal resistance prediction results of

this period. From the Table 4 we can observe that the prediction model based on the optimized SVMs has

more satisfactory results than the improved asymptotic model and T-S model. Fig. 8 demonstrates the

predicted fouling thermal resistance evolution based on the optimized SVMs model and the improved

asymptotic model.

Table 4 fouling thermal resistance prediction results of the 85th cleaning period

Running

time Ta

(hour)

Operating conditions Measuring

values Rf

(K.m2/kW)

Prediction values (K.m2/kW) Relative error

v(m/s) Ti(?) Ts(?)

SVMs

model

T-S

model

Asymptotic

model

SVMs

model

T-S

model

Asymptotic

model

2830 2.0 2.5 24.9 0.1594 0.1698 0.1632 0.1058 0.99 2.38 33.63

2835 2.0 2.6 25.4 0.2467 0.2541 0.2345 0.1950 0.64 4.95 20.96

2840 2.0 2.8 26.3 0.3055 0.3065 0.2827 0.2583 0.01 7.46 15.45

2845 2.0 2.6 27.1 0.3410 0.3414 0.3177 0.3032 0.12 6.83 11.09

2850 2.0 2.5 27.7 0.3696 0.3702 0.3548 0.3351 0.17 4.01 9.33

2855 2.5 2.4 28.1 0.3891 0.3932 0.3776 0.3578 1.05 2.96 8.04

2860 2.5 2.4 29.2 0.4005 0.4003 0.3883 0.3648 0.05 3.05 8.92

2865 2.5 2.6 29.6 0.4087 0.4075 0.3979 0.3738 0.28 2.64 8.54

2870 2.5 2.7 29.4 0.4138 0.4132 0.4062 0.3852 0.14 1.84 6.91

7 Conclusions

The fouling accumulating process in condensers is very complex, which is influenced by a lot of

operating conditions. Hence, fouling prediction is a very difficult task, and the traditional techniques, such

as asymptotic fouling model and heat transfer coefficient method, are not efficient to describe it. In this

paper, the condenser fouling forecasting model based on SVMs was proposed. The parameters of the SVMs

model were optimized by an improved differential evolution. Practice application experiment results show

that, compared with traditional asymptotic model, the forecasting model based on optimized SVMs is able

to effectively eliminate the influence of residual fouling to forecasting accuracy, has good dynamic

self-adaptive ability and can obtain satisfactory forecasting accuracy when the condenser operating

conditions varied in a large range. The proposed approach is also compared with the T-S fuzzy logic model,

experiment results in four different cleaning periods present that the optimized SVMs model has higher

forecasting accuracy than T-S model.

The relevance of these results shows that the technique of support vector machines is able to estimate

well the evolution of the fouling in thermal power plant condensers without being much affected by the

operating conditions change and periodic residual fouling, and the general principles of application of

SVMs developed here would be useful in other problems of fouling.

Acknowledgement: This work was partly supported by the National High Technology Research and

Development Program of China (Grant no. 2008AA04Z214, 2007AA04Z244), the State Key Program of

National Natural Science Foundation of China (Grant no. 60835004, 60775047), the Scientific Research

Fund of Hunan Provincial Education Department (08C337), and the Program for New Century Excellent

Talents in University.

References: