ЗАЩИТА МЕТАЛЛОВ, 2007, том 43, № 4, с. 439-444
== == МЕТОДЫ ИЗУЧЕНИЯ ФИЗИКО-ХИМИЧЕСКИХ СИСТЕМ, СТРУКТУРЫ И СВОЙСТВ МАТЕРИАЛОВ И ПОКРЫТИЙ
УДК 541.138.3
AN IMPROVED GREY MODEL FOR CORROSION PREDICTION
OF TANK BOTTOM1
© 2007 r. S. M. Liu and G. Y. Zhang
Institute of Soil Science, Chinese Academy of Sciences, Nanjing, 210008, China E-mail: smliu@issas.ac.cn, Tel.: +86 (0)25 86881033 Received December 17, 2006
Leaking storage tank caused by bottom corrosion easily leads to soil and underground water contamination. It is important for tank operators to pay attention to corrosion controlling and monitoring of tank bottom underground. Though it is difficult to understand the process of underground corrosion of tank bottom, the progress of corrosion depth can be predicted by forecasting models. An improved first-order one-variable grey prediction model (IGM (1, 1) model) was proposed for the corrosion prediction of tank bottom. The IGM model used the synthesizing grey correlation degree as weights to modify the results from GM (1, 1) models based on long and short son data sequence from the identical original data. The IGM model was applied to simulate the progress of corrosion depth of tank bottom from 1994 to 1999. Results showed that the IGM dynamic model was more precise than the existing GM (1, 1) model and was an effective tool in the corrosion prediction of tank bottom. And the values of corrosion depth of tank bottom from 2000 to 2001 were predicted.
PACS: 81.15.-z
INTRODUCTION
Storage tanks are the usual facilities for petroleum products and hazardous chemical substances in refineries and chemical plants. Such many tanks constructed of steel are buried beneath the land surface. As the tank bottom is supported by the ground and is subjected to only hydrostatic pressures, the bottom can be made of thinner metal than is used for pipelines, which operate under pressure. But the tank bottom is subjected to the same corrosion issues as are buried pipelines [1]. Since the metal of tank bottom is thinner, it can be more easily perforated by even low rates of corrosion [2]. Nondestructive testing results also demonstrate that tank bottom is a severer corrosion location [3]. Corrosion of tank bottom usually leads to crack and finally results in leaking. Both crude oil spills from storage tanks into bunds at a Kaohsiung, Taiwan refinery in 2002 and at a Fawley, Hampshire, UK refinery were caused by the corrosion of tank bottom [4]. The proximity of these tanks to soil and groundwater poses substantial environmental risk. Leaking storage tank becomes one of the major sources of soil and underground water contamination with petroleum products and hazardous chemical substances into soil. They cause approximately 40% of all groundwater contamination in the US [5]. Of the more than 19.000 documented leaking tanks in Texas, 6000 have impacted groundwater [6].
Much attention must be paid to the corrosion state of storage tank bottoms. Normally, it is difficult to under-
1 Financed by Chinese Academy of Sciences (Grant No. KSCX2-YW-N-038) and National Nature Science Foundation of China (Grant No. 5049933X-N).
stand the corrosion process of tank bottom underground, but the progress of corrosion depth can be predicted by forecasting models. The main purpose of prediction is to clear up the uncertain future of tanks and to offer to the administrators the related information. The administrators can make appropriate decision for the future safe operation of storage tanks according to the accurate prediction.
In this paper, an improved GM (1, 1) model (IGM model) was proposed for corrosion prediction of tank bottom. The IGM model was applied to simulate the progress of corrosion depth of storage tank bottoms from 1994 to 1999. And the values of corrosion depth of tank bottom from 2000 to 2001 were predicted. This work may be beneficial to tank operators and engineers.
THEORETICAL ANALYSIS
Factors influencing the corrosion of tank bottom are variant and random, such as soil, air, water, storage oil, corrosive gas, temperature and the defects of plate and welding line. The relationship between various factors described above is unclear. This is a complex and multivariate system. Such system often implies poor, incomplete, and uncertain information. For the complex environment of tank bottoms, practical and experimental data are difficult to obtain and too much scatter to analyze. The conventional prediction method based on classical statistics, requiring not only a large number of history data but also typical distributions, are not efficient enough to solve such complex problems and find out the corrosion laws. So we should walk out from the shadow of large sample statistics.
In 1982, Deng proposed grey system theory [7] to study the uncertainty of a system. In this theory, a system with partial information known and partial information unknown is called as grey system. Grey forecasting theory focuses on model uncertainty and information insufficiency in analyzing and understanding system via research on prediction and decision making. It avoids the inherent defects of conventional, large sample statistical methods, and only requires a limited number of discrete data to estimate the behavior of a system with incomplete information. To date grey theory has been widely applied to many research fields such as economics, sociology, engineering, and so forth [8-11].
Among the family of grey forecasting models, GM (1, 1) model is the simplest and the most frequently used grey prediction model to describe a grey system. However the existing GM (1, 1) model has a defect. It always uses the whole original data to establish the forecasting model and then directly applies the model to predict the future data. But we all know that the predicted future values have great relation only with near most up to date original data. The existing GM (1, 1) model does not completely utilize the up to date information contained in the original data in different period. This leads to a result that the prediction accuracy may not be always satisfactory. So many studies were reported on how to improve the accuracy of the GM (1, 1) model [12-16]. However, to build a reasonable forecasting model, most of these methods were either rather complex or requiring a large number of data points. Especially they concentrated on revising the GM (1, 1) model with residual error modification. With few exceptions, researchers paid less attention to improve the existing GM (1, 1) model.
In this paper, an improved GM (1, 1) model (IGM model) was proposed. The IGM model used the synthesizing grey relational degree as the weights to modify the results from existing GM (1, 1) models based on different periods of son data series from the identical original data series. Thus the last information in different periods was all taken into account. So the predicted values were optimized and the prediction accuracy was improved. The IGM model was applied to simulate the progress of corrosion depth of storage tank bottoms from 1994 to 1999. And the values of corrosion depth of tank bottom from 2000 to 2001 were predicted.
GM (1, 1) MODEL
Assume that X(0) = {x(0)(1), x(0)(2), ..., x(0)(n - 1), x(0)(n)|n > 4} is the original non-negative data series taken in consecutive order and at equal time interval. The procedures for applying the GM (1, 1) model to predict the future value x(0)(« + =) with = > 1 can be described as follows [17, 18]:
Step 1 Form a new data series X1 = {x(1)(1), x(1)(2), ..., x(1)(« - 1), x(1)(«)} by an accumulated generating operation (AGO) technique:
x(1)(=) = £x(0)(i) .
(1)
Step 2 Establish the grey differential equation as below:
x(0)(k) + az(l\k) = b, k = 2, 3, n , (2)
where
z (k) = 0.5x J(k) + 0.5x (k -1 ).
(3)
Step 3 Estimate the developing coefficient a and the grey input b in Eq. (2) by the least squares method (LSM):
where
p = [ a, b ] = [ btb r1 BTYn,
-z( 1)( 2 ), 1
(4)
B =
-z( 1)( 3 ), 1
(5)
and
-z( ^ n ), 1
Y = [x(0) x(0) x(0)]T 1 n La2 > A3 > • • • > n J •
(6)
Step 4 Establish the following whitened first-order differential equation for future prediction:
dx{l) (i) , —— + ax = b .
(7)
Step 5 Solve the grey Eq. (7) and obtain the predicted values for the data X(1) from AGO as
x( 1)( k +1 ) = f x( 0)( 1 ) -b! e-a= + b,
= V ay a
k = 0, 1, 2, n - 1.
(8)
By applying the inverse AGO (IAGO), x(0)(= + 1) = = (x(1)(= + 1) - x(1)(=) to Eq. (8), the predictions for the original data series can be obtained:
x (0)(= +1 ) = ( 1- ea )V x( 0)( 1 ) -0-) e~"=\
= = 1, 2, ..., n - 1, in which, x( 0) (1) = x(0)(1).
i = 1
By Eq. (9), the data series {x(0) (1), x(0) (2), ..., x(0) (n)} are called fitted series, while series {X(0) (n + 1), X(0) (n + + 2), ..., X(0) (n + k)} are called predicted series.
IGM (1, 1) MODEL
When taking different periods of son data series from an identical sequence as modeling data, different GM (1, 1) models are established as the grey information and developing tendency contained in long or short son data series are diverse. Corresponding to a given original datum, a series of fitted data are obtained from different models. This just demonstrates the compact relationship between the fitted results and the "far or near" information reflected in long or short data series. Namely the predicted data have a certain relation with the original data series. Grey correlation degree can exactly reflect the approximate correlation of different sequences [19, 20]. So we propose an improved GM (1, 1) model (IGM model) for forecasting. The IGM model uses the synthesizing grey correlation degree as the weights to modify the predicted results from the existing GM (1, 1) model based on different periods of son data series from the identical original data. Forecasting data obtained from IGM model are the weighting combination of that from existing GM (1, 1) model based on long and short data series. The IGM model
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