Fault Diagnosis of Sucker-Rod Pumping System Using Support Vector Machine

Transcription

1 Fault Dagnoss of Sucker-Rod Pumpng System Usng Support Vector Machne Jln Feng, Maofa Wang,, Yheng Yang 2, Fangpng Gao, Zhan Pan, Wefeng Shan, Qngje Lu, Quge Yang, and Jng Yuan Department of Informaton Technology, Insttute of Dsaster Preventon, Bejng 060, P.R. Chna 2 College of Appled Scence, Bejng Informaton Scence and Technology Unversty, Bejng 000, P.R. Chna wangmaofa2008@26.com, fengjln@cdp.edu.cn Abstract. It s nterestng thng to dagnose the faults of Sucker-rod Pumpng System by analyzng dynamometer cards, whch s nexpensve and easy to obtan. But conventonal statstcal methods are often neffectve to dagnose the faults of Sucker-rod Pumpng System. One reason s that there are more parameters or smaller sample szes than a smple model n analyzng dynamometer cards, so that ther degree of freedom s reduced. Another reason s that the accurate structure of the conventonal statstcal model dagnosng the faults of Sucker-rod Pumpng System s hard to make certan, because of that statstcal crtera are crucally dependent on such assumptons as normalty, homogenety, ndependence. In the paper, we present a SVM-based approach for fault dagnoss of Sucker-rod Pumpng System by analyzng dynamometer cards. Wth the method, we can get the workng status of the Sucker-rod Pumpng System. Keywords: Support Vector Machne, Classfer, Pattern Recognton, Dynamometer card, Cross-valdaton, Ol well rod pump. Introducton Sucker-rod pumpng s the most wdely used means of artfcal lft. It s reported n [] about 85% to 90% of all producng wells n the USA use rod-pumped system. Thus, a relable method of analyzng these pumpng systems s a necessty. For many years, a down-hole dynamometer has been used to analyze sucker-rod systems. Recently, a varety of useful statstcal methods have been rased n analyzng down-hole dynamometer cards of sucker-rod pumpng system. Wth these study methods, people wsh to get more work status nformaton of sucker-rod pumpng system and clearly know whch type of fault s occurrng, so that we can take a well-tmed nterventon when some fault s occurrng. For example, artfcal neural network have been proposed to establsh pattern classes of down-hole dynamometer cards n ol well rod pump systems such as shown n [2, 3, 4] and so on. Correspondng author. M. Zhao and J. Sha (Eds.): ICCIP 202, Part II, CCIS 289, pp , 202. Sprnger-Verlag Berln Hedelberg 202

2 80 J. Feng et al. Tradtonally, neural networks were proved to be a powerful method on ntellgent fault dagnoss. Relable tranng methods have been developed manly thanks to nterdscplnary studes and nsghts from several felds ncludng statstcs, systems and control theory, sgnal processng, nformaton theory and so forth. Despte many of these advances, there stll remans a number of weak ponts such as, dffculty n choosng the number of hdden unts, over-fttng problem, exstence of many ocal mnmal solutons, and usually needng an abundant tranng samples. In order to overcome those tough problems, major breakthroughs have been made at ths pont wth a new class of neural net-works called support vector machnes (SVM), whch developed n the area of statstcal learnng theory and structural rsk mnmzaton. SVM has many advantages, such as automatc selecton of model complexty, few local mnmal solutons, no loss of dmensonalty, good generalzaton performance. SVM has been recently ntroduced for solvng pattern recognton and functon estmaton problems. In ths paper we wll dscuss the faults classfcaton of Sucker-rod Pumpng System by analyzng down-hole dynamometer cards wth SVM. Throughout the whole processng course, there are manly the followng approaches: preparng down-hole dynamometer cards data sets whch s a set of vector ponts whose two correspondng weghts are load and dsplacement, extractng characterstc values from nput dynamometer cards data, transformng the characterstc values to the format of an SVM method, conductng smple scalng on the data, consderng to use the RBF kernel X, usng cross-valdaton to fnd the best parameter C and γ, usng the best parameter C and γ to tran the whole tranng set, gvng the faults classfcaton labels of testng data set. 2 SVM Classfcaton Algorthm A lot of studes about SVM Classfcaton algorthm have been proposed n [5, 6, 7, 8], and the summary of the SVM Classfcaton algorthm s descrbed as the followng: Suppose we have gven Observatons. Each observaton object conssts of a par of data: a vector F and a class label y { +, } for each vector x. We say x belongs to class І f y = +, and x belongs to class ІІ f y = -. These data pars buld the tranng data sets. For lnearly separable data, we can determne a hyper-plane f ( x ) that separates the data sets. For a separatng hyper-plane f ( x) 0 f the nput x belongs to the postve class, and f( x ) < 0 f x belongs to the negatve class. m j= f( x) = w x+ b = w x + b () j j yf( x) = y( wx + b) 0, =,, (2) where w s an m-dmensonal vector and b s a scalar. w x s the nner producton of w and x. If we addtonally requre that w and b s such that the ponts closest to the hyper-plane has a dstance of/ w, the equaton (2) should be wrtten as

3 Fault Dagnoss of Sucker-Rod Pumpng System Usng Support Vector Machne 8 y ( w x + b), =,, (3) The separatng hyper-plane has the maxmum dstance between the hyper-plane and the nearest data,.e. the maxmum margn, s called the optmal separatng hyper-plane. The classfcaton ablty s maxmzed wth the optmal hyper-plane. A sketch of an optmal separatng hyper-plane of two data sets s shown n Fg., where H s the optmal separatng hyper-plane. The optmal hyper-plane can be obtaned by solvng the followng convex quadratc optmzaton problem: Fg.. A sketch of an optmal separatng hyper-plane 2 mnmze / 2 w (4) subject to y ( w x + b), =,, (5) Ths problem can be transformed nto the equvalent Lagrange dual problem as: where α ( α α α ) ( ) α /2 α α k k k =, k= (6) Q a = y y x x subject to yα = 0, α 0, =,, (7) = =, 2,, s the Lagrange factor. Each sample has correspondng α, =,,. Those samples for whose α > 0 are called support vectors, and are ones where the equalty condton holds n equaton (5). All other tranng samples havng α = 0 can be removed from the tranng set wthout affectng the optmal hyper-plane. Let us assume that optmal soluton of α for the dual problem s α, the soluton of w and b are w and b whch can be gven by: = αk = α = sup port vectors (8) w y x y x b w x for x wth y = = (9)

4 82 J. Feng et al. After tranng, the classfer can be used to classfy a new sample x by the decson functons: ( ) = ( α j j j + ) (0) j= f x sgn y x x b class I, f f ( x) =+ x { class II, f f ( x) = () Fg. 2. A sketch of mappng from the orgnal data space X to the feature space F The SVM classfer s based on a non-lnear kernel functon, whch maps the orgnal data space X where the samples may be non-lnear classfed to a data space F where the samples can be lnearly classfed. The new data space F s called the feature space of X. Ths s depcted n Fg. 2. ow, usng the non-lnear vector functon Φ ( x) = ( Φ( x),, Φ l ( x)), whch maps the d-dmensonal nput vector x nto the l-dmensonal feature space, the lnear decson functon n dual form s gven by: ( ) = ( α Φ( ) Φ ( ) + ) = f x sgn y x x b (2) otcng that n Equaton (2) as well as n the optmzaton problem Equaton (6), we only concern the nner productons of samples. So, n the hgher dmensonal space (feature space), we only deal wth the data n the form of the nner product Φ( x) Φ ( z). If the dmenson of F s very large, then t wll be dffcult or very expensve computatonally. However f t s possble to fnd a knd of functon to calculate nner products of feature space n the orgnal data space, ths functon s called a kernel functon, kxz (, ) =Φ( x) Φ ( z). Then we use ths kernel functon n place of kx ( ) =Φ( x) Φ ( z) everywhere n the optmzaton problem and never need to know explctly what Φ s. Usng a kernel functon, the decson functon wll be: f( x) = sgn y ( x, x) + b α (3) sup port ve ctors class I, f f ( x) =+ x { class II, f f ( x) = (4)

5 Fault Dagnoss of Sucker-Rod Pumpng System Usng Support Vector Machne 83 However, all kernels do not correspond to nner products n some feature space F. Wth a so-called Mercer theorem t s possble to fnd out f a kernel K depcts an nner product n that space where Φ s mapped. Some typcal kernel functons are: d Polynomal Functon : Kxy (, ) = ( xy + ), d=, 2, (5) Radal Bass Functon : K( x, y) = exp( x y / 2 σ ) (6) [ ] Sgmod Functon : K( x, y) = tanh b( x y) c (7) In our research, Radal Bass Functon s adopted. 3 Preparng Down-Hole Dynamometer Cards Data Sets For many years, the surface dynamometer has been used to analyze sucker-rod systems. Interpretaton of actual pump condtons from surface dynamometer cards s often dffcult, even not mpossble. Results obtaned from surface cards are strctly qualtatve and are dependent on the analyzer's expertse. The deal analyss procedure would be to measure the actual pump condtons wth a down-hole dynamometer. However, ths stuaton s not economcally feasble. Therefore, an accurate method of transformng down-hole pump cards from measured surface cards s needed. Several transformng methods have been wdely studed [9,, 0-]. In our research, we can calculate down-hole pump cards usng a quck recurson arthmetc whch s depcted by Peng et al. [2]. There occurs about vector ponts n each one down-hole pump card, where these vector ponts ndcate the dsplacement and the tenson values of the sucker rod n tme sequence n one stoke course. Then we gve out the class label of every one down-hole pump card n an emprcal manner. In expermental stage, a part of dynamometer cards need to be dvded nto three classes: the frst fault class s normal operaton, the second s no enough flud supplyng, and all other types of faults are regarded as the thrd fault whch ncludes leakng standng valve, worn out pump, stuck pston, gas nterference and so on. Those dvded dynamometer cards data wll be used to tran the classfcaton model. In the followng applcaton, the tranng dynamometer cards could be dvded nto more classes. The detal nformaton extractng from a tranng dynamometer cards fnally s saved as a text table, whose structure s shown as table. From whch, we can gan the man dynamometer card parameters ncludng fault class-label, mnmal and maxmal tenson of sucked-rod and, mnmal and maxmal dsplacement of sucked-rod, and common vector ponts of dsplacement and tenson. A true value of class-label can be only one of, 2 and 3, whch are respectvely correspondng to 3 types of work operatons: normal, no enough flud supplyng, and other types of fault. For a gven well, all the card parameters have the same values n mnmal and maxmal tenson, mnmal and maxmal dsplacement. If tranng data sze s too small, the representatve of samples wll be dubous, and f tranng data sze s too large, a heavy workload wll be need to emprcally and manually dvde the samples to dfferent classes. So that for each type of fault, preparng tranng data tables of dynamometer cards s feasble.

6 84 J. Feng et al. Table. The detal nformaton extractng of a tranng dynamometer 4 Extractng Characterstc Values from Per Input Dynamometer Card Because of that there are vector ponts of dsplacement and tenson n per down-hole dynamometer card, so the whole calculaton quanttes wll be very large f we use drectly these vector ponts data as parameters to tran and get classfcaton model. Therefore, we need extractng relatvely few and effectve feature parameters from vector ponts of down-hole dynamometer cards as nput parameters of classfcaton functon, whch s mportant for the accuracy of the fnal fault dagnoss. Accordng to analyss to abundant typcal down-hole dynamometer cards, we wll extract 3 dmenson feature parameters from a dynamometer card n total. The detaled extracton arthmetc of the 3 dmenson feature parameters s depcted as the followng: 4. ormalzng Dynamometer Card To enhance the robustness of parameters and get rght classfcaton model whch can use dynamometer cards from dfferent wells, we frstly need normalze all the dynamometer cards from dfferent wells. Eq. (8) and Eq. (9) shows the normalzaton processng. ds = ( ds PLQDs)/( maxds-plq Ds) (8) WHQ WHQ PLQ7HQ PD[ 7HQ PLQ7HQ (9) where ds s the dsplacement of one pont n a dynamometer card, WHQ s the tenson, PLQ Ds, max Ds, PLQ7HQ and PD[7HQ respectvely corresponds to mnmal dsplacement, maxmal dsplacement, mnmal tenson and maxmal tenson of a well.

7 Fault Dagnoss of Sucker-Rod Pumpng System Usng Support Vector Machne Calculatng the 5 Ponts Havng Maxmal Curvature Values of Every One Tranng Down-Hole Dynamometer Card Suppose there are three adjacent ponts: A(x, y ), B(x 0, y 0 ), C(x 2, y 2 ) n a dynamometer card where x and y represents dsplacement and tenson respectvely. And pont A s the pror pont of pont B, C s the backward pont of B, and the curvature of pont B can be calculated such as Eq. (20-24)): area = /2 x x0 y y0 (20) x x y y α = ( x x ) + ( y y ) (2) 0 0 β = ( x x ) + ( y y ) (22) δ = ( x x ) + ( y y ) (23) 2 2 cur = 2.0 area /( α β δ) (24) where area s the area of the trangle ABC, α s the length of lne segment AB, β s the length of lne segment BC, δ s the length of lne segment AC, and cur s the fnal curvature result of pont B. After calculatng curvature values of each one pont n a down-hole dynamometer card, we choose the 5 feature ponts: P, P 2, P 3, P 4, P 5, whose curvature values are larger than all other ponts. Physcally, the fve ponts corresponds to the shut and open postons of standng valve and travelng valve respectvely. 4.3 Calculatng 3 Dmenson Feature Parameters for Every Tranng Dynamometer Card Usng the 5 Feature Ponts Frstly, the dsplace values of the 5 feature ponts are used as the frst 5 dmenson feature tranng parameters. After that, we calculate the 5 slope values between every two adjacent ponts n the 5 feature ponts, and the 5 slope values are used as the second fve dmenson feature parameters. Eq. (25) gves the slope formula. slope = ( y y ) / ( x x ), =, 2,3, 4,5 (25) + + where f = 5, we gve the formula + the value of. Further, we need to calculate the means of tenson n the up and down stroke respectvely. And the two means are used as other two dmenson tranng feature parameters. Fnally, the area of pentagon PPPPP s calculated as the 3th dmenson feature parameter by the Eq. (26) and Eq. (27).

8 86 J. Feng et al. area = /2 x x y y = 2,3,4,5 x x y y + + (26) 4 area = area (27) where area s the area of the trangle PPP +, = 2,3, 4,5 of pentagon =, and area s the area PPPPP. The feature extracton transforms an observaton space of dmenson m, whch s the down-hole dynamometer cards data, nto the characterstc space of dmenson q, where q<m, n order to smple the classfcaton task. The structure of nput data of classfcaton functon s shown as table. Table 2. The tranng class-label and 3 dmenson feature parameter of a well Class-label 3 dmenson feature parameter 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Tranng and Testng In our research, we have developed our own software wrtten wth C++ language to dagnose faults of sucker-rod pumpng system, where the open-source code lbsvm wrtten by Chang and Ln [3] has been modfed and ntegrated nto our software system, and the chosen type of the SVM Classfcaton s C-SVC, and Radal Bass kernel functon s adopted. 5. Rescalng Feature Extracton Data Set The structure of orgnal tranng and testng data are shown as table 2, and the test data allows of no class-label weght. And these data may be too huge or small n range, thus we can rescale them to the proper range so that tranng and predctng.

9 Fault Dagnoss of Sucker-Rod Pumpng System Usng Support Vector Machne 87 The scale functon s descrbed by Ln (2009), and the upper lmt of scalng s, and the lower lmt s -. Table 3 s the scale date set of a tranng feature extracton data set, where range of every dmenson parameter s between - and. Table 3. The scale date set of a feature extracton data Class-label 3 dmenson feature parameter 3 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6: : : : : : : : Cross Valdatng to Get Optmzed SVM Tranng Parameters The two most crtcal parameters for SVM tranng model are cost and gamma. After gettng nce parameters: cost and gamma, we wll use them to tran the tranng data to get the best tranng mode and use the mode for fnal predcton on test data wth unknown class-label. The whole process to get nce parameter cost and gamma s called cross valdaton. In our research, the cross valdaton can be formalzed to the followng sx-step procedure: a) Splt the tranng data to 3 sets of pre-classfed data randomly. b) Obtan a par of cost and gamma by the followng: for =, 2, 3, 20 do for j =, 2, 3,0 do L F, M J c) Tran two sets of pre-classfed data wth parameters c and g obtaned by Eq. (27), and predct thrd set of data to calculate the accuracy. d) Choose another two sets of pre-classfed as tranng data, the other data as predct set, repeat step 3. So that we wll get the average accuracy wth ths set of parameter: cost and gamma. e) Repeatng Step 2, Step 3, and Step 4, we can obtan the maxmum of accuracy wth a gven set of parameter: cost and gamma, whch s the best tranng parameters. f) Tran all the tranng data wth above calculated parameter: cost and gamma to obtan the tranng model. The structure of the tranng model fle s descrbed as n Chang and Ln. (200).

10 88 J. Feng et al. 5.3 Predct the Testng Data The testng data mght have no class-labels. In the last step, we predct ther labels usng the obtaned tranng model. The average accuracy rate of tranng s hgher than 87%, whch are calculated by the consstent percent of orgnal label to predct label. The structure of predct result fle s shown as Table 4. Table 4. The predct result of a well Accuracy Orgnal class-label Predct class-label 9% Concluson and Prospect The present study have get good classfcaton effects even wth a few tranng sample, and more hopeful to analyze the faults of sucker-rod pump n slope well later. We also wrte the correlatve software wth c++ codes, whch apples the above processng algorthms. Usng the software, we can easly and automatcally gve out the class-label of the faults, whch could actually occurs n hundreds of meters deep underground. If more detaled fault classfcaton s requred such as no less than 3 fault classes, gvng out the fault class-label of tranng data n an emprcal manner wll become dffcult even f the tranng data sze s no so large. So the study about how to automatcally label the faults of trannng data s gong on now n our team. Acknowledgment. Two of Authors (Jln Feng, Maofa Wang) thank the fnancal support from fundng program of scentfc researchng for teachers of Chna sesmology bureau ( ). References. Evertt, T.A., Jennngs, J.W.: An Improved Fnte-Dfference Calculaton of Downhole Dynamometer Cards for Sucker-Rod Pumps. SPE Producton Engneerng 7, 2 27 (992) 2. Abello, J., Houang, A., Russell, J.: A Herarchy of Pattern Recognton Algorthms for the Dagnoss of Sucker Rod Pumped Wells. In: Conference on Computng and Informaton- ICCI, pp (993) 3. Bezerra, M.A.D., Schntman, L., Barreto Flho, M.D.A., Felppe De Souza, J.A.M.: Pattern Recognton for Downhole Dynamometer Card n Ol Rod Pump System usng Artfcal eural etworks. In: Internatonal Conference on Enterprse Informaton Systems- ICEIS, pp (2009)

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