Soft sensor for cobalt oxalate synthesis process in cobalt hydrometallurgy based on hybrid model Shuning Zhang, Fuli Wang, Dakuo He & Fei Chu
Neural Computing and Applications ISSN 0941-0643 Neural Comput & Applic DOI 10.1007/s00521-012-1096-x
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Author's personal copy Neural Comput & Applic DOI 10.1007/s00521-012-1096-x
ORIGINAL ARTICLE
Soft sensor for cobalt oxalate synthesis process in cobalt hydrometallurgy based on hybrid model Shuning Zhang • Fuli Wang • Dakuo He • Fei Chu
Received: 13 March 2012 / Accepted: 17 July 2012 Ó Springer-Verlag London Limited 2012
Abstract In the cobalt oxalate synthesis process in cobalt hydrometallurgy, the key end-product quality index, average particle size of cobalt oxalate, needs to be monitored and controlled. It is difficult to measure such particle size online by existing hardware sensors. Soft sensor technique has been widely used for estimating product quality or other important variables when online instruments and sensors are not available. In this paper, a hybrid modeling approach for cobalt oxalate synthesis process in cobalt hydrometallurgy is proposed by combining simplified first principle model with stacked LSSVR model. The former based on population balance equations and mass conservation equation with some assumptions is used for description and analysis of synthesis process in general; and the latter is developed to compensate the unmodeled characteristic and to enhance model generalization capability. Furthermore, a model output offset compensation strategy is also employed to increase the model prediction accuracy. Applications to a cobalt hydrometallurgy pilot plant demonstrate that the proposed approach is more precise and effective than the other conventional models. Keywords Hybrid model Least squares support vector regression Simplified first principle model Cobalt oxalate synthesis process Cobalt hydrometallurgy
S. Zhang (&) F. Wang D. He F. Chu College of Information Science and Engineering, Northeastern University, Shenyang 110004, Liaoning Province, People’s Republic of China e-mail:
[email protected] F. Wang D. He State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110004, People’s Republic of China
1 Introduction Cobalt oxalate synthesis process, which is an important composition unit of cobalt hydrometallurgy, is a batch crystallization process in fact. Batch crystallization is specially used in the production of high-value-added products, mainly because it offers flexible and simple processing steps for plants with frequently changing recipes and product lines [1, 2]. In synthesis process, product purity and particle size are of great importance. Furthermore, the cobalt oxalate crystals produced through a crystallization process have a decisive influence on the downstream processing, and therefore, the particle size should be reproducible in each operation and as regular as possible [3]. Hence, in the industrial practice, optimizing and tracking the average particle size of cobalt oxalate are the main concern in control. The performance of advanced process control and optimization depends on an accurate and efficient online measurement or prediction of average particle size. However, direct measurement may not always be available due to the financial burden. Hence, one has to resort to the offline measurement that is usually time-consuming and cannot satisfy the requirements of real-time optimal control. The measurement delay of the average particle size can be addressed through the use of soft sensor technique. And soft sensor systems have been developed for predicting product properties and indirectly controlling the end-product qualities by using the measurable secondary process variables, such as temperature, flow rate and rotational speed. The most popular soft sensor modeling methods are first principle model [4, 5], data-driven model [6] and hybrid model [7]. The first principle model can reflect the process and be explained, but it is difficult to develop in industrial practice and may not be suitable for
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agile responsive manufacturing. Data-driven model can be a very useful alternative in this case. However, the datadriven model cannot be explained and easily prone to over fitting. By combining the simplified first principle model and data-driven model, hybrid model can complement both methods to obtain good performance because the simplified first principle model can improve the extrapolation capability and the data-driven model can increase the prediction accuracy. However, most of the data-driven models (e.g., LSSVR models) for quality prediction using historical data are single model, which is strongly influenced by the availability of training data. In order to build an accurate LSSVR model, ideally a large amount training data should be made available. However, in cobalt oxalate synthesis process, the collection of sufficient, appropriate good quality data is still a real problem. In addition, data for offline measured quality variables are usually limited due to the cost in laboratory analysis. Limited process data are a serious problem in the development of accurate LSSVR models. To address the problem of limited process data, stacked neural networks also known as aggregated neural networks have been shown to possess better generalization than single neural network [8–10]. In the method, a set of neural networks is developed on bootstrap resampling replication of the original training data. Then, the trained individual neural networks are combined. However, the selection of parameters and the structure determination for neural networks are difficult. LSSVR can obtain the global solution without above-mentioned problems. Therefore, a stacked LSSVR model is proposed to compensate the unmodeled characteristic of the simplified first principle model in this paper. The main purpose of this paper is to present a hybrid modeling approach for cobalt oxalate synthesis process in cobalt hydrometallurgy by combining simplified first principle model with stacked LSSVR model. The simplified first principle model that captures the dominant characteristics of the synthesis process is built to predict the average particle size of cobalt oxalate crystals. The stacked LSSVR model is utilized to compensate the unmodeled characteristic of the simplified first principle model and to enhance the model generalization capability. Finally, a model output offset compensation strategy is also employed to rectify the output of hybrid model. The rest of this paper is organized as follows. A stacked LSSVR based modeling method is introduced in Sect. 2. Section 3 describes cobalt oxalate synthesis process in cobalt hydrometallurgy. The procedures of establishing the hybrid model for cobalt oxalate crystals and model output offset compensation strategy are developed in Sect. 4. The application of the proposed approach to a cobalt hydrometallurgy pilot plant is given in Sect. 5. Finally, Sect. 6 draws some concluding remarks of this paper.
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2 Stacked LSSVR 2.1 Least squares support vector regression LSSVR is briefly reviewed firstly in order to derive a stacked LSSVR. LSSVR first map the data from the original space to the feature space U : x ! H, and then the following optimization problem is formulated to get the wanted model f [11]: n 1 cX min Jðx; eÞ ¼ xT x þ e2 x;b;e 2 2 k¼1 k ð1Þ s:t yk ¼ xT uðxk Þ þ b þ ek ;
k ¼ 1; . . .; n:
where {xk, yk}nk=1 is a training set, x is weight vector, b is bias term, ek is error variable, and c is penalty parameter. The Lagrangian for the optimization problem is constructed as: n X Lðx; b; e; aÞ ¼ Jðx; eÞ ak ½xT uðxk Þ þ b þ ek yk k¼1
ð2Þ where ak are Lagrange multipliers. The conditions for optimality can be written as the solution to the following set of linear equations after eliminating the variables x and e: 0 1T b 0 ¼ ð3Þ 1 X þ cI a y where y ¼ ½y1 ; . . .; yn T ; 1 ¼ ½1; . . .; 1T ; a ¼ ½a1 ; . . .; an T , and I is an identity matrix. The kernel trick is applied here as following: Xkl ¼ uðxk ÞT uðxl Þ ¼ Kðxk ; xl Þ k; l ¼ 1; . . .; n
ð4Þ
where K(xk, xl) is the kernel function, the RBF kernel function is used in the paper, that is, Kðxk ; xl Þ ¼ 2
lk expð kxk x Þ; r is the width for RBF kernel. r2 Then, we can get a and b from Eq. (3). Therefore, the result of LSSVR model is n P ð5Þ f ðxÞ ¼ ak Kðx; xk Þ þ b
k¼1
2.2 Stacked LSSVR The accuracy and robustness of LSSVR models depend upon the amount and appropriateness of the available training data. However, the collection of sufficient, appropriate good quality data is still a real problem in many industrial plants. Thus, the trained LSSVR model gives unsatisfactory performance on unseen data that are not used in the training process. Several techniques have been developed to improve LSSVR generalization
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capability on future predictions. Among these techniques, combining multiple LSSVR models is a very promising approach to improving model predictions on unseen data. When building LSSVR models, it is quite possible that different models perform well in different regions of input space. By combining multiple LSSVR models, prediction accuracy on the entire input space could be improved [8, 12]. A diagram of stacked LSSVR is shown in Fig. 1, where several LSSVR models are developed to model the same relationship. Then, these individual LSSVR models are combined together (i.e., the overall output of the stacked LSSVR is a weighted combination of the individual LSSVR model outputs.) to improve the model accuracy and robustness. This can be described as: m P f ðxÞ ¼ wj fj ðxÞ ð6Þ j¼1
where f ðÞ is the stacked LSSVR predictor, fj ðÞ is the jth LSSVR model, wj is the stacking weight for combining the jth LSSVR model, m is the number of LSSVR models, and x is a vector of model inputs. Proper determination of the stacking weights is important for good modeling performance. A simple approach is to take equal weights for the individual LSSVR model, that is, the stacked LSSVR output is obtained by averaging of the outputs of multiple member models (LSSVR). Perrone and Cooper [13] show that combining m independent predictors by simple averaging can reduce the mean squared prediction error by a factor of m. However, since the individual LSSVR models are highly correlated, stacking weights can be obtained through principal component regression [14]. Stacked LSSVR proceeds in two separate steps: (1) each individual LSSVR model is trained on a bootstrap resampling replication of the original training data; and (2) the trained individual LSSVR models are combined using PCR. Through the two steps, the stacked LSSVR can be expected to solve over-fitting or under-fitting problem that may occur during the training process, since the individual LSSVR is trained separately on bootstrap resampling replications of the original training data. Hence, it can enhance the stability of the predictions.
Fig. 1 A stacked LSSVR
3 Cobalt oxalate synthesis process A cobalt oxalate synthesis process in cobalt hydrometallurgy is a liquid phase reaction of cobalt chloride and ammonium oxalate, leading to the desired cobalt oxalate crystals. Cobalt oxalate and ammonia chloride are produced: CoCl2 þ ðNH4 Þ2 C2 O4 ! CoC2 O4 # þ 2NH4 Cl A
B
P
D
The process flow sheet is shown in Fig. 2. The process consists of one ammonium oxalate dissolver and one crystallizer. The evolution of cobalt oxalate crystal is carried out in the crystallizer operated with continuously stirring. In order to keep the constant temperature of the reaction, a heating jacket is mounted in the crystallizer. A fixed volume of cobalt chloride is first charged to the crystallizer, after which ammonium oxalate is fed. During the operation, the setpoint for temperature, feed flow rate and agitation speed are constant. The final batch time is fixed. The process is very complex which includes a large number of interacting operating variables. The evolution of crystal size is dependent upon the reactor temperature, feed flow rate of ammonium oxalate, concentrations of cobalt chloride and ammonium oxalate, and agitation speed. In the plant operation, the online measurements including reactor temperature (Tr), flow rate of ammonium oxalate (Fao) and agitation speed (Na) are collected. The initial concentration of cobalt chloride and the concentration of ammonium oxalate are constants in each batch. However, the average particle size of cobalt oxalate is measured by an offline laboratory analyzer that is expensive and timeconsuming. The interval of sampling and analysis is 24 h (i.e., one time per day); hence, large time delay exists in obtaining the average particle size values and makes it difficult to meet the requirements of the online close-loop advanced control. Average particle size as well as particle size distribution (PSD) of cobalt oxalate crystals has to meet stringent specifications, as it has a decisive influence on the
Fig. 2 Schematic diagram of the cobalt oxalate synthesis process
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downstream processing, and therefore, the PSD should be reproducible in each operation and as regular as possible. However, controlling the crystallization process is quite challenging [15], because the evolution of the PSD is governed by the growth, nucleation and coagulation phenomena that interact strongly with each other. Furthermore, there is lack of appropriate instrument to study the phenomena (i.e., growth, nucleation and coagulation) in crystallizer. Despite the recent advance in instrumentation of online measurement for crystallization process such as laser diffraction spectroscopy [16], laser backscattering [17], they are still far away from being used for performing online feedback control. Therefore, the purpose of our present work is to offer an effective approach for synthesis process to predict the average particle size of cobalt oxalate crystals.
4 Hybrid model 4.1 Simplified first principle model for synthesis process Cobalt oxalate synthesis process is a batch crystallization process, which is an ancient unit operation and is widely used in fact. The dynamic behavior of a crystallization process can be captured by a population balance equation (PBE), along with conservation equations and kinetic relations. According to Puel et al. [18], the PBE is firmly established as a basic theoretical framework for all crystallization processes. Furthermore, nucleation and crystal growth dominate the crystallization kinetics. Hence, based on the following basic assumptions [22]: the suspension is perfectly mixed; crystal agglomeration or breakage phenomena are neglected; and growth of crystals is size-independent, the PBE for cobalt oxalate synthesis process is described as: 1 oðNðL; tÞVs ðtÞÞ oNðL; tÞ þG ¼ Rn Vs ðtÞ ot oL
ð7Þ
where Vs represents the suspension volume, N(L, t) represents the number density at a characteristic length, L and time, t, G represents the crystal growth rate, and Rn represents the crystal birth rate. For the sake of simplicity, the expressions for the growth rate and crystal birth rate can be written, respectively, as: Ka Rn ¼ Kn exp ð8Þ Nac ðDCÞa Tr Kb ð9Þ G ¼ Kg exp ðDCÞb Tr where Kn is birth rate coefficient, Kg is growth rate coefficient, Ka and Kb are constants, Tr is the reactor
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temperature, Na is agitation speed, a and c are birth rate exponent, and b is growth rate exponent. As is evident from Eq. (7), the population balance equation is a hyperbolic partial differential equation, which does not possess an analytical solution in most cases. Therefore, numerical methods such like method of moments, discretization techniques and finite elements method [18] have been established for PBE. Among these techniques, discretization sizing techniques appear to be robust. Marchal et al. [19] developed the method of classes, a method that recasts the PBE into a set of computationally affordable reduced-order ordinary differential equations (ODEs). Hence, under the assumption of constant number density function at each granulometric class, the PBE can be transformed into a set of ODEs using the method of classes, as represented in Eq. (10). 8 dN1 1 dVs G G > > dt ¼ Rn Vs dt N1 2DL2 N2 2DL1 N1 > > .. > > > > > . > .. > > > : dNM 1 dVs G G dt ¼ Vs dt NM þ 2DLM NM þ 2DLM1 NM1 where Ni is the number of crystals per unit volume in the ith class (Li) at time t; DLi is the width of the ith class, and M is the number of granulometric classes. As the nucleation and growth of crystals rely on the liquid phase properties, the mass balance is used to explain the concentration change of the solute and can be written as: dCP Fao CB VA ¼ 3qP Kv Gl2 dt ðVA þ Fao tÞ2
ð11Þ
where VA is the volume of cobalt chloride, CB is the concentration of ammonium oxalate, qP is the density of crystal, Kv is the shape factor, and l2 is the second moment of the PSD. Until recent now, the particle technology, especially the particle synthesis from a solution, has not been well developed. Some semi-experiential models are difficult to establish in practical situation and also make the detailed first principle model difficult to be solved online. Therefore, the simplified first principle model is developed to describe the process in general, and then stacked LSSVR approach proposed in Sect. 2.2 is utilized to compensate the unmodeled characteristics. 4.2 Model output offset updating strategy As mentioned previously, there are many phenomena that cannot be explained in practical crystallization processes, the aforementioned hybrid model is not adequate to express
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the process. Furthermore, due to the uncertainty of the reliability of process data and the existence of other unknown factors in hybrid model, the model can only capture the general trend of the synthesis process and may not predict the process with acceptable accuracy. Hence, the hybrid model needs to be rectified to eliminate the model output offset between prediction and laboratory analysis values. In this paper, a model rectification method [20] is used to correct the model output for the current kth batch by adding the overall offset d(k) defined in Eq. (12) which is the weighted sum of the current batch bias d0(k) and the average model prediction offset of all previous batch davg(k - 1): dðkÞ ¼ dd0 ðkÞ þ ð1 dÞdavg ðk 1Þ
ð12Þ
where d is a weighting factor between 0 and 1 and davg(0) = 0. The current bias d0(k) and davg(k - 1) are defined as: d0 ðkÞ ¼ ylab ðk 1Þ yhyb ðk 1Þ k1 1 X ðylab ðiÞ yhyb ðiÞÞ davg ðk 1Þ ¼ k 1 i¼1
2.
3. 4.
5.
Determine the parameters such as the number of LSSVR models m, weighting factor d. Predict the average particle size of cobalt oxalate crystals yfpm using simplified first principle model and save the prediction results and correspondent inputs x in database. Calculate the prediction error e0 = ylab - yfpm and use the inputs x building the stacked LSSVR model. Calculate the staked LSSVR model output e and predict the average particle size of cobalt oxalate crystals yhyb. If a new sample is available, calculate d0 of the new data and correct the output yhyb using the model output offset updating method in Sect. 4.2, obtain the modified output ycor.
5 Results and discussion
ð13Þ ð14Þ
where ylab and yhyb are the assay value from the offline laboratory and the hybrid model output, respectively. The modified prediction ycor,k of hybrid model can be obtained: ycor ðkÞ ¼ yhyb ðkÞ þ dðkÞ
1.
ð15Þ
4.3 The architecture of hybrid model The architecture of hybrid model is shown in Fig. 3; where e is output of the stacked LSSVR model to compensate the difference between the laboratory and the output of the simplified first principle model. In this paper, the inputs to train stacked LSSVR are x = [Tr, Fao, Na]. Further, the model output updating method is used to correct the hybrid model output. The main steps of hybrid model calibration are summarized as follows:
Fig. 3 Architecture of hybrid model of cobalt oxalate synthesis process
The proposed hybrid model has been used to predict the average particle size of cobalt oxalate crystals produced from an industrial cobalt oxalate synthesis process in a cobalt hydrometallurgy pilot plant. Ninety samples were collected in 90 days with the interval 24 h from September 12, 2008 to December 10, 2008. Due to the interval of sampling and analysis is 24 h (i.e., one time per day), large time delay exists in obtaining the average particle size values and makes it considerable difficult to perform the online close-loop advanced control. However, the variables that include the reactor temperature, the flow rate of ammonium oxalate and the agitation speed can be obtained continuously with the interval of 15 s. Therefore, the three variables (i.e., the reactor temperature, the flow rate of ammonium oxalate and the agitation speed) are chosen as inputs and the average particle size is chosen as output for the LSSVR model in the present paper. In this study, the first 50 samples were used to develop LSSVR models and the remaining 40 samples were used as unseen testing data. In the stacked LSSVR model, the number of member models (m) is an important parameter that affects the stability and accuracy of a stacked LSSVR. In this case, m from 1 to 60 was investigated. For each m, a stacked LSSVR model is developed and the model is then used to predict the testing data. For training each LSSVR model, bootstrap resampling with replacement [21] was used to generate a replication of the 50 samples of process data. Thus, the training data for member LSSVR models are different due to bootstrap resampling so as to ensure that different member LSSVR models are obtained. In each LSSVR model, the most popular kernel function, RBF kernel, is selected and the tuning parameters c and r are
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sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1P ðyi y^i Þ2 RMSE ¼ n i¼1
Fig. 4 Variation of RMSE with the number of member LSSVR models in stacked LSSVR
determined by fivefold cross-validation. Values of the two tuning parameters are c = 10 and r2 = 0.8, respectively, in the study. In the optimization of parameter m, the root mean squared error of prediction (RMSE) of the testing set is used as an evaluation criterion. Fig. 5 Comparison of the actual with prediction results from the four methods. a Simplified first principle model, b LSSVR model, c stacked LSSVR model and d hybrid model. (dotted lines measurements, bold lines predictions)
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ð16Þ
where y^i is the prediction of the ith sample, yi is the measurement of the ith sample, and n is the number of samples in testing set. To determine the number of member LSSVR models, the variation curve of RMSE with different m is drawn in Fig. 4. It can be seen from the figure that, at the beginning, RMSE is comparatively large. Then, however, with the increase of m, RMSE reaches almost constant with little fluctuation. And also in order to reduce the computation burden, m = 30 is used for further study. In order to demonstrate the efficiency of the developed hybrid model, the prediction performances are compared with the simplified first principle model, the LSSVR model and the stacked LSSVR model. For the stacked LSSVR model, the number of member LSSVR models is 30 and the first 50 samples are used to train the model. For the hybrid model, the number of the member LSSVR models is the same as the stacked LSSVR model. And the weighting factor (d) is 0.2. The comparisons of the average particle size values of the cobalt oxalate crystals measured offline and predicted from the four methods are shown in Fig. 5.
Author's personal copy Neural Comput & Applic Fig. 6 Comparison of the relative errors from the four methods. a Simplified first principle model, b LSSVR model, c stacked LSSVR model and d hybrid model
And the corresponding relative prediction errors is also given in Fig. 6. It can be observed that poor prediction accuracy is obtained; however, the trend of the process is achieved for the simplified first principle model (Fig. 5a) and the LSSVR model (Fig. 5b). Compared with the above two methods, the stacked LSSVR model predictions accuracy is improved, but it still deviates from the measurements(Fig. 5c). Therefore, the three methods are not adequate in predicting the behavior of the cobalt oxalate synthesis process. For the hybrid model (Fig. 5d), it is clearly that the proposed approach not only well tracks the process trend but also predicts the average particle size of cobalt oxalate crystals with the best accuracy. Furthermore, the root mean squared error of prediction (RMSE) and maximal absolute error (MAE) are also considered: MAE ¼ maxðjyi y^i jÞ
ð17Þ
the performance results of the four methods on the testing data are summarized in Table 1. It can be seen from Table 1 that the proposed method results in the lowest MAE and RMSE among all of the four methods. It is well known that process control and optimization highly rely on the accuracy of the soft sensor model. Thus, the hybrid model proposed in this paper is adequate to the online
Table 1 Comparison of MAE and RMSE result based on different modeling methods No.
Method
MAE
RMSE
1
Simplified first principle model
0.0714
0.0335
2
LSSVR
0.1108
0.0412
3
Stacked LSSVR
0.0919
0.0274
4
Hybrid model
0.0399
0.0174
measuring requirement of cobalt oxalate synthesis process in cobalt hydrometallurgy.
6 Conclusions Due to direct online measurements of average particle size of cobalt oxalate in synthesis process may not be possible, soft sensors as a valuable alternative method can be implemented instead of the traditional measurement instruments. In this paper, a soft sensor model for cobalt oxalate synthesis process in cobalt hydrometallurgy has been proposed. It combines the simplified first principle model with a stacked LSSVR model in the proposed approach. The first principle model using the population
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balance equations and mass conservation equation with some assumptions is established to predict the average crystal size, which possesses low computational complexity and is suitable to apply online. The stacked LSSVR model is built to compensate the unmodeled characters of cobalt oxalate synthesis processes, which has the advantages of improved computational efficiency, limited training data and enhanced model generalization capability. A model output offset updating strategy is also employed to correct the final output of hybrid model, which can enhance the prediction accuracy. The proposed model has been used to predict average crystal particle size for an industrial cobalt oxalate synthesis process in cobalt hydrometallurgy. Compared to other soft sensor models, the prediction results determinate that the proposed soft senor model has better prediction performances. Therefore, the presented hybrid model structure can be successively extended to other industrial processes. Acknowledgments This work was supported by the National High Technology Research and Development Program of China (No. 2011AA060204), National Natural Science Foundation of China (Nos. 61074074, 61174130 and 61004083), Project 973 of China (No. 2009CB320601) and the Fundamental Research Funds for the Central Universities (No. N100604008).
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