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Multivariate projection methods applied to flotation columns
Minerals Engineering 18 (2005) 721–723
Technical note
This article is also available online at: www.elsevier.com/locate/mineng
Multivariate projection methods applied to flotation columns L.G. Bergh *, J.B. Yianatos, A. Leo´n Chemical Engineering Department, Santa Maria University, P.O. Box 110-V Valparaiso, Chile Received 4 November 2004; accepted 10 December 2004
Abstract On line monitoring and diagnosis systems of process operating performance are becoming important part of industrial programs, leading to improve process operation and therefore product quality over time. In processes, such as flotation, a large number of input variables, highly correlated, are presented. These characteristics usually pose more difficulties in modelling the process for monitoring and diagnosis purposes. Multivariate statistical projection methods have been proposed to effectively deal with these situations. The application of these ideas to a column flotation process is discussed here. In this work, the detection of measurement problems in relevant variables, and the identification of the set of variables responsible for driving the process outside its normal operating region, are demonstrated. Furthermore, the use of these techniques gave considerable information for the correction of the operating problem. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Column flotation; Process control; Modelling; Artificial intelligence; Simulation
1. Introduction
1.1. Flotation column control
In the last two decades the use of pneumatic flotation columns became wide-spread throughout the mineral processing industry of metallic, non-metallic and coal ores, as well as in waste removal and recycling processes, in the world. Columns out perform conventional mechanical cells in cleaning operations (better product grade) due to their particular froth operation, discussed in detail by Finch and Dobby (1990). More degrees of freedom in operating variables of flotation columns have led to large variations in metallurgical performance and have therefore provided much scope for improving their control. A more complete discussion can be found in Bergh and Yianatos (2003).
The primary objectives are column recovery and concentrate grade, which represent the indices of process productivity and product quality. A common practice is to control secondary objectives, such as pH at the feed, froth depth, air flow rate and wash water flow rate. These are usually implemented as local controllers or under distributed control systems (DCS). Ideally, when primary objectives are measured, the control strategy is to change the set points of the controllers under DCS, in order to achieve a good process performance. This is usually implemented in the form of expert systems (Bergh and Yianatos, 2003). 1.2. Multivariate statistical techniques
*
Corresponding author. Tel.: +56 32 654 229; fax: +56 32 654 478. E-mail address: [email protected] (L.G. Bergh).
0892-6875/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2004.12.008
The key feature of principal component analysis (PCA) method (Jackson, 1991) is its ability to mathematically project high dimensional process and quality
722
L.G. Bergh et al. / Minerals Engineering 18 (2005) 721–723
data into smaller dimensional summary data sets, via the development of linear models. Some of the numerous advantages that PCA method has over traditional monitoring and prediction technologies are: provision for data dimension reduction and robustness to highly correlated, noisy and missing data (Kourti and MacGregor, 1995). More complete description of PCA methods, which includes discussion of numerical estimation techniques, are provided by Wold et al. (1987). Following Wise et al. (2003), PCA can be graphically explained in Fig. 1. Assume a set of collected data of three variables measured on a process, as shown in this three dimensional plot. It may be that the samples all lie on a plane, enclosed by an ellipse. Some samples vary more along one axis of the ellipse than along the other. The first principal component (PC) describes the direction of the greatest variation in the data set, which is the major axis of the ellipse. The second PC describes the direction of the second greatest variation, which is the minor axis of the ellipse. In this case, a PC model (the scores and loadings vectors and associated eigenvalues) with two principal components adequately describes all the variation in the measurements. Q is a measure of the Euclidean distance from a point off the plane containing the ellipse. A sample with a large Q is shown in the upper left side of Fig. 1. The Q limit defines a distance off the plane that is considered unusual based on the data used to form the PCA model. On the other hand, the HottelingÕs T2 statistic is a measure of the distance from the multivariate mean, which is the intersection of the PCs in Fig. 1, to the projection of the sample onto the 2 PCs. The T2 limit defines an ellipse on the plane within which the data normally project. A sample with a large T2 value (but small Q) is shown on the upper right side of Fig. 1. Once a PC model has been built with the normal operating data, a set of steady state data can be judged to be in each of the following states: (i) normal operation if the new set of data satisfies the Q and T2 test, (ii) abnormal operation if only the Q test is satisfied and (iii) measurement problem or PC model representa-
tion problem if only T2 test is satisfied. In this way a diagnosis of the operation can be accomplished for steady state data. Furthermore, the residuals are informative of the principal process variables affecting the abnormal situation.
2. Column flotation simulation 2.1. Nonlinear model from experimental data A flotation column model was developed, based on Finch and Dobby (1990). First the gas holdup, the bias rate and the kinetic constants for each mineralogical specie are estimated from empirical models, depending on operating variables such as feed flow rate, gas flow rate, wash water flow rate and froth depth. Then, dispersion number, residence times, froth and collection recovery are estimated. Finally concentrate and tailings grades are predicted. The empirical model parameters were fitted using experimental data. More details can be found in Bergh et al. (1998). 2.2. PC Models Using the nonlinear model predictions a complete set of data was constructed, allowing the exploration of multivariate statistics techniques to obtain PC models. The first step was to define the set of normal data. According to the number of principal components chosen to explain the variability of the data, several PC models can be obtained. A model with 2 PCs explains 62.5 % of the variability shown in the data. This interpretation increased to 78.1% with a 3 PCs model and to 89.1% with a 4 PCs model. More PCs will marginally affect the model representation. For example, if a 2 PCs model is used; all data included in the 95% confidence interval will be considered normal data, as Fig. 2 shows.
5 4
Point with large Q Unusual variation outside the model
1st PC
Point with large T2 Unusual variation inside the model
Variable 2
Scores on PC2 (25%)
3
95% confidence limit
24 26
2
21 13
1 16 6
0
7
-1 -2
12
15 23 14 4 28 32 27 3 5 29 1
17
19
30 31 22
18 20
2
10 11 8 9
-3
25
-4
2nd PC Variable 3
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
Scores on PC1(37%)
Variable 1
Fig. 1. Data representation by principal components.
Fig. 2. A two PCs model representing normal and abnormal sets of data.
% of points detected
L.G. Bergh et al. / Minerals Engineering 18 (2005) 721–723 100 90 80 70 60 50 40 30 20 10 0 -60
723
this point to the normal region, the previous information suggests that a reduction of gas flow rate and a combined increment of the froth depth and decrement of the wash water flow rate, are the appropriate corrective actions. In fact, doing that the point will be that shown in Fig. 2 under the label ‘‘29’’, which is well inside the normal region. -50
-40
-30
-20
-10
0
10
20
30
40
50
60
% error in feed grade
3. Conclusions % out 2PC
% out 3PC
% out 4PC
Fig. 3. Sensitivity of Q test to detect measurement errors in feed grade.
2.3. Testing new data sets Once the PC model has been built, new sets of steady state data can be tested in order to identify an abnormal situation. For each operating condition, data with errors of minus 50% to plus 50% for feed grade were tested for models with two to four PCs. After proper scaling and PCs calculation, the Hotelling T2 and residuals Q test were applied. The results are summarized in Fig. 3. Similar results can be found if the same set of data is corrupted by measurement errors in other process variable. The sensitivity of each test will depend on the importance of the variable in the model. For example, if errors in the wash water flow rate are considered, even with a 4 PCs model, measurement errors should be greater than 100% to detect only one over ten abnormal situations. Sometimes the measurements present small errors, however some process variables or combination of them produce results considered outside the normal area. If an abnormal operating case is detected, then the next step is to identify the variables that cause the problem, in order to start a remediation procedure. To study this kind of problems a normal point of operation is deviated modifying one process variable. For example, when the gas velocity was increased from 1.65 to 2.3 cm/s, in the set of data shown under label 16 in Fig. 2, the nonlinear steady state model predicted that the recovery increased from 55.6 to 69.9% and that the concentrate grade decreased from 31.6 to 30.1%. This new operating point data was scaled and the PCs were calculated. After that, the Hotelling T2 and residuals Q test were implemented. The new point lies now outside the 95% confidence region of the normal space for the 4 PCs model. The abnormal operating point was due to a positive large deviation of the gas velocity, and in minor way to positive deviation of the wash water flow rate and to negative deviation of the froth depth, resulting in a large positive deviation of the process recovery and a moderate decrement of the concentrate grade. To move
The quality of flotation column control is strongly depending on the accuracy of key measurements as feed, concentrate and tailings grades, and also on operating variables such as gas and wash water flow rate, and froth depth. Thus methods leading to detect and isolate measurement errors are very important and difficult to implement. PC models based on normal steady state data can provide an excellent opportunity to aid in the solution of this complex problem. PC models based on steady state data can help not only to detect abnormal operations but also to identify the main variables that are provoking the problem, suggesting in this way how to remedy the operation and go back to a normal operation. One advantage is that this monitoring tool can work on-line and therefore can be part of a more complex supervisory control scheme. However, one limitation is that this method is applicable to analyze steady state data.
Acknowledgement The authors would like to thanks Conicyt (Project Fondecyt 1020215) and Santa Maria University (Project 270322) for their financial support.
References Bergh, L.G., Yianatos, J.B., 2003. Flotation column automation: state of the art. Control Engineering Practice 11, 67–72. Bergh, L.G., Yianatos, J.B., Leiva, C., 1998. Fuzzy supervisory control of flotation columns. Minerals Engineering 11 (8), 739–748. Finch, J.A., Dobby, G.S., 1990. Column Flotation. Pergamon Press. Jackson, J.E., 1991. A UserÕs Guide to Principal Components. John Wiley & Sons, New York, NY. Kourti, T., MacGregor, J.F., 1995. Process analysis monitoring and diagnosis using multivariate projection methods—a tutorial. Chemometrics and Intelligent Laboratory Systems 28, 3–21. Wise, B., Gallagher, N., Bro, R., Shaver, J., 2003. PLS_Manual, PLS_Toolbox 3.0. Eigenvector Research Inc. Wold, S., Esbensen, K., Geladi, P., 1987. Principal component analysis. Chemometrics and Intelligent Laboratory Systems 2, 37–55.
Technical note
This article is also available online at: www.elsevier.com/locate/mineng
Multivariate projection methods applied to flotation columns L.G. Bergh *, J.B. Yianatos, A. Leo´n Chemical Engineering Department, Santa Maria University, P.O. Box 110-V Valparaiso, Chile Received 4 November 2004; accepted 10 December 2004
Abstract On line monitoring and diagnosis systems of process operating performance are becoming important part of industrial programs, leading to improve process operation and therefore product quality over time. In processes, such as flotation, a large number of input variables, highly correlated, are presented. These characteristics usually pose more difficulties in modelling the process for monitoring and diagnosis purposes. Multivariate statistical projection methods have been proposed to effectively deal with these situations. The application of these ideas to a column flotation process is discussed here. In this work, the detection of measurement problems in relevant variables, and the identification of the set of variables responsible for driving the process outside its normal operating region, are demonstrated. Furthermore, the use of these techniques gave considerable information for the correction of the operating problem. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Column flotation; Process control; Modelling; Artificial intelligence; Simulation
1. Introduction
1.1. Flotation column control
In the last two decades the use of pneumatic flotation columns became wide-spread throughout the mineral processing industry of metallic, non-metallic and coal ores, as well as in waste removal and recycling processes, in the world. Columns out perform conventional mechanical cells in cleaning operations (better product grade) due to their particular froth operation, discussed in detail by Finch and Dobby (1990). More degrees of freedom in operating variables of flotation columns have led to large variations in metallurgical performance and have therefore provided much scope for improving their control. A more complete discussion can be found in Bergh and Yianatos (2003).
The primary objectives are column recovery and concentrate grade, which represent the indices of process productivity and product quality. A common practice is to control secondary objectives, such as pH at the feed, froth depth, air flow rate and wash water flow rate. These are usually implemented as local controllers or under distributed control systems (DCS). Ideally, when primary objectives are measured, the control strategy is to change the set points of the controllers under DCS, in order to achieve a good process performance. This is usually implemented in the form of expert systems (Bergh and Yianatos, 2003). 1.2. Multivariate statistical techniques
*
Corresponding author. Tel.: +56 32 654 229; fax: +56 32 654 478. E-mail address: [email protected] (L.G. Bergh).
0892-6875/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2004.12.008
The key feature of principal component analysis (PCA) method (Jackson, 1991) is its ability to mathematically project high dimensional process and quality
722
L.G. Bergh et al. / Minerals Engineering 18 (2005) 721–723
data into smaller dimensional summary data sets, via the development of linear models. Some of the numerous advantages that PCA method has over traditional monitoring and prediction technologies are: provision for data dimension reduction and robustness to highly correlated, noisy and missing data (Kourti and MacGregor, 1995). More complete description of PCA methods, which includes discussion of numerical estimation techniques, are provided by Wold et al. (1987). Following Wise et al. (2003), PCA can be graphically explained in Fig. 1. Assume a set of collected data of three variables measured on a process, as shown in this three dimensional plot. It may be that the samples all lie on a plane, enclosed by an ellipse. Some samples vary more along one axis of the ellipse than along the other. The first principal component (PC) describes the direction of the greatest variation in the data set, which is the major axis of the ellipse. The second PC describes the direction of the second greatest variation, which is the minor axis of the ellipse. In this case, a PC model (the scores and loadings vectors and associated eigenvalues) with two principal components adequately describes all the variation in the measurements. Q is a measure of the Euclidean distance from a point off the plane containing the ellipse. A sample with a large Q is shown in the upper left side of Fig. 1. The Q limit defines a distance off the plane that is considered unusual based on the data used to form the PCA model. On the other hand, the HottelingÕs T2 statistic is a measure of the distance from the multivariate mean, which is the intersection of the PCs in Fig. 1, to the projection of the sample onto the 2 PCs. The T2 limit defines an ellipse on the plane within which the data normally project. A sample with a large T2 value (but small Q) is shown on the upper right side of Fig. 1. Once a PC model has been built with the normal operating data, a set of steady state data can be judged to be in each of the following states: (i) normal operation if the new set of data satisfies the Q and T2 test, (ii) abnormal operation if only the Q test is satisfied and (iii) measurement problem or PC model representa-
tion problem if only T2 test is satisfied. In this way a diagnosis of the operation can be accomplished for steady state data. Furthermore, the residuals are informative of the principal process variables affecting the abnormal situation.
2. Column flotation simulation 2.1. Nonlinear model from experimental data A flotation column model was developed, based on Finch and Dobby (1990). First the gas holdup, the bias rate and the kinetic constants for each mineralogical specie are estimated from empirical models, depending on operating variables such as feed flow rate, gas flow rate, wash water flow rate and froth depth. Then, dispersion number, residence times, froth and collection recovery are estimated. Finally concentrate and tailings grades are predicted. The empirical model parameters were fitted using experimental data. More details can be found in Bergh et al. (1998). 2.2. PC Models Using the nonlinear model predictions a complete set of data was constructed, allowing the exploration of multivariate statistics techniques to obtain PC models. The first step was to define the set of normal data. According to the number of principal components chosen to explain the variability of the data, several PC models can be obtained. A model with 2 PCs explains 62.5 % of the variability shown in the data. This interpretation increased to 78.1% with a 3 PCs model and to 89.1% with a 4 PCs model. More PCs will marginally affect the model representation. For example, if a 2 PCs model is used; all data included in the 95% confidence interval will be considered normal data, as Fig. 2 shows.
5 4
Point with large Q Unusual variation outside the model
1st PC
Point with large T2 Unusual variation inside the model
Variable 2
Scores on PC2 (25%)
3
95% confidence limit
24 26
2
21 13
1 16 6
0
7
-1 -2
12
15 23 14 4 28 32 27 3 5 29 1
17
19
30 31 22
18 20
2
10 11 8 9
-3
25
-4
2nd PC Variable 3
-5 -4
-3
-2
-1
0
1
2
3
4
5
6
Scores on PC1(37%)
Variable 1
Fig. 1. Data representation by principal components.
Fig. 2. A two PCs model representing normal and abnormal sets of data.
% of points detected
L.G. Bergh et al. / Minerals Engineering 18 (2005) 721–723 100 90 80 70 60 50 40 30 20 10 0 -60
723
this point to the normal region, the previous information suggests that a reduction of gas flow rate and a combined increment of the froth depth and decrement of the wash water flow rate, are the appropriate corrective actions. In fact, doing that the point will be that shown in Fig. 2 under the label ‘‘29’’, which is well inside the normal region. -50
-40
-30
-20
-10
0
10
20
30
40
50
60
% error in feed grade
3. Conclusions % out 2PC
% out 3PC
% out 4PC
Fig. 3. Sensitivity of Q test to detect measurement errors in feed grade.
2.3. Testing new data sets Once the PC model has been built, new sets of steady state data can be tested in order to identify an abnormal situation. For each operating condition, data with errors of minus 50% to plus 50% for feed grade were tested for models with two to four PCs. After proper scaling and PCs calculation, the Hotelling T2 and residuals Q test were applied. The results are summarized in Fig. 3. Similar results can be found if the same set of data is corrupted by measurement errors in other process variable. The sensitivity of each test will depend on the importance of the variable in the model. For example, if errors in the wash water flow rate are considered, even with a 4 PCs model, measurement errors should be greater than 100% to detect only one over ten abnormal situations. Sometimes the measurements present small errors, however some process variables or combination of them produce results considered outside the normal area. If an abnormal operating case is detected, then the next step is to identify the variables that cause the problem, in order to start a remediation procedure. To study this kind of problems a normal point of operation is deviated modifying one process variable. For example, when the gas velocity was increased from 1.65 to 2.3 cm/s, in the set of data shown under label 16 in Fig. 2, the nonlinear steady state model predicted that the recovery increased from 55.6 to 69.9% and that the concentrate grade decreased from 31.6 to 30.1%. This new operating point data was scaled and the PCs were calculated. After that, the Hotelling T2 and residuals Q test were implemented. The new point lies now outside the 95% confidence region of the normal space for the 4 PCs model. The abnormal operating point was due to a positive large deviation of the gas velocity, and in minor way to positive deviation of the wash water flow rate and to negative deviation of the froth depth, resulting in a large positive deviation of the process recovery and a moderate decrement of the concentrate grade. To move
The quality of flotation column control is strongly depending on the accuracy of key measurements as feed, concentrate and tailings grades, and also on operating variables such as gas and wash water flow rate, and froth depth. Thus methods leading to detect and isolate measurement errors are very important and difficult to implement. PC models based on normal steady state data can provide an excellent opportunity to aid in the solution of this complex problem. PC models based on steady state data can help not only to detect abnormal operations but also to identify the main variables that are provoking the problem, suggesting in this way how to remedy the operation and go back to a normal operation. One advantage is that this monitoring tool can work on-line and therefore can be part of a more complex supervisory control scheme. However, one limitation is that this method is applicable to analyze steady state data.
Acknowledgement The authors would like to thanks Conicyt (Project Fondecyt 1020215) and Santa Maria University (Project 270322) for their financial support.
References Bergh, L.G., Yianatos, J.B., 2003. Flotation column automation: state of the art. Control Engineering Practice 11, 67–72. Bergh, L.G., Yianatos, J.B., Leiva, C., 1998. Fuzzy supervisory control of flotation columns. Minerals Engineering 11 (8), 739–748. Finch, J.A., Dobby, G.S., 1990. Column Flotation. Pergamon Press. Jackson, J.E., 1991. A UserÕs Guide to Principal Components. John Wiley & Sons, New York, NY. Kourti, T., MacGregor, J.F., 1995. Process analysis monitoring and diagnosis using multivariate projection methods—a tutorial. Chemometrics and Intelligent Laboratory Systems 28, 3–21. Wise, B., Gallagher, N., Bro, R., Shaver, J., 2003. PLS_Manual, PLS_Toolbox 3.0. Eigenvector Research Inc. Wold, S., Esbensen, K., Geladi, P., 1987. Principal component analysis. Chemometrics and Intelligent Laboratory Systems 2, 37–55.