Minerals Engineering 56 (2014) 1–9

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Minerals Engineering

journal homepage: www.elsevier .com/locate /mineng

Visualization of the controller states of an autogenous mill from timeseries data

0892-6875/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.mineng.2013.10.018

⇑ Corresponding author at: Department of Minerals Engineering and ExtractiveMetallurgy, Western Australian School of Mines, Curtin University of Technology,Perth, WA, Australia. Tel.: +61 8 9266 4349; fax: +61 8 9358 4912.

E-mail address: chris.aldrich@curtin.edu.au (C. Aldrich).

C. Aldrich a,b,⇑, J.J. Burchell b, J.W. de V. Groenewald c, C. Yzelle b

a Department of Minerals Engineering and Extractive Metallurgy, Western Australian School of Mines, Curtin University of Technology, Perth, WA, Australiab Department of Process Engineering, University of Stellenbosch, Private Bag X1, Matieland 7602, Stellenbosch, South Africac Anglo Platinum Management Services, PO Box 62179, Marshalltown 2107, South Africa

a r t i c l e i n f o

Article history:Received 3 June 2013Accepted 14 October 2013Available online 14 November 2013

Keywords:ComminutionModelingNonlinear time series analysisNeural networks

a b s t r a c t

The operational variables of an industrial autogenous mill were embedded in a low-dimensional phasespace to facilitate visualization of the dynamic behavior of the mill. This was accomplished by use of amultivariate extension of the method of delay coordinates used in nonlinear time series analysis. In thisphase space, the controlled states of the mill could be visualized as separate regions or clusters in thephase space.

Comparison of the correlation dimension of the state variable of the mill (the load) embedded in phasespace suggested that the dynamic behavior of the mill could not be represented by a linear stochasticmodel (Gaussian or otherwise). The low dimensionality (62) of the correlation dimension further sug-gested that the mill load depended on a few variables only and that the underlying generative processhad a significant deterministic component.

In addition, the operational variables could be used as reliable predictors in a neural network model toidentify the controlled states of the mill. As a complementary approach to visualization of the operationof the mill, a different neural network model could be used to reconstruct a corrected power load curveby compensating for the effect of varying operating conditions.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Energy consumption in ore milling processes constitutes up to40% of the energy expenditure in concentrator plants, but is notmore than 15–25% efficient, depending on the approach used tocalculate the energy required to reduce the size of the ore particles(Fuerstenau and Abouzeid, 2002). As a consequence, the reductionof energy consumption in comminution circuits has drawn consid-erable interest over an extended period of time.

Energy efficient operation of mills is known to depend to a largeextent on the behavior of the mill load (Van Nierop and Moys,2001), but reliably accounting for the effect of load on mill powerconsumption is not an easy task. Other process variables, such asmedia shape (Lameck et al., 2006), mill speed (Powell et al.,2009), ore type (Powell et al., 2009) and mill filling (Kiangi andMoys, 2008) also play a significant role that may be difficult toquantify from first principles.

This is especially the case with autogenous and semi-autoge-nous grinding (SAG) mills, which are popular in new plants, owing

to their favorable capital and operating costs. SAG mill operation,in particular, is strongly affected by the type and size of the orebeing ground, which makes stable control difficult. In these mills,grinding is driven by the mill load (Powell et al., 2009).

One approach to mill control is based on the use of power–loadcurves (Powell et al., 2001, 2009). This approach involves establish-ing empirical grind curves that reveal at which mill load the cir-cuit’s power draw peak. These curves have a typical concaveparabolic shape when plotted with power and load as the ordinateand abscissa respectively. Once these curves are established, theymay be used to guide milling operation with respect to achievinga desired throughput or grind quality.

In this context, for example, current control strategies areaimed at the maintenance of maximal mill power draw at or nearthe top of the power load curve (Powell and Mainza, 2006) bymanipulation of the mill feed rate and other process variables,such as the coarse to fine ore feed ratio and the water inlet rate.This aim is challenging, since the power load curve is not staticfor a particular milling circuit. Instead, the curve is affected byother process variables, such as the ore characteristics, and itmay therefore be difficult to track the designated operating pointwith sufficient accuracy. Therefore, accounting for the effect ofmultiple variables on the controlled states of the mill would bea major advantage.

Nomenclature

Symbol Meaninga, b, c, d, h, # threshold values used by expert control system

(–)e0 scaling parameter (–)K diagonal eigenvalue matrix (–)ti, tj ith and jth point of a time series embedded in phase

space (–)x frequency (–)C correlation matrix (–)CN correlation count (–)dc correlation dimension (–)f ðxÞ Fourier transform (–)I(A,B) mutual information of two random variables A and

B (–)IðA;BÞ average mutual information of two random vari-

ables A and B (–)k embedding lag of phase space (–)kj embedding lag of jth variable in phase space (–)Lnorm normalized load (–)Lavg average load (ton)Lmax maximum load (ton)Lmin minimum load (ton)LROC load rate of change (ton h�1)m number of variables (–)M embedding dimension of phase space (–)Mj embedding dimension of j’th variable in phase

space (–)N number of samples in a time series (–)

NE number of rows in an embedded time series matrix(–)

pA probability density function of random variable A (–)pAB joint probability density function of random vari-

ables A and B (–)Pnorm normalized mill power consumption (–)Pavg average mill power consumption (kW)Pmax maximum mill power consumption (kW)Pmin minimum mill power consumption (kW)PROC mill power rate of change (kW h�1)q number of eigenvectors in a reduced dimensional-

ity principal component model (–)P eigenvector matrix (–)Pq loading matrix with q columns (–)t time (s)Tq score matrix with q columns (–)XE matrix of embedded variables (–)xE

j vector of jth embedded variable (–)x1 fine ore feed rate (ton h�1)x2 coarse ore feed rate (ton h�1)x3 power consumption (kW)x4 mill load (ton)x5 water inlet rate (ton h�1)xt time series observation at time t (–)Xt random variable at time t (–)yt surrogate data observation at time t (–)z1, z2 discriminant functions (linear combination of vari-

ables x1 to x5) (–)

2 C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9

In this paper, analysis of the dynamic behavior of an autogenousmill is considered, based on reconstruction of the mill variables inphase space, which allows time series variables and their history tobe considered simultaneously within a state space context. It isshown that most of these states can be predicted from the opera-tional variables of the mill.

2. Autogenous grinding circuit

The milling circuit analyzed here consisted of a fully autoge-nous closed-loop mill with a recycling load that overflowed froma screen unit, as indicated in Fig. 1. The circuit was under expertcontrol, which manipulated the mill’s control variables, namelyits water inlet, coarse ore feed rate and fine ore feed rate, to controlthe output variables of the mill, i.e. the mill load and the powerconsumption of the mill. Five mill variables were monitored, viz.the fine ore feed rate (x1), coarse ore feed rate (x2), power con-sumption (x3), mill load (x4), and the inlet water flow rate (x5).

FAG MILL

SCREEN FEED

Fig. 1. Basic flow diagram of grinding circuit.

Each individual time series (xi, i = 1, 2, . . . 5), consisting of32,564 records, was normalized to zero mean and unit variance be-fore further analysis. Typical time series measurements of thesevariables normalized between 0 and 1 are shown in Fig. 2. Sincesamples were collected approximately every 10 s, the time seriesspan a period of almost 89 h of continuous operation. Moreover,state labels were logged by the mill controller for each sample col-lected in the dataset. These logged controller states, represented bythe data in Table 1 and Fig. 3, are related to the operating condi-tions during the period under investigation, and also reflect thecollective, heuristic knowledge of the operation of the comminu-tion circuit. Note that not all the states are necessarily displayed,hence the labeling starting at no 3 and skipping some states in be-tween, as not all the mill states occurred over the period of obser-vation. Moreover, normal operating conditions were not definedformally, but could be considered the default state of the mill, i.e.when none of the other conditions were flagged by the controller.

In Fig. 3, operational variables are indicated by FO = fine ore,CO = coarse ore, LD = load, PW = power and WI = water inlet, whilethe different mill controller states are represented by differentcolor1 lines, as indicated in the legend of the figure. As can be seenfrom the radar chart (Fig. 3), mill states 3 and 6 have virtually iden-tical profiles and would be difficult to distinguish reliably from theoperational variables. This is also shown in Table 1, where the devi-ations of the variables from the normal state are shown in the lastcolumn in order from left to right for the fine ore (FO), coarse ore(CO), mill load (LD), power consumption (PW) and water inlet (WI).

The mill variables were embedded into a phase space to facili-tate analysis of the mill dynamics, as described in more detail inthe next section.

1 For interpretation of color in Figs. 3, 5 and 12 the reader is referred to the webversion of this article.

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.5

1

Time

Fine

Ore

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.5

1

TimeC

oars

e O

re

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.5

1

Time

Pow

er

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.5

1

Time

Load

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.5

1

Time

Wat

er In

let

Fig. 2. Typical realizations of the scaled mill variables considered in the analysis.

Table 1Mill states logged by controller and deviations of associated operating conditions from the norm, in sequence from left to right of fine ore (FO), coarse ore (CO), power (PW), millload (LD) and water inlet (WI).

Mill state Description Deviations of operating variables

3: Normal operating conditions Normal operating conditions, the default state of the mill4: High loading of mill Normalized load exceeding specified threshold Lnorm ¼ ðLavg�LminÞ

ðLmax�LminÞ > a5: Feed disturbance Combined ore feed rate (x1 + x2) in excess of the average feed rate by a set threshold

6: Feed limited LROC < 0 and combined ore feed rate x1 + x2 in excess of a set threshold7: Low mill loading Having a normalized load smaller than a specified threshold Lnorm ¼ ðLavg�LminÞ

ðLmax�LminÞ > b

8: High power consumption Having a normalized power larger than a specified threshold Pnorm ¼ ðPavg�PminÞðPmax�PminÞ > c

11: Recycle limited –12: Sand loading PROC 6 �d and ðP=LÞROC 6 �h and Circulating Feed Rate

Total Feed Rate P #

C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9 3

3. Phase space reconstruction of grinding circuit variables

It is generally accepted that the space in which to view thestructure of the dynamics of a system is not that of the one dimen-sional observations, or time series, but rather a space of higherdimensionality (Abarbanel, 1996). This higher dimensional spacecan be identified by means of a phase space reconstruction, andconsists of coordinates composed of the original time series andtheir time delayed copies. The task of identifying the phase spacefrom the original observations is addressed by the embedding the-orem attributed to Takens (1981). Essentially, the reconstructiondepends on two parameters, viz. the time delay or lag and thedimensionality of the phase space. More formally, if datamatrix X represents N observations on m mill variables, asindicated in Eq. (1), then this matrix of variables is expanded (XE)to contain the variables, as well as a number of lagged copies ofeach variable, as indicated by Eqs. (2) and (3). For each variableXE

j , the lag kj, as well as the embedding dimension Mj, is determinedindividually, before concatenation into the aggregated laggedtrajectory matrix XE.

X ¼ x1 x2 � � � xm½ � ¼X11 � � � xm1

..

. . .. ..

.

X1N � � � xmN

2664

3775 ð1Þ

XE ¼ xE1 xE

2 � � � xEm

� �ð2Þ

where

XEj ¼ xj;t x1;t�k;j � � � x1;t�ðMj�1Þkj

� �ð3Þ

3.1. Estimation of embedding lag parameters of variables

Optimal lags kj, for j = 1, 2, . . . m, were determined by the aver-age mutual information criterion (Gallager, 1968), while the meth-od of false nearest neighbors (Kennel et al., 1992) was used todetermine the embedding dimension of the variables.

The mutual information (I) is a measure of the amount of infor-mation gained with regard to variable bj, drawn from set B, whenvariable ai, drawn from set A, is measured:

Fig. 3. Radar chart of the mill control states and operating variables summarized inTable 1 scaled to a range of [0, 1].

Table 2Embedding of mill variables.

Variable Description Units k M

x1 Mill load [ton] 1376 4x2 Mill power [kW] 1017 5x3 Inlet water [m3 s�1] 1432 5x4 Fine ore feed rate [ton s�1] 1477 5x5 Coarse ore feed rate [ton s�1] 1379 4

Fig. 4. The eigenspectrum of the principal components of the embedded millvariables.

4 C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9

IðA;BÞ ¼ log2pABðai; bjÞ

pAðaiÞpBðbjÞ

� �ð4Þ

In Eq. (4), pAB (ai,bj) is the joint probability density of measure-ments of the random variables ai 2 A and bj 2 B. pA(ai) and pB(bj)are the individual or marginal densities of measurements ai 2 Aand bj 2 B. The average over all the measurements is known asthe average mutual information and can be expressed as:

IðA;BÞ ¼Xai ;bj

pABðai; bjÞlog2pABðai; bjÞpðaiÞpBðbjÞ

� �ð5Þ

The ideal time lag was selected to correspond with the samplelag k, at which the average mutual information between Xc and Xt�k

reached a minimum.

IðXt ;Xt�kÞ ¼X

xt ;xt�k

pðxt; xt�kÞlog2pðxt ; xt�xÞ

pðxtÞpðxt�kÞ

� �ð6Þ

3.2. Estimation of embedding dimensions of variables

The false nearest neighbor (FNN) algorithm assesses whetheror not neighboring points in the phase space are close to one an-other owing to their dynamic origins, in which case they areconsidered true neighbors, or whether they appear to be neigh-bors, owing to the suboptimal unfolding of the phase spaceattractor as a result of an incorrect embedding dimension. Ifso, these points are considered to be false neighbors. The idealembedding dimension is chosen as the dimension at which thepercentage of false nearest neighbors reaches a minimum. Asindicated in the last two columns of Table 2, the lags (k) andembedding dimensions (M) of the variables were very similarin magnitude.

3.3. Multivariate embedding of mill variables

The aggregate trajectory matrix XE of the mill variables had adimensionality of NE = N �max(kj) = 22,363 rows xME ¼

P5j¼1Mj = 23 columns, which still required further decorrela-

tion to account for the cross-correlation between the different vari-ables. This was accomplished by use of principal componentanalysis, the principal components of which then constituted thephase space coordinates of this multivariate embedding, i.e. in gen-eral for data set X, the covariance matrix C is constructed, i.e.

C ¼ 1NE � 1

XET

XE ð7Þ

Following this, the eigenvectors P and eigenvalues ^ are calcu-lated for the covariance matrix C using the eigenvaluedecomposition.

C ¼ 1NE � 1

XET

XE ¼ P ^ PT ð8Þ

Finally, the principal component scores constituting the phasespace are calculated using principal components Pq (q columns ofeigenvector matrix P), based on the reduced dimensionality qwhich captures significant variance, according any of a numberof such criteria, such as the Kaiser criterion (Yeomans and Golder,1982), scree tests, and some specified level or variance to be cap-tured (Joliffe, 2011).

CTq ¼ XEPq ð9Þ

As indicated from the eigenspectrum (K) of the principal com-ponents of this embedding in Fig. 4, a significant portion of the var-iation (approximately 61.5%) could be captured by the first threecomponents, resulting in a three-dimensional subspace. This is suf-ficient for visualization of the phase space.

4. Tracking and controlling the mill dynamics

4.1. Visualization of controlled states

The reconstructed state space of the mill can be augmented bysuperimposing the controlled states of the mill onto the plot. Thisis shown in Fig. 5, for a phase space that was reconstructed fromthe data, after removal of a number of sections where close-to-zeromill power consumption was recorded. The controller states con-sidered here are the states associated with normal operating con-ditions (NOC), or default state, and states associated with highmill loads, feed disturbances, feed limited conditions, low mill loads,recycle limited conditions, as well as sand loading. In this case, thefirst three principal component coordinates collectively account

Fig. 5. Reconstructed phase space of the autogenous mill with controlled statesdetermined by the mill controller superimposed.

C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9 5

for approximately 73.6% of the variance of the data, indicating areasonably reliable projection of the data.

In Fig. 5, it can be seen that some of the same states occur in dif-ferent regions of the phase space. The controller state associatedwith sand loading, for example, occurs as two clusters (red mark-ers) in the phase space, together with several smaller spots in be-tween. The same goes for the low mill load controller state.

4.2. Transitions between controlled states

To complement Fig. 5, Table 3 summarizes the time spent ineach state over a period of approximately 66 h. The entries in thetop left to bottom right diagonal of the table are the samples (at10 s intervals) associated with each state, while the off-diagonalentries represent the frequencies of transitions between states.For example, from the NOC state, the mill control state moved toa high mill load state on three occasions and to a feed limited stateon 57 occasions (1st row of the table). From a high mill load state,it moved to the NOC state only once, and twice to a state associatedwith a feed disturbance.

4.3. Analysis of mill load dynamics

In order to better understand the dynamic behavior of the pri-mary state variable of the mill, i.e. the load, this variable was alsoembedded into a phase space by itself. The embedding was basedon a lag of k = 500 sample intervals, as estimated by use of Eq. (6),as well as an embedding dimension of m = 4, which was the lowestembedding dimension where the number of false nearest neigh-bors was negligible. The reconstructed attractor is shown in threedimensions in Fig. 6 (scaled data), which is an approximation of the

Table 3Controlled states of the mill with diagonal entries (bold) showing time spent in a particparticular state in the left hand column to the state in the upper row.

Mill states NOC High mill load Feed disturbance

3. NOC 16,320 3 44. High mill load 1 669 25. Feed disturbance 4 0 10996. Feed limited 51 0 28. Low mill load 7 0 011. Recycle limited 3 0 012. Sand loading 21 0 0

Total 16,407 672 1107

four-dimensional phase space into which the variable wasembedded.

Hypotheses about the nature of the underlying model that havegenerated the mill data can be tested by means of surrogate dataanalysis (Theiler et al., 1992; Barnard et al., 2001). In this case,the null hypothesis that the load time series data were generatedby a Gaussian process undergoing a static transform was tested.The alternative hypothesis would suggest a nonlinear (possiblydeterministic) process.

The hypotheses can be tested by generating surrogate data thatpreserve the linear correlational structure of the original time ser-ies, as well as its marginal distribution. With the iterative ampli-tude adjusted Fourier transform (IAAFT) algorithm originallyproposed by Schreiber and Schmitz (1996), a two-stage processis applied. It starts with a time series constituting white noise,and in the first stage, the Fourier amplitudes of the time seriesare replaced by the Fourier amplitudes of the original time series.In the second stage, the derived time series is reordered to the ori-ginal marginal distribution until convergence of both power spec-trum and marginal distribution is attained. Termination of thealgorithm is triggered by complete convergence (same reorderingin two consecutive steps), or if a maximum number of iterationsis reached.

More formally, if {xt} is a time series of length N and scaled tohave a zero mean, then its Fourier transform is (Eq. (10)):

f ðxÞ ¼ 1ffiffiffiffiffiffiffiffiffiffi2pNp

XN

t¼1

e�ixtxt; for � p 6 x 6 p ð10Þ

For discrete frequencies, xj ¼ffiffiffiffi2pN

q; j ¼ 1;2 . . . N the transforma-

tion can be calculated according to Eq. (11), and the original timeseries can be recovered by back transformation.

xt ¼ffiffiffiffiffiffiffi2pN

r XN

j¼1

eixj t f ðxjÞ; for t ¼ 1;2; . . . ;N ð11Þ

Fourier transform surrogates construct new time series (yt) withthe same periodogram as the time series, xt. The Fourier ampli-tudes are fixed, while the Fourier phases are replaced by uniformlydistributed random numbers, urandðxjÞ 2 ½0;2p�. The AAFT surro-gate time series (yt) is realized as

yt ¼ffiffiffiffiffiffiffi2pN

r XN

j¼1

eixj t jf ðxjÞje�iurandðxjÞ; for t ¼ 1;2; . . . ;N ð12Þ

After generation of the AAFT surrogates, the periodogram valuesare replaced by the periodogram of the original time series and thephases are retained. The amplitudes are subsequently adjusted inthe time domain and the two steps are iterated until the

ular state and off-diagonal entries showing frequency of state transitions from the

Feed limited Low mill load recycle limited Sand loading

57 0 3 200 0 0 01 0 0 33316 2 0 110 569 0 40 0 303 08 9 0 908

3382 880 306 946

Fig. 6. Reconstructed attractor of the mill load time series data.

Fig. 7. Correlation dimensions of the mill load time series data (solid curve) andIAAFT surrogate data derived from the mill load time series (broken curves).

6 C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9

periodograms and the amplitude distributions of the original timeseries and its iterated AAFT surrogate are approximately equal.

Fig. 8. Prediction of the mill load with a m

When embedded in phase space, the topology of these time ser-ies can be represented in a robust way by their correlation dimen-sions (Grassberger and Procaccia, 1983a,b) and for this reason thecorrelation dimension is used very widely as a test statistic to com-pare time series data in phase spaces. In this paper, a version of theanalysis proposed by Judd (1994) is used, where the correlationdimension (dc) is plotted for different values of a scale parameter(e0).

A plot of the correlation dimensions of the original load timeseries data (solid curve) and 20 IAAFT surrogate time series (bro-ken curves) derived from the original load time series is shownin Fig. 7. Since the correlation dimensions of the original dataand their surrogates, dc(e0), are markedly different, the nullhypothesis that the time series was generated by a stochastic pro-cess (Gaussian or otherwise) undergoing a static transform can berejected.

It is also interesting to note that the correlation dimension ofthe load (the solid curve in Fig. 7) is low (less than or equal to 2).Since the dimension of the correlation dimension is an indicationof the number of dominant variables present in the evolution ofthe underlying system dynamics (Boon et al., 2008; Hill et al.,2008), it also suggests that only a few (two) variables would beneeded to model the behavior of the mill load. This should notbe construed as a generalization for the modeling of load behaviorin autogenous mills, as in this case, it is applicable to a compara-tively short period of observation of the particular mill run underset conditions of control.

ultilayer perceptron neural network.

Table 4Classification of mill states with a multilayer perceptron on validation data.

Class Sandloading

NOC Highmillload

Feeddisturbance

Feedlimited

Lowmillload

Overall

No of samples 61 118 72 177 113 59 600Correct (%) 54.1 84.7 94.4 94.9 91.2 81.4 86.7

Fig. 9. Linear projection of mill states showing maximum separability between thestates.

C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9 7

Moreover, the identities of the dominant variables cannot be in-ferred from the analysis, and it can only serve as a guideline to theconstruction of models. Such a low dimensionality could also beindicative of a strong deterministic component in the data.

dc ¼ lims0!0

limN!1

log CN

log e0

� �ð13Þ

CNðe0Þ ¼N

2

� ��1 XN

06i<j6N

I k#i � #j < e0Þ ð14Þ

Although this would need to be confirmed by more extensiveanalysis beyond the scope of the present study, the predictabilityof the load behavior supports the indication of a deterministic

sand loading

NOC

low loading

feed

disturbance

high loading

NOC

0.5 1 10

0.2

0.4

0.6

0.8

1

Time

Scal

ed L

oad

[-]

0.5 1 10

0.10.20.30.40.50.60.7

Time

Scal

ed P

ower

[-]

Fig. 10. Scaled load (top) and power (bottom) measurements with

system. Fig. 8 shows the results obtained with of a multilayer per-ceptron with six hidden layer nodes, trained by the Levenberg–Marquardt algorithm, which was used to predict 500 sample pointsahead, based on four inputs, viz. the load at time t, t-500, t-1000and t-1500. The number of nodes in the hidden layer of the neuralnetwork was optimized by use of the Schwarz–Christoffel informa-tion criterion (Schwarz, 1978). The neural network was trained onthe first 38,128 samples of the time series data, and the predictionshown in Fig. 8 is for the last 5448 sample points.

5. Prediction of mill states from operational variables

The mill states were predicted from the operational variables inTable 2 by use of a multilayer perceptron neural network config-ured as a classification model. The network had an input layer withfive nodes, corresponding to five inputs x1, x2, . . . x5, as summarizedin Table 2, a single hidden layer with nine bipolar sigmoidal activa-tion units, and six output nodes with unipolar sigmoidal activationfunctions coding for the six mill states. A subset of the data consist-ing of 4000 samples was randomly divided into a training, test andvalidation data set, respectively containing 70%, 15% and 15% of thesamples. The validation data set was used to assess the accuracy ofthe model after training with the training and test data sets. Thenetwork was trained with the Broyden–Fletcher–Goldfarb–Shanno(BFGS) gradient descent algorithm and a sum of squares error cri-terion and could predict the state of the mill with 86.7% accuracyon the validation data.

A breakdown of the results are given in Table 4, where it can beseen that the model was not able to predict the state labeled SandLoading meaningfully, but could predict the other states with anaccuracy of at least 80%. The mill states are shown in Fig. 9, whichshows a linear projection of the variables generated to maximizethe separability between the different groupings or states by opti-mizing the ratio of the between scatter of the samples in the clas-ses to the within scatter of the samples in the classes. From thesedata, it can be seen that the Sand Loading samples shown by smallsolid diamonds, are spread across several other groups. Since thisclass is defined by the rates of change of the power and the powerto load ratio (decreasing more than set thresholds), as well as theratio of feed rates, it is difficult to capture by instantaneous mea-surement of the variables only (as also suggested by the inabilityof the neural network model to identify the class).

feed limited

unknown

.5 2 2.5x 104

Index

.5 2 2.5x 104

Index

logged mill states representing a period of 89 h of operation.

Fig. 11. Power–load curve of the mill with logged controller states.

8 C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9

6. Relationship between mill load and power consumption

Fig. 10 shows the load and power measurements as time seriesdata, with the logged controller states superimposed on the data.The relationship between the mill load and power consumptionis shown in Fig. 11, again with the logged mill controller statessuperimposed on the data. As can be seen from Fig. 11, the curveis not well-defined, owing to the effect of the other mill variablesnot accounted for in the power–load plane.

Fig. 12. Relative importance of variab

Fig. 13. Corrected power–load curve generated by neural network model (bottom left) w(top left) with 10-sample localized mean values for all inputs (broken red line). The breferences to color in this figure legend, the reader is referred to the web version of thi

The effect of disturbances on the power–load curve in Fig. 11can be eliminated by using neural network models to estimatethe curve at fixed (set point) values of the control values. A multi-layer perceptron model trained for this purpose managed to ac-count for approximately 71% of the variation in the powerconsumption, using mill load as a single input only. More than92% of the variance in the power consumption could be explainedby the model, when the water inlet, fine ore and coarse ore feedwere also taken into account.

More specifically, the model used to generate this smoothpower–load curve had six logsigmoidal units in its single hiddenlayer and a tansigmoidal output unit and was trained with theBroyden–Fletcher–Goldfarb–Shanno (BFGS) gradient descent algo-rithm and a sum of squares error criterion, as before. A randomsample of 70% of the data was used as training data set, whilethe remainder of the data was used to test and validate the model.

Of these additional variables, the water inlet had the mostimportant influence, as shown in Fig. 12. The contributions of thevariables were determined by removal of the variable from themodel and noting the effect on the model output. The broken redline in Fig. 12 coincides with the 99% significance level for the var-iable contributions to the model and was generated by use ofMonte Carlo simulation of the effects of random input variablesin the model, as discussed in more detail by Auret and Aldrich(2011).

The solid black curve in Fig. 13 shows the correction when themultilayer perceptron neural network model was used to predict

les in power–load curve model.

ith set point inputs for control variables (solid black line) and neural network modellue dots are the raw data used to validate the models. (For interpretation of the

s article.)

C. Aldrich et al. / Minerals Engineering 56 (2014) 1–9 9

the power–load curve at the mean values of the control variables(fine ore feed, coarse ore feed and water inlet). Alternatively, theoutputs of the model can be averaged over local regions tosmooth the curve, as indicated by the broken red line in Fig. 13with a moving average of 10 samples. However, this approachis not as reliable as that based on the use of a neural networkmodel.

7. Conclusions

In this study, the dynamic behavior of an autogenous mill wasanalyzed by embedding the mill variables in a multivariate phasespace where all the variables could be simultaneously consideredtogether with information on the operating states of the mill. Inaddition, the mill load was also embedded in the phase space byitself. Based on qualitative (visualization) and quantitative (surro-gate data) analysis of the phase space embeddings, the followingconclusions can be made.

� The controlled states of the mill could be visualized as separateclusters in low-dimensional (2D or 3D) mappings of the phasespace that consisted of multivariate embeddings of the opera-tional variables of the mill. This validates the labels in the millcontroller assigned to the different states.� These controlled states of the mill could be predicted with a sat-

isfactory degree of accuracy by multilayer perceptron neuralnetworks.� Comparison of the correlation dimensions of the load time ser-

ies data with random time series that had the same power spec-trum and marginal distribution as the load data suggest that theload data do not conform to a linear stochastic model. Instead,the data could have a significant deterministic component andwere influenced by relatively few (perhaps as few as two) dom-inant variables.� Neural network models were constructed that could predict the

power load curve with a high degree of accuracy (R2 > 0.92) and,in principle at least, these models could be used to eliminate theeffect of disturbances on the power load curve, which couldthen be used to assist plant operators to control the mill.

Acknowledgements

The authors gratefully acknowledge:

� Anglo American Platinum’s Advanced Process Control Group fortheir inputs, as well as making plant data available to theauthors.

� The Anglo American Platinum Centre for Process Monitoring atthe University of Stellenbosch in South Africa for use of theircomputational infrastructure.

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