Potential use of model predictive control for optimizing the column flotation process

Int. J. Miner. Process. 93 (2009) 26–33

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Int. J. Miner. Process. j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i j m i n p r o

Potential use of model predictive control for optimizing the column flotation process M. Maldonado a,⁎, A. Desbiens a, R. del Villar b a b

Department of Electrical and Computer Engineering, LOOP (Laboratoire d'observation et d'optimisation des procédés), Université Laval, Québec City, Canada G1V 0A6 Department of Mining, Metallurgical and Materials Engineering, LOOP (Laboratoire d'observation et d'optimisation des procédés), Université Laval, Québec City, Canada G1V 0A6

a r t i c l e

i n f o

Article history: Received 16 January 2009 Received in revised form 28 April 2009 Accepted 13 May 2009 Available online 27 May 2009 Keywords: Model predictive control MPC Flotation column Optimization

a b s t r a c t A constrained model predictive control (MPC) strategy is proposed to deal with the problem of optimizing flotation column operation using secondary variables. Froth depth, collection zone gas hold-up and bias rate are selected as secondary variables to be controlled whereas tailing, wash-water and gas flow rate are used as manipulated variables. The control problem was formulated in order to minimize the tracking error of the gas hold-up and bias rate by maintaining gas flow rate, wash-water flow rate and bias rate within their operational limits. In particular, a strategy was conceived to optimize the column flotation process based on establishing an unreachable high set point for the gas hold-up (which is equivalent to maximizing the bubble surface area available for particle collection at a given flotation reagent dosage and thus recovery), while simultaneously satisfying operational constraints (such as ensuring a positive bias rate to prevent gangue entrainment and therefore concentrate grade deterioration). Several other operational constraints on washwater, gas rate, gas hold-up and bias rate were considered, their use being justified from a processing point of view. Since this study deals with the hydrodynamic characteristics of flotation columns, a pilot flotation column working with a two-phase system is sufficient to demonstrate the advantages of using predictive control for this process optimization. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The metallurgical performance of a flotation column is determined by the valuable-mineral concentrate grade and recovery, often called primary variables. Whereas the first of these two variables can be measured on-line using an X-ray on-stream analyzer (OSA), the latter must be estimated from steady-state material balance, which strongly limits its use for regulatory control purposes. Moreover, the long sampling times of these OSA devices, usually multiplexed, favour the use of a cascade control to reject the frequent disturbances occurring in this type of process. Although real-time optimization (RTO) offers a direct method of maximizing an economic objective function, steady-state assumptions limit the optimization frequency, often leading to suboptimal economic performance (Engell, 2007). These factors, along with the high cost of the instrumentation associated with the implementation of OSA devices for each flotation column of a plant, make it more convenient to indirectly control these primary variables through secondary variables which can be measured on-line at a reasonable cost. In this work, the following secondary variables have been selected: (a) the froth depth (hf) (pulp–froth interface position), responsible for the collection zone volume as well as the hydraulic entrainment reduction when deeper than 1 m (Yianatos et al., 1987), (b) the ⁎ Corresponding author. E-mail address: [email protected] (M. Maldonado). 0301-7516/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2009.05.004

collection zone gas hold-up (εg), which is related to the gas flow rate and the total bubble surface area available for particle collection, and (c) the bias rate (JB), partly responsible for the cleaning action of the froth zone. Clingan and McGregor (1987) reported a monotonically increasing relationship between the recovery and the superficial gas velocity (Jg) at Magma Copper Co. This result can be explained by the fact that the bubble surface area available for particle collection increases with the gas rate for a given and constant flotation reagent concentration. Recently developed techniques for measuring some gas dispersion variables in flotation processes (Gomez and Finch, 2007) allow the estimation of the bubble surface area flux. These techniques could be used for control purposes when an on-line version of the corresponding sensor becomes commercially available. Among these variables, the collection zone gas hold-up provides important information, while having the advantage of being measured on-line by commercial sensors (Gomez et al. 2003; O'Keefe et al., 2007). Gas hold-up is usually modified in plant practice by manipulating the gas rate, because it is simple and low-cost procedure which provides an effect rapidly. Although frother type and concentration also affect gas hold-up, pulp chemistry (collectors, frothers, activators, etc.) is usually kept unaltered in mineral processing plants, unless ore characteristics or throughput change. Thus, no effort was made in this work to use it as a manipulated variable to modify gas hold-up. Gas flow rate is often limited in practice by various operating conditions, for instance, by the amount of bias rate required to ensure hydraulic entrainment reduction (and thus concentrate grade

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Table 1 Description of operational constraints. Constraint

Function description

Jg ≤ Jgmax

To To To To To To To To To To

Jg ≥ Jgmin Jw ≤ Jwmax Jw ≥ Jwmin JB ≤ JBmax JB ≥ JBmin

prevent hydraulic entrainment prevent loss of the interface prevent froth “burping” keep solid in suspension avoid froth mixing avoid wash-water short-circuiting promote froth stability facilitate transfer of collected particles into the concentrate avoid reduction of collection residence time for valuable minerals perform cleaning action thus reducing gangue entrainment

deterioration). Another similar situation might arise from the fact that excessive gas flow rate could lead to interface loss and/or the presence of large gas bubbles at the froth surface commonly referred to as ‘burping’ (Finch and Dobby, 1990), conditions that must be avoided. Other operational constraints must also be taken into account in flotation column operation. For instance, an excessive wash-water rate (Jw) would increase froth mixing and water short-circuiting to the concentrate, thus decreasing the froth cleaning action (Yianatos and Bergh, 1995). A high bias rate would diminish the residence time of valuable minerals in the collection zone and therefore must be avoided.

Fig. 2. Validation of froth depth estimation.

In this regard, Finch and Dobby (1990) claim that a bias rate less than 0.1 cm/s is usually sufficient for adequate cleaning purposes. On the other hand, a minimum value of wash-water is necessary to promote froth stability and to carry the collected particles into the concentrate launder. Table 1 summarizes these operational constraints.

Fig. 1. Pilot column flotation set-up.

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Fig. 3. Bias rate relationship validation.

Fig. 5. Step-change response identification: gas hold-up (εg) v.s gas rate (Jg).

Predictive controllers are now more accepted in many industrial plants, and several successful applications have been reported (Qin and Bagdwell, 2003). One of the most appealing features of predictive controllers is their ability to handle process constraints, an important issue considering that optimal process operating points often lie at the operational constraints. Despite the important role played by process control in improving many process operations, relatively few applications to industrial flotation columns have been reported. Among those mentioned in technical literature, those based on expert rules seem to be the most popular (Bergh and Yianatos, 1993). At the regulatory level, only the froth depth is usually controlled using PID controllers. Nothing has been reported on industrial bias rate control or gas hold-up control. A few attempts have been made by academic researchers. Pu et al. (1991) implemented a predictive control using the Dynamic Matrix Control (DMC) formulation to a three-phase pilot flotation column. Froth depth and gas hold-up were estimated using pressure transducers and respectively controlled using tailing flow rate and gas flow rate as manipulated variables. The system was represented as an upper-triangular 2 × 2 transfer matrix, from which it is evident that closed loop stability can be achieved with relative ease. Reasonably good results were obtained, an expected achievement considering the structure of the matrix transfer function. Unfortunately, one of the main advantages of predictive control, effective constraint handling, was not exploited in this work. Nunez et al.

(2006) later implemented a constrained predictive control based on the Global predictive control (GlobPC) formulation (Desbiens et al., 2000), for a three-phase pilot flotation column. Results reassert the capabilities of MPC to deal with constraints. One flaw of this study was that constraints were treated just as a mathematical formalism rather than using a process point of view. Considering that decentralized control is often preferred in the mineral industry because of its ease of implementation, fault tolerance and re-tuning simplicity, delVillar et al. (1999) and Bouchard et al. (2005) have explored the application of such a control working on a pilot flotation column on, respectively, two and three-phase systems. The selected variable pairings were froth depth-tailing flow rate and bias rate–wash-water flow rate. Although good results were reported, other important variables related to recovery such as gas hold-up and gas rate were omitted from the testing. Maldonado et al. (2008a) extended these works by incorporating a control loop for gas hold-up by manipulating the gas flow rate. Under this scenario, results showed that the bias control loop performance is strongly dependent on gas flow rate. This fact had been already pointed out by (Bergh and Yianatos, 1994) who indicated that bias rate is affected by the gas flow rate in that it changes the fraction of the wash-water directly going to the concentrate (short-circuiting). Another decentralized control was reported by Persechini et al. (2004), which consisted of a 3 × 3 multivariable system whose input– output pairings were obtained by using the relative gain array (RGA) analysis. The resulting pairings were the following: (a) froth depth —

Fig. 4. Relationship between bias rate and volumetric fraction of wash-water.

Fig. 6. Step-change response identification: bias rate (JB) vs. wash-water flow rate (Jw).

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The problem was formulated so as to minimize the tracking error of gas hold-up and bias rate by maintaining the gas flow rate, wash-water flow rate and bias flow rate within their respective operational limits. To reflect the recovery-grade trade-off by using secondary variables, a high gas hold-up set point was imposed (high enough not to be reached, as suggested by (Aske et al.2008)), which increases the bubble surface area available for collection of valuable particles, thus increasing collection recovery for a given reagent concentration. A constraint on the lower bound value of bias rate was used to provide cleaning action (positive bias) on the froth, thus reducing concentrate deterioration due to gangue entrainment. Other operational constraints on wash-water flow rate, gas rate and gas hold-up were also considered. 2. Experimental set-up

Fig. 7. Step-change response identification: bias rate (JB) vs. gas rate (Jg).

wash-water flow rate, (b) gas hold-up — gas flow rate and (c) bias rate — tailing flow rate. Only the first two control loops were implemented, leaving the bias rate loop in manual mode. This choice of variable pairings for control purposes has an important drawback as there is no direct relationship between the collection zone height (total column height–froth depth) and wash-water, however there is a relationship between collection zone height and the bias rate, i.e., the net water flow rate crossing the interface. Consequently, froth depth stabilization might become difficult when a high gas flow rate is used (wash-water shortcircuiting to concentrate) and impossible for negative bias values. Under these conditions, the froth depth could diminish indefinitely without any external intervention (Hyma and Salama, 1986). In order to solve this problem, Carvalho and Durão (2002) proposed a control strategy based on fuzzy inference, whereby if the froth depth is far from its reference value, evaluated through the membership functions ‘high’ and ‘low’, the tails and wash-water flow rates are simultaneously adjusted. Consequently, a coupled non-linear system is obtained. In the present paper, a constrained multivariable predictive controller was implemented in a pilot flotation column working with a two-phase system (water–gas mixture, no solids). Considering the fact that a change of 50 cm in froth depth only leads to a 5% change in the collection zone volume, for a 10 m collection zone height, the impact of froth depth on column optimization is not obvious and therefore its value was fixed at 80 cm in order to damp-down gangue entrainment.

The column set-up used for this work is located at COREM (Mineral Processing Research Consortium) pilot plant in Québec City, Canada. The column is made of 5.1 cm internal diameter polycarbonate tubing for a total height of 732 cm. A cylindrical porous stainless-steel sparger (38 cm2 area) is located at the bottom of the column for bubble generation. Gas flow rate is controlled through a local PID loop, as shown in Fig. 1. To measure froth depth and gas hold-up, the pilot flotation column is instrumented with 15 stainless-steel conductivity electrodes. Nine of these electrodes are 5.1 cm external diameter by 1.5 cm height rings, flush-mounted inside the column wall at 10 cm intervals, for measuring the conductivity profile in the upper part of the column, across the interface. The remaining six electrodes, all ring type having 2 cm external diameter and 1 cm height, are installed in pairs (5 cm apart) in flow-cells installed in the wash-water, column feed pipes and in the lateral branch of the gas hold-up sensor, for measuring the conductivity of the corresponding streams. Wash-water is added at the top of the column using a peristaltic pump and its flow rate is measured using a turbine flow meter. Column feed and tailings are handled with peristaltic pumps and their flow rates are measured by magnetic flow meters. Local control loops (PID) are implemented to regulate feed, tailings, wash-water and gas flow rates at their respective set points. Data acquisition is performed by a SCADA software (iFIX™) working under a Windows XP™ operating system. 3. Measuring the operating variables 3.1. Froth depth (hf) The method used for froth depth estimation is based on the semianalytical procedure developed by Perez (Perez et al., 1993) and later modified by Grégoire (delVillar et al., 1999). The calculation algorithm

Fig. 8. Global predictive control (GlobPC) structure.

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(k⁎) is given by Eq. (3) where ki is the conductivity of the gas–water mixture in the cell immediately below the cell containing the interface.



k = ki

kl kl–g

! ðμS=cmÞ:

ð3Þ

To relate the bias rate to the fraction of wash-water just underneath the interface, the following empirical relationship, is used: JB = 0:003966 · ɛw − 0:03409 ðcm=sÞ:

ð4Þ

Fig. 3 presents the validation of Eq. (4) using a different set of data. It can be seen that reasonably good results are obtained using the model proposed. Fig. 4 shows the static relationship between the bias rate measured through the additivity-rule equation (Uribe-Salas et al., 1991a), under perfectly controlled steady-state conditions and that obtained with the present method. More details of this method are presented elsewhere (Maldonado et al., 2008c). 4. Process dynamics identification Fig. 9. Control system performance under active constraints on the bias rate (lower bound) and the wash-water flow rate (upper bound).

has recently been improved through a non-linear interpolation between the two probable values of froth depth determined by the ‘maximum slope method’ (Uribe-Salas et al., 1991b). This improved sensor has been described elsewhere (Maldonado et al., 2008b). Fig. 2 shows the validation results of the proposed method.

As will be explained in the next section (Control algorithm: GlobPC), the control scheme chosen for this work is composed of a PI controller for froth depth and a 2 × 2 model predictive control for bias rate and gas hold-up. Continuous process models described as first-order lag plus deadtime systems were identified using the MatLab® System identification toolbox. In this case, the model parameters (gain, time-constant and time-delay) were determined by minimizing the mismatch between model and process step-responses. Then, discrete-time models were obtained using a zero-order hold block and a sampling time of 30 s.

3.2. Collection zone gas hold-up (εg) The gas hold-up in the column collection zone can be derived from Maxwell's equation for electrolyte mixtures (Maxwell, 1892), as proposed by Tavera (Tavera et al., 2001). This method relates the relative proportions of the dispersed (non-conducting) phase (i.e. gas bubbles) and the continuous (conducting) phase (liquid or pulp) to their respective electrical conductivities as shown in Eq. (1), where kl and kl–g are the conductivities of liquid and liquid–gas mixture, respectively. ɛg = 100

kl − kl–g kl + 0:5kl–g

! ð%Þ:

ð1Þ

3.3. Bias rate (JB) Bias rate is estimated from the prevailing volumetric fraction of wash-water just beneath the froth zone(εw), which for a two-phase system can be estimated using the relationship shown below, where kf and kw are the conductivity of the feed water and wash-water streams.   k − k ɛw = 100 f ð%Þ kf − kw

ð2Þ

The conductivity of the liquid only (no gas) contained in the ith conductivity-cell, immediately below the cell containing the interface

Fig. 10. Control system performance under active constraints on the bias rate (upper bound) and the wash-water flow rate (lower bound).

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The input–output relationship is represented by a discrete-time linear time-invariant system:

Based on the receding horizonprinciple, a predictive control law is obtained at every sampling time k by minimizing the following cost function:

yðzÞ = GðzÞuðzÞ

J=

ð5Þ

Ny HP   2  X X wj r̂j ðk + i =kÞ− ŷ j ðk + i=kÞ

ð7Þ

j = 1 i = HS

where for this case of study y(z) = [ΔJB (z)Δεg (z)]T, u(z) = [ΔJw (z)ΔJg (z)]T and

2

0

ð0:468z + 0:916Þz zðz − 0:948Þ

Nu HX C −1 X

2

λp Δup ðk + iÞ +

p=1 i=0

Ny X

2

ρj ν j

j=1

where:

3

6 7 −7 − 11 7 6 + 0:0419Þz ð0:0228z + 0:0172Þz GðzÞ = 6 ð0:0185z − 7: zðz − 0:896Þ zðz − 0:939Þ 4 5

+

– wj, λp and ρj weight the tracking errors, control variations and the ‘slack variables’ vj used to soften the output constraints (Maciejowski, 2002). – Δuj (k + i) = uj (k + i) − uj (k + i − 1) are the control increments over the control horizon HC. – r̂j are the predicted set points (it was assumed in this work that all future set points are equal to the present set points). – ŷj(k + i / k) are the predicted controlled variables over a future horizon HS to HP and – Ny and Nu are the number of outputs and inputs of the system to be controlled.

ð6Þ

−2

These dynamic models were obtained under constant froth depth, controlled by the manipulation of the tailing flow rate. Since the effect of wash-water flow rate on collection zone gas hold-up proved to be negligible, for simplicity it was assumed to be zero for the rest of the work. Figs. 5–7 show the process step-responses of the process used for identification purposes as well as those of the identified discrete-time models given in Eq. (6). 5. Control algorithm: GlobPC As anticipated in the previous section, since the control of froth depth does not represent a major problem it was decided to use a simple PI controller, as is often the case in industry. For the remaining two secondary variables, gas hold-up and bias rate, a MIMO model predictive control scheme (Eq. (6)) was retained, owing to its proven ability for dealing with existing time delays and constraints.

The output predictions are the sum of deterministic and stochastic predictions: ŷj ðk + i = kÞ = ŷDj ðk + i = kÞ + ŷSj ðk + i = kÞ:

ð8Þ

Deterministic predictions (ŷDj(k+i/k) are based on the model G(z) (Eq. (6)), whereas stochastic predictions (ŷSj(k+i/k) are based on a stochastic model that usually contains an integrator to represent non-stationary disturbances, hence introducing an integral action in the control law. The minimization of this cost function is subjected to the following constraints: uj ―

≤uj ðkÞ; …; uj ðk + HC − 1Þ

≤ uj

y − νj

≤ŷj ðk + 1 = kÞ; …; ŷj ðk + HP = kÞ

≤ yj + νj

―j

≤νj ̲ ̲ where u- j, uj, y j and yj are the minimum and maximum values of the control actions and output predictions respectively. The GlobPC formulation (Desbiens et al., 2000) was selected for the implementation of the multivariable constrained predictive control. The GlobPC is based on the internal model control (IMC) structure (Garcia and Morari, 1982) as shown in Fig. 8. The following notations are used in Fig. 8: 0

½ ½  ½ ½  r̂JB ðk jk + HS Þ

⋮ r̂JB ðkjk + HP Þ

R=

−− − − − −U= r̂ɛg ðk jk + HS Þ

⋮ r̂ɛg ðk jk + HP Þ

YŜ =

ŷDJ ðkjk + HS Þ B ⋮ ŷDJ ðk jk + HP Þ

− − − − − − YD̂ = ŷSɛ ðk jk + HS Þ

− − − − − − ŷDɛ ðkjk + HS Þ

ŷSɛ ðkjk + HP Þ

ŷDɛ ðkjk + HP Þ

B

g

Y ̂ =YD̂ +YŜ :

⋮

ð10Þ

uJB ðkjk + HS Þ ⋮ uJB ðkjk + HP Þ −− − − − − uɛg ðkjk + HS Þ ⋮ uɛg ðkjk + HP Þ

ŷSJ ðkjk + HS Þ B ⋮ ŷSJ ðk jk + HP Þ

g

Fig. 11. Control system performance under active constraints on the gas hold-up (lower bound) and gas flow rate (upper bound).

ð9Þ

ð11Þ

B

g

⋮

g

ð12Þ

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6. Experimental results The following MPC parameters were chosen by simulation: HP = 30, HS = 30, HC = 1, λ1 = λ2 = 0, w1 = w2 = 1, and ρ1 = ρ2 = 105. All the tests were conducted at 50 ppm of frother concentration (Dow froth, 250C), constant feed rate and constant froth depth. The following three figures present the result of the various control tests conducted on the pilot column running under this control approach. Set points are indicated by dashed lines whereas constraints are represented by dashed-dotted lines. Fig. 9 shows the performance of the implemented predictive control under changes in gas hold-up set points. At about 20 min, a set point change in the gas hold-up was made. As a result, the predictive controller increases the gas flow rate and the wash-water rate to compensate for the effect of the gas rate on the bias rate. It can be seen that wash-water rate saturates to its upper bound (0.26 cm/s). Consequently, the gas hold-up increases until reaching its set point and bias rate decreases, clearly indicating that the wash-water rate could not compensate for the requested gas rate variation. Then, a new set point change in the gas hold-up is introduced at about 63 min. The controller increases the gas flow rate, which in turn decreases the bias rate, reaching its lower operational limit (0.04 cm/s). As a result, the gas hold-up never reaches its new set point. It is observed that for optimizing the flotation column operation, a high (unreachable) gas hold-up set point could be implemented, taking into account the constraints on the bias rate. In particular, the bias rate lower-limit imposes an upper bound on the required superficial gas velocity to prevent a concentrate grade deterioration due to hydraulic gangue entrainment. It can also be noticed (Fig. 9) that for a constant feed rate, the steady-state tailing rate decreases with the gas rate, which is in agreement with the bias rate estimates. However, smoother bias rate estimates are obtained using the method based on conductivity measurements, as compared with what would have been obtained from a collection zone water balance. Fig. 10 shows the opposite situation of the previous figure. In this case, a sequence of decreasing gas hold-up set points was implemented. As a result, the gas rate and wash-water rate decreased, the latter reaching its lower bound. It can be observed that the bias rate imposes a lower bound to the gas hold-up, to prevent an excess of wash-water in the collection zone. Finally, if a small positive bias rate is not enough to ensure an adequate concentrate grade, an upper level constraint on the gas rate or equivalently on the gas hold-up (assuming constant pulp chemistry) could be implemented. Fig. 11 shows the control system performance under an active upper level constraint on the gas rate and a lower level constraint on the gas hold-up. 7. Discussion The use of this approach in a three-phase system working in a 6 in. diameter by 8 m height column is part of an ongoing CRD-NSERC research project. The column will be installed at a partner concentrator where it is going to be fed with a given plant slurry stream. Although new system identification tests need to be conducted, it is expected that the underlying dynamic characteristics will hold. For instance, the integrating dynamic characteristic between froth depth and tailing flowrate will prevail even though solid percentage affects the total height pressure over the tailing pump. Recent studies conducted by the authors showed that gas hold-up gas rate dynamic relationship varies with frother concentration. However these variations can still be handled using an appropriately designed linear timeinvariant controller. Concerning the use of conductivity based sensors, this sort of instruments have been extensively plant-tested by McGill mineral processing group (Gomez et al., 2003) and COREM (Bartolacci et al.,

2008) to measure gas hold-up so it does not seem to us a major concern.

8. Conclusions A combined PI and multivariable predictive control strategy was implemented in a two-phase (water–gas) pilot flotation column. The froth depth was regulated using a PI controller while a predictive controller was formulated as the minimization of the tracking errors of the gas hold-up and bias rate, keeping several operating constraints between their upper and lower limits. In particular, a high set point of gas hold-up was established in order to maximize the bubble surface area available for particle collection, while maintaining the bias rate above a minimum value required for froth cleaning. Other practical operational constraints such as upper and lower limits in the washwater, gas flow and gas hold-up were similarly taken into account. The proposed strategy may indirectly help optimize flotation column operation by using secondary variables, avoiding the assumption of steady-state conditions in standard real-time optimization methods.

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