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An integrated system for supervision and economic optimal control of mineral processing plants
Minerals Engineering, Vol. 12, No. 6, pp. 627--643, 1999
Pergamon 0892--6875(99)00048-5
© 1999 Elsevier Science Ltd All fights reserved 0892-6875/99/$ - see front matter
AN INTEGRATED SYSTEM FOR SUPERVISION AND ECONOMIC OPTIMAL CONTROL OF MINERAL PROCESSING PLANTS
C. MUI~IOZ and A. CIPRIANO Faculty of Engineering, Catholic University of Chile, P.O. Box 306, Santiago 22, Chile E-mail: [email protected] (Received 1 August 1997; accepted 21 January 1999)
ABSTRACT
This work tackles the problem of dynamically optimising the performance of a mineral concentration plant, taking into account economic profits and technical constraints. The paper proposes a two-level control strategy with regulatory control and the optimisation of an objective function. The regulatory control employs linear model based multivariable predictive controllers with constraints on controlled and manipulated variables, while the optimiser maximises the economic profits using non.linear dynamic models and linear constraints. The results, drawn from simulations, show the proposed strategy to lead to a significant improvement in economic profits when compared against an exclusively regulatory strategy. © 1999 Elsevier Science Ltd. All rights reserved. Keywords Grinding; froth flotation; process optimisation; process control; simulation
INTRODUCTION Control of grinding and flotation plants has been widely studied both through simulation and, directly, through plant experiments with regard to the complexity of the process. Proposals for grinding control strategies include, for example, PID regulators, multivariable control systems [1], adaptive control [2], and optimal control [3]. When looking at flotation strategies, the application of expert control [4], multivariable predictive control [5] and fuzzy control [6] stand out. Works tackling either the grinding or flotation optimisation, have generally considered technical criteria, such as maximising mineral circulation flow, maximising mineral output, minimising product particle size or operating a flotation plant at minimum tailings given a minimum ore concentrate. This work proposes the use of a combination of technical and economical criteria in order to determine the most effective control. Initially, there is a description of the grinding-flotation plant, upon which the study is developed. This is followed by the presentation of the dynamic simulator about which the new strategy will be evaluated, l.~en, expressions for costs, incomes and profits will be outlined, as well as the technical constraints on different operating variables. After that, the optimal control problem is addressed, and a two-
Presented at Minerals Engineering '97, Santiago, Chile, J u l y - A u g u s t 1997
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level control strategy is proposed. The first is a multivariable predictive regulator, while the second level is for optimisation, employing a Hammerstein model based optimiser which supplies the predictive controller's set-points. Finally, the test results are given both for regulatory and optimising control under typical ore concentrate plant disturbances.
T H E PLANT AND ITS SIMULATION Figure 1 shows a concentration plant with three grinding sections and a single flotation section. The ore, fresh from a storage pile, is water treated in the mill forming a slurry. The discharge from each mill is stored in a sump, to which water is added with the slurry then pumped on to a battery of classifiers. A fraction of the classifier discharge is fed back to the mill while the remainder continues on its path to the flotation plant, where the high copper content particles are separated from the rest. The basic grinding control consists of: a) fresh ore PI control manipulating three belt speed, b) water flow PI control manipulating the aperture of the valves, and c) PI control of the sump pump.
GrindingSection 1 Rougher Flotation and Concentrate
Crushing ore
ore
Grinding Section 2
FirstCleanning Tails
ore
Grinding Sectio~ 3
Fig. 1 Diagram of the concentration plant. The phenomenological model for each grinding section consists of: the mill model, the sump model and the pump-cyclone circuit model [7]. The mill model comprises ten non-linear differential equations, providing mass balances for each class of particle size. The sump model is a mass balance for class of ore and a general balance for the flow of stored pulp. Finally, the pump--cyclone circuit model is based on the relationship developed by Lynch and Rao. The flotation section includes both rougher and first cleaning flotation each of which is made up of a series of cells. The basic flotation control consists of PI controls of the level and ratio control of reactives respect to the total fresh ore feed. The flotation process is modelled using flow balances and relationships that consider two phases (slurry and foam) and two classes of ore (rich and poor) [8]. The main relationships are mass balance equations, by ore type, for each of the phases that provide five differential equations per cell. As the plant only consists of two cells, the dynamic flotation model comprises ten non-linear differential equations. The concentrator plant simulator also includes, for each grinding section, an ore feed controller, a water/ore ratio controller at the mill feed, a water flowrate controller to the sump and a sump level controller determining the pump velocity. It also considers local cell level and flotation reactant flow controllers. Level references will be fixed while the references of reactant flows shall be proportional to the amount of o r e processed, The simulator was programmed in MATLAB/SIMULINK, using a third-order Runge Kutta integration routine.
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OBJECTIVE FUNCTION The objective function, F, represents the profits, in US$/hr, of a grinding-flotation plant and can be formulated as [9]: F = I - %,i. - %,,-
with I the fraction of the revenues from the sale of fine copper produced, where fine copper refers to the amount of copper, in ton/hr, present in the concentrate ore flow of the flotation plant, that will be finally marketed, Cgrin the variable grinding costs, Cltot the variable notation costs, and Cfzx the plant's fixed costs. Income during the time span (to, ty), is given by: t:
1=
fKPycF/~(t+x:) at
(2)
to
where K is a factor indicating the percentage of revenues drawn from concentration of the ore, Pfc is the price per tonne of fine copper, Ffc is the fine copper in the concentrate ore flow, and xf is the mineral processing delay in the grinding-flotation plant. The fine copper flow, Ffc, is expressed as a function of the recovery in smelting, Rsme, the feed grade gf and the tailing grade gt [10]: = R
(3)
in which T(t) represents the fresh feed rate to the grinding, in ton/hr, and, Xg is the mineral processing delay in the grinding plant. Therefore, given a constant feed grade, income takes the form: If
(4) t0
where
f~ = KPIcR,,.,gl
(5)
~2 = g-PfcRsme
(6)
Only the costs associated with the steel and the energy consumed are considered in regard to the grinding costs. It is also assumed, according with industrial plant operation experience, that the variable consumption of steel is proportional to the energy used. Therefore, tf
Csri. = ~TW(t + "~g)dt
(7)
Io
where
= g, Rs,]+ + e .
(8)
with Pst the cost per tonne of steel used, Rst/w the rate of steel used per unit of energy consumed by the grinding plant, Pw the cost of electricity per kW and W the power demand of the mills. The costs due to the steel corrosion, that are independent of the energy consumption, are considered in the fixed costs, Cfvc, of equation (1).
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In determining an expression for the variable cost of flotation, only the most significant cost is taken into account, that is that of the reactants: the lime, the collector, and the foaming agent. These reactants are administered according to feed weight, and, as such the flotation cost is given by: If
C$o, = ~dpZ(t) at
(9)
where to
(lo)
= P,,.R,,. + Poo,Rco, +
with Plim, Pcol and Pfoam the cost of the reactants (lime, collector, and foaming agent) and Rlira, RcoI and efoam their respective addition rates. On using (4), (7) and (9) it is possible to evaluate (1), starting with the parameters given and the following variables: feed weight, tailing grade and the power demand. Finally, transforming the objective function in terms of the discrete time variable k and expressing it in its predictive form, the function that should be maximised by the economic optimal control strategy is achieved:
(11)
F=~" k=l
~,-d~-~2~,(k o + k + K / ) r(k o + k ) - y~.,~(k o +k+Ks) -C),, i=1
where ~, y wi are the predictions of the tailing grade and the power consumed in each of the grinding sections, k0 is the current time, ~Cgand Kf are the mineral processing times Xg and xf expressed in terms of the number of samples, Nf is the prediction horizon and N s is the number of grinding sections.
OPERATING CONSTRAINTS The operating constraints are limits of the measured and manipulated variables that are placed to ensure the non-occurrence of undesired effects. In a plant such as the one under study, the group of operating constraints having to be considered include restrictions per grinding section and also global restrictions for both the grinding and flotation plant. These are specified as follows: Constraint R 1. Lower limits for grinding power demand, to avoid overfilling, and upper limits to avoid motor overload. Constraint R 2. Maximum and minimum limits on the sump, to avoid spillage or blockage of the pipes to the classifier. Constraint R 3. Upper limits on the percentage of pulp solids in the classifier feed to avoid blockages. Constraint R 4. Upper limit on the particle size of the grinding's product, measured as a percentage of particles over 100 mesh, in order to avoid classifier blockage. Constraint R 5. Upper and lower limits on each section's feed rate, due to the operating range of the feeders. Constraint R 6. Upper and lower limits on the rate of addition of water to the sump due to pipe limitations. The six constraints are crucial as the infringement of any of the conditions would lead to a sequence of operations that would significantly reduce the plant performance. These constraints should be considered for Nf samples, as follows.
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Constraint RI:
1
Wlupp <- ~.pp [,
k=-l,2,..,Ns (12)
~,o.j L~Cko + k / ko)J A
where W.(k o +k/ko) is the prediction of the power demand for section i, set at instant kO. Constraint R2:
~,o./ _
k=-l,2,..,N I
(13)
L3,o.j LZ~Cko+k/ko)j LZ~.p.j where
L(k o+k/ko) is the prediction of the sump level for section i, set at instant ko.
Constraint R3:
k=l,2,...,Nf
(14)
D3(ko + k / ko)J LD3.ppj with 19i(k o +k/ko) being the prediction of the percentage of solids of section i, set at instant ko. Constraint R4:
~=,o. <_la~C~o+~/~o)l<_ ~...,
a3,o.] L@k,,+k/ko)j
with
k=l,2,...,Nf
(15)
a~...j
Gi(k o+k/ko) the prediction of the particle size for section i, set at instant kO.
Constraint R5:
k-=l,2,...,Nf
A~,o.j L~3Cko+k/ko)j with Ai(ko+/ak o)
LA3.ppj
the prediction of water flow used in section i, set at instant/%
(16)
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Constraint R6:
L,,,,,p 1., k0)/ <. IT2up L,ow L Cko+k/ o)J r ,o.
l :ko + /
k=-l,2,...,Nf
(17)
with T(k o +k[ko) the prediction of the feed rate sent to section i, as set at instant ko.. It is necessary to impose the following constraints upon the flotation plant: Constraint R 7. Upper limits on the pulp flows to the flotation plant in order to avoid flotation cell overflow. Constraint R 8. Lower and tipper limit on the percentage of solids to the flotation plant, that enables the plant to operate within a pre-determined density range and avoid sanding problems in flotation cells. Constraint R 9. Lower limit on the concentrate grade arising from an operating requirement of the smelting plant. Constraint RIO. Upper limit on the tailings grade in order to comply with the requirements of the tails treatment plant. In order to satisfy restriction R 7 one needs to restrict the flow of pulp to the flotation section. It is, however, necessary to estimate the flow as the large quantity of bubbles make measurement impossible. Given that the dynamics of the flotation plant are much slower than that of the grinding it is reasonable to estimate this flow Q through the following relationship:
Q(k)
A(k + r~g) + T(k + r.g) 5,,
(18)
where A is the total water added, T is the total tonnage of mineral fed to the grinding and 8m is the density of the mineral. A relationship estimating the percentage of solids in the slurry is determined the same way: S(k) =
T(k + Ks) T(k + ~¢g)+ 8wA(k + r.g)
(19)
where 5 w is the density of the water. While the restriction on the percentage of solids in the flotation slurry
S>Slo w is non-linear, it can be transformed into a linear restriction:
1 -SiowT(k +Ks) > S(k) = 8.,A(k + Ks)
(20)
Using the relationships (18) and (20), constraints R 7 to RI0 are established for each sampling within the interval defined as the current time and the prediction horizon N2 as follows: Constraint R7:
k=l,2,...,N; i=I
(21)
Constraint R8: Ns
1-S,ow _, ,(ko +k/ko) i=l
N.~
>
J,(ko + k / k 0 ) , k=-l,2,..,Nf i=1
(22)
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Constraint R9: ~, ( k 0 + k / k 0) 5.-"g, ,pp, with
k =-1,2,...,Nf
(23)
~i(ko+k/ko) the prediction of the tailings grade set at instant ko.
Constraint R10:
gc,o,, < L(ko + k / ko), k=-l,2,...,N where
(24)
~,c(ko+k/ko) is the prediction of the concentrate grade set at instant k0.
CONTROL STRATEGY It is possible to build several strategies based on the previous analysis which satisfy pre-determined requirements. Two strategies follow that are compared in terms of the amount of computing effort required.
Centralised control strategy As has just been outlined before, in order to optimise the objective function keeping the whole plant operating well it is necessary to impose constraints on certain variables. The most direct way of satisfying these requirements is to design an optimiser subject to the constraints that calculates optimal set-points for the final controllels of each section (tonnage, pump speed, and water). This optimiser, which we call centralised, has to maximise the objective function (11) while satisfying constraints R 1 to R10 under both steady and dynamic conditions. Appropriate proce,;s control requires the strategy to determine a new set of set-points for the local controllers with much shorter responses than the system's most significant time constants [11]. Considering that the power responds to changes in the flow of water to the tank at a time constant of 5 to 7 minutes it would be desirable to take references from the local controllers at least once a minute. It is also important that the optimiser is able to solve the optimal control problem such that the constraints are satisfied throughout the prediction horizon. The prediction horizon should be fixed so that the process operates a great deal of the time in a steady state. As the effect of manipulated variables such as tonnage is only noted at times of over three hours in the flotation section, the prediction horizon should be of at least that order of magnitude. How, therefore, should six specific constraints for each of the grinding sections (R 1 to Rt) and four global plant constraints be satisfied while considering that in this case the circuit contains three grinding sections, bringing the total number of constraints to consider up to 22, which has to multiplied by 180 (3 hours at an optimisation interval of one minute), meaning that the optimal control problem has to analyse 3,960 constrains every iteration. In order to calculate the control horizon we need to bear in mind th~Ltthis has to be the least dynamic component of the process, which in this case is the tailings grade faced with loading changes that take approximately one hour. As the optimiser's sampling time is one minute and there are nine references to determine in the optimisation (tonnage, water and pump speed for each of the three grinding sections) it raises the number of optimisation variables to 540. Nowadays there exist computers available with enough speed rates to achieve this requirement. However, trying to group these tasks in a single algorithm elevates its complexity. Troubleshooting identification turns on a hard task, malting it inappropriate for industrial use. In other hand, the centralization of the all tasks makes the strategy less robustness, and any failure of the algorithm would leave the optmisation control strategy ineffectual.
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Decentralised control strategy
Figure 2 shows an alternative strategy, which incorporates a regulatory control stage for each grinding section whose objective is to ensure that each section's critical variables are held at values fixed by hierarchical level above, that act upon references provided by local controllers for each of the grinding sections. As these variables need to satisfy both range and change constraints, the best way of implementing the multivariable controller is through the use of predictive control [12].
Economic [ Decentralized ¢ Optimisation
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Fig.2 Hierarchical control strategy. The optimisation stage provides each of the grinding sections with references concerning the amounts of regulation, as shown in Figure 2. This disaggregation is based on the different response times of the grinding and flotation plant as well as differences in sensor sampling times associated with each plant. It is reasonable to assume that, when a change in the input variables is produced, the flotation plant responds more slowly than the grinding sections, in such a way that, while the flotation plant is still responding, the grinding variables have reached its steady state. Therefore, it is natural to set a one minute sampling time for the grinding plant regulators, which is consistent with commonly used data acquisition systems, and an optimisation period of twelve minutes. That is reasonable considering the times required for the analysis of the flotation grade. The objective of the upper hierarchical level is to maximise the objective function, but without allowing the infringement of the plant's global constraints (constraints R 7 to RIO). As the control horizon is one hour and the prediction horizon is three hours upon considering an updating time of twelve minutes, the optimal control problem takes on 285 constraints and 30 optimisation variables, and demands significantly less computing effort than the centralised system. Regulation in the decentralised control strategy
As the pump speed directly affects the sump level, while its effect on other significant variables is only transitory, the use of a PI regulatory controller is recommended for the pump speed to control the sump level. This controller contributes to the avoidance of R 2 constraint infringements since its manipulated variable, the pump speed, generally exhibits a great degree of variability. The remaining significant variables, power, percentage of solids and particle size need to be controlled in order to satisfy operating
Supervision and economicoptimalcontrolof mineralprocessingplants
635
restrictions through the manipulation of the tonnage and sump water references. Here, operating constraints R I, R 3 to R 6 are expressed as: Range and change constraints on the manipulated variables; tonnage and sump water. Range constraints on the controlled variables; power, particle size and percentage of solids. Within the framework of the imposed restrictions the grinding process exhibits a near linear behaviour and a reduced sensitivity to disturbances. This makes it sufficient, for regulation purposes, to use linear models that are robust against disturbances, and is the reasing behind proposing the algorithm used which also includes possibilities for parameter adaptation. MPCT, a set of routines for developing predictive control strategies in the MATLAB environment [13], was used for testing arLd evaluating regulatory control via simulation. As MPCT does not determine optimal control in situations where regulation of the problem presents no feasible solution, the method is complemented with a set of rules that gradually, relax the constraints upon the controlled variables in a hierarchical manner. The highest priority is assigned to the power component as oversupply or mill overfilling are critical events. A lower priority is assigned to the particle size, while the lowest is assigned to the percentage of solids.
Decentralised control strategy optimisation As we mentioned before, the response time of the grinding plant is of the same order of magnitude that the flotation sampling period, which has been taken as the time interval for the optimal control iteration. Therefore, for purposes of optimisation, in each new optimization sample, the grinding plant can be considered to be in a steady state. This means that the optimising control only requires steady power, percentage of solids and particle size models for each of the grinding sections. For the evaluation of the objective and the dynamic monitoring of the flotation constraints, R 9 and RIO, dynamic tailings and concentrate grades models should be used. As these are non-linear relationships, it is proposed to use Hammerstein models [14] that include both a non-linear static component and a linear dynamic component in identifying the flotation relationships.
Models The linear multiv~riable dynamic model, of each mill, relates mill power consumption, W, percentage of solids, D, and size ore distributon, G, with fresch feed ore, T, and surnp water adition, A. The model is formulated as:
-zX ( k )AD,(k) AG,(k)
= M...,(z
_, ,[ A T j ( k - 1 ) )[A,4,(k-
-1
1)/_]
(25)
where matrix Mmi includes transfer function components in Z-1. For the tailings and concentrate grades multivarible, non-linear Harmnerstein-type models are used. The models relates the grades gc and g~ with the total tonnage, T, and a static non-linear function of the average particle size of product, v. The model has the following structure:
Ck)l rB,,,cz-,) B,,oCz-l)]rrCk-U
BoGCz-')JLv(k-1)]
(26)
in which Agt, Agc, Bt,T, Bt,G, Be.l; Be,G are polynomials of z-~, and v is given by:
v(k) = f (G(k))
(27)
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C. Mufioz and A. Cipriano
The total tonnage T and the average particle size of product G are: Ns
Ns
T(k)=ZT~(k) and G(k)= ZG'(k)Tj(k) i=l i-~l
Predictors at K steps
(28)
Ns
~ Tj(k) i=1
Expressions need to be developed for the predictions for power, percentage of solids and the flow of water consumed in each of the grinding sections and for the tailings and concentrate grades in the flotation plant. It is then possible to evaluate the objective function and constraints predictively, and later formulate the optimal control algorithm. The former expressions are derived from equations (25) to (28).
The optimisation problem We can formulate the objective function (11), starting from the predictions for power and the tailings grade, in terms of the optimisation variables, which are the future particle size and tonnage reference values in each of the sections. The numeric solution of the optimisation problem (11) with constraints also requires the consideration of constraints R 1 and R 3 to R10 in terms of the optimisation variables.
Calculation of the mill regulator references Once the optimal controls {G i, Ti} have been determined, it is necessary to obtain the new power and percentage of solids references, using the static grinding models. These references remain constant during the optimisation interval.
Optimisation algorithm It is possible to construct a flow diagram of the optimisation algorithm (Figure 3), from the previous analysis, essentially revealing four tasks: 1. 2. 3. 4.
Obtain future tonnage and particle size values that optimise the objective function (11) while taking the constraints into account. With the particle size and tonnage values calculated at instant k+l, find the corresponding regulatory control set-points using the static grinding models. The values obtained are applied as the references for the grinding's multivariable regulators. Wait until time period Atopt has elapsed before undertaking the next optimisation.
Solution of the optimisation problem AG*,AT* =arg(max {F(AG,AT) }) subject to RI and R3 to RI0
I
AG*,AT*
Calculation of the optimal references W*, D* and G , using the grinding static models ,
~ W*, D* y G* Transfer of IW', D* and G* as references for regulatory control
÷ Waiting At~wtime required for an optimisation cycle
I Fig.3 Flow chart of the optimiser.
Supervision and economic optimal control of mineral processing plants
6 3 7
SIMULATION RUNS System response without disturbances Figures 4 to 7 show the evolution of the main variables and economic indicators for the plant over time (PR-1 test). Initially, the plant is operated using a regulatory control strategy for 190 minutes, at which point the optimising control strategy is activated. 250
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Fig.7 Grades and economic indicators under regulatory and optimising control. In the three grinding sections, the strategy holds the controlled variables (power, particle size and percentage of solids) at around the pre-determined reference values, see Figures 4 to 6, for which both the tonnage and the water flowrate for each section are modified independently. Due to this, the economic profits also presents variabilities (see Figure 6). No constraints are activated during the test while both the manipulated and controlled variables exhibit very little variation.
Supervision and economicoptimalcontrolof mineralprocessingplants
639
From the moment the optimiser is activated, at minute 190, all of the process's variables produced high variability, although none exceeds the constraints. Figures 4 to 6 show the optimal control altering the tonnage in each section so that one of the tonnages is always at its maximum permitted value, another at an intermediate point with the third at a lower value. A similar situation occurrs with both the particle size and the percentage of solids. This all produces significant changes in the dynamic of the plant, however, as average particle size values and values for the total power and tonnage are maintained the plant's grades appears almost unaltered. The enormous ch~mges in all of the process variables upon activating the optimal control strategy has a period equal to that of the optimisation time. This is due to the fact that the objective function does not have a term weighting the control energy spent.explicitly incorporated, resulting in the plant responding in a very similar fashion to a "dead-beat" type regulator [15]. A term in the objective function, representing the control energy, could be incorporated to improve plant response. This, however, would significantly undermine the objective function with it thereby losing its "economic profits" connotation. An alternative is to impose additional constraints on the variation of optimisation variables, forcing the controller to perform slower optimisation, solving the problem of extreme variability. The additional constrains fix the solution to a one local optimum, avoiding jumps between diferent solutions. This method is called "improved optimising control". Figures 8 and 9 show the results obtained from section 1, the respective grades and economic indicators when xunning test PR-1, and considering improved optimising control. It is possible to see the appreciable reduction in the variation, in the Figures, providing a better response while achieving similar costs and profits to those obtained with the original version of the optimiser.
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i ..... i:-- i...i
50 100 Mill power
150
200
250
300
350
50 11111 Particle ore size
150
200
250
300
350
50 % solids
100
150
200
250
300
350
511
100
150
200
250
300
350
15 35
Minutes
Fig.8 Section 1 variables under regulatory and improved optimising control.
System response with disturbances Test PR-2 is developed in order to prove that the strategy works well under disturbances. This test includes the following events:
640
c. Mufiozand A. Cipriano The regulatory control strategy is activated after 10 minutes. After 70 minutes the value of parameter Q is changed from 105 to 95, indicating that the mineral is harder. After 130 minutes the value of parameter Q is readjusted from 95 to 105. The optimising control strategy effects after 190 minutes. After 250 minutes the value of parameter Q is changed from 105 to 95 After 310 minutes the value of parameter Q is readjusted from 95 to 105 The test ends after 370 minutes had elapsed.
In this description Q is a parameter of the mill model representing the hardness characteristic of the mineral [7].
Concentrate grade
02 o.ls
i
i
;
" :
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450
5500
i . . . . .
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:
200
250
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ti
. ;. . . . . . . . 50 100 Grinding costs
150
~i 200
i
i
i
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100
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i i
i
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250
300
350
i
;
i
250
300
350
250
300
350
250
300
Income=
450O 40OO 3000
1500
I
I 200 Minutes
Fig.9 Grades and economic indicators under regulatory and improved optimising control. Figure 10 shows the evolution of the main variables of section 1 as the test developed, while Figure 11 shows the grades and economic indicators over the same time period. As soon as the disturbance is issued, generated by the increase in mineral hardness, the levels fell, although by a lesser amount than that without using optimal control. While regulatory control responds to disturbances by reducing the feed to the grinding sections and increasing the water supply in order to maintain the particle size constant, the optimising control strategy reduces the total tonnage processed very little accepting a slightly higher particle size which in turn produces a small impact on the process relationships and an almost insignificant reduction in the profits. The most significant effect of increasing the mineral hardness under optimised control conditions is the increase in average power consumed to almost the maximum level permitted. Table 1 presents the maximum profits under ideal circumstances, calculated under steady state conditions, for both tests PR-1 and PR-2. It also shows data for both tests drawn from regulatory and optimising control strategies. In the PR-1 test the regulatory control strategy only reaches average profits of $ 2,262 per hour against a maximum of $ 2,550 per hour. The profits achieved with the optimising control strategy are much closer to the ideal level, and even more so when considering the alternative that took into account changes in the optimisation variables. Optimising control also exhibits much better behaviour when the plant is submitted to disturbances in mineral hardness (test PR-2).
Supervision and economic optimal control of mineral processing plants
250
Fresh ore feed
~50100[ ' " ' ......... ~ - " : i ":. . . . . . . . . . " 50 250
:i: ......... ::l. . . . . . . . .
100
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i:........ i:
"i . . . . . . . . . .
250
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641
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100
:
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.
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150
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200
250
300
350
200
2~
300
350
Particle o¢e size
20 15 35
50 % solids
100
150
50
100
150
Minutes
Fig. 10
Section 1 variables under regulatory and improved optimising control upon mineral hardness disturbances.
o.2 oA5
C(m~rlr~e
~'ade
[
j
',
J
0,
:,
2.5
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.......
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i
i
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:
'
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i
i
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50
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:
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5oo0
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i .
i ............
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i
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~'.... 350
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Fig. 1 1
Grades and e c o n o m i c indicators under regulatory and improved optimising control upon changes in mineral hardness.
642
C. Mufiozand A. Cipriano TABLE 1 Profits evaluation (S/hr.)
PR- 1 PR-2
Ideal 2,550 2,500
Regulatory control Optimising control 2,262 2,494 2,187 . --
Improved optimising control 2,530 2,477
CONCLUSIONS This work evaluates the application of an optimising economic control technique on a simulated mineral grinding-flotation plant. It has exhibited good performance in keeping economic variables at higher levels than those obtained using only regulatory control. This is due to compensation in changes to the profits with greater regulated variable flexibility while maintaining the variables within operating ranges as defined by technical constraints. The strategy employs predictive control principles based on linear models for the regulation and non-linear models for the optimisation. The objective function used for the regulatory control of each grinding section is a predictive controller standard-type that weights error minimisation between the outputs and the references, as in control energy. The optimising control strategy only considering economic aspects exhibits improved indicators, but generates erratic responses. In order to solve this problem further restrictions were added to optimisation variable changes which smooth the evolution of the variables greatly, but without significantly affecting the economic indicators. These positive results are due to the fact that the optimising control strategy maintains the feed at its greatest value by using the water flowrate to the holding sump to compensate for disturbances in mineral hardness. Only when this is no longer possible is the feedrate reduced in order to avoid infringing the operating constraints.
ACKNOWLEDGEMENTS The authors are grateful for FONDECYT grant 1960394 and Mr. Alex Crawford for the English translation of this paper.
REFERENCES .
2. 3.
.
5.
.
Hulbert, D. et al., Multivariable Control of an Industrial Grinding Circuit. In Proceedings of the 3rd IFAC Symposium on Automation in Mining, Minerals and Metal Processing, 1980, 311-322. Hodouin, D., Najim K., Adaptive Control in Mineral Processing. In Canadian Mineral Processing Division of CIM, 1985, 85, N ° 965, 70-78. Gonzfilez, G., Barrera R., Cartes J., Larenas, G., Cipriano A. Feddersen A. and Zamora C., A Multivariable Dynamic Suboptimal Control Strategy for Mineral Processes with Uncertainties in Model and Inputs. In Proceedings of the 6th IFAC Symposium on Automation in Mining, Minerals and Metal Processing, 1989, 78-83. Manrique, R., Fuzzy Expert Control of Flotation Plants. MSc. thesis, Catholic University of Chile, 1996. Zavala, E., P~rez, R., Mufioz, C. and Cipriano, A., Heuristic and Model Predictive Control Strategies for a Simulated Flotation Circuit. In Proceedings of the 8th IFAC International Symposium on Automation in Mining, Minerals and Metal Processing, 1995, 59--65. Cipriano A. and Manrique, R., Comparison by Simulation of Fuzzy Control Strategies for Mineral Flotation Plants. In Proceedings of the 7 ° Latinoamerican Congress of Automatic Control, 1996, 25-31.
Supervisionand economicoptimalcontrolof mineralprocessingplants .
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.
10. 11. 12. 13. 14. 15.
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Cipriano, A. and Mufioz, C., (1995). A Dynamic Low Cost Simulator for Grinding-flotation Plants. Proceedings of the 4th IFA C International Symposium on Low Cost Automation, LCA'95, Buenos Aires, September 13-15, 294-299. P&ez-Cotrea, R., Gonz~lez, G., Casali, A., Cipriano, A., Barrera, R. and Zavala, E., (1998). Dynamic Modelling and Advanced Multivariable Control of Conventional Flotation Circuits. Minerals Engineering, 11(4), 333-346. Domic, T., Technical-economic Optimisation of a Mineral Flotation Plant. In Minerales, 1980, 35(151), 13-22. Wills, B., (1988). Mineral Processing Technology, 1988, Pergamon Press. Bordons, C., Generalized Predictive Control and Applications. Ph.D. thesis, University of Sevilla, 1994. Camacho E. and Bordons, C., Model Predictive Control in Process Industry, 1995, SpringerVerlag. Morari,/vI. and Ricker, N., Model Predictive Control Toolbox for Use with MATLAB, 1994, The Mathworks, Inc. Haber, R. and Unbehauen, H., StructureIdentification of Non-linear Dynamic Systems--a Survey on InpuffOutput Approaches. In Automatica, 1990, 26(4), 651-678. Isermann, R., Digital Control Systems, 1981, Springer-Verlag.
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Pergamon 0892--6875(99)00048-5
© 1999 Elsevier Science Ltd All fights reserved 0892-6875/99/$ - see front matter
AN INTEGRATED SYSTEM FOR SUPERVISION AND ECONOMIC OPTIMAL CONTROL OF MINERAL PROCESSING PLANTS
C. MUI~IOZ and A. CIPRIANO Faculty of Engineering, Catholic University of Chile, P.O. Box 306, Santiago 22, Chile E-mail: [email protected] (Received 1 August 1997; accepted 21 January 1999)
ABSTRACT
This work tackles the problem of dynamically optimising the performance of a mineral concentration plant, taking into account economic profits and technical constraints. The paper proposes a two-level control strategy with regulatory control and the optimisation of an objective function. The regulatory control employs linear model based multivariable predictive controllers with constraints on controlled and manipulated variables, while the optimiser maximises the economic profits using non.linear dynamic models and linear constraints. The results, drawn from simulations, show the proposed strategy to lead to a significant improvement in economic profits when compared against an exclusively regulatory strategy. © 1999 Elsevier Science Ltd. All rights reserved. Keywords Grinding; froth flotation; process optimisation; process control; simulation
INTRODUCTION Control of grinding and flotation plants has been widely studied both through simulation and, directly, through plant experiments with regard to the complexity of the process. Proposals for grinding control strategies include, for example, PID regulators, multivariable control systems [1], adaptive control [2], and optimal control [3]. When looking at flotation strategies, the application of expert control [4], multivariable predictive control [5] and fuzzy control [6] stand out. Works tackling either the grinding or flotation optimisation, have generally considered technical criteria, such as maximising mineral circulation flow, maximising mineral output, minimising product particle size or operating a flotation plant at minimum tailings given a minimum ore concentrate. This work proposes the use of a combination of technical and economical criteria in order to determine the most effective control. Initially, there is a description of the grinding-flotation plant, upon which the study is developed. This is followed by the presentation of the dynamic simulator about which the new strategy will be evaluated, l.~en, expressions for costs, incomes and profits will be outlined, as well as the technical constraints on different operating variables. After that, the optimal control problem is addressed, and a two-
Presented at Minerals Engineering '97, Santiago, Chile, J u l y - A u g u s t 1997
627
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c. Mufiozand A. Cipriano
level control strategy is proposed. The first is a multivariable predictive regulator, while the second level is for optimisation, employing a Hammerstein model based optimiser which supplies the predictive controller's set-points. Finally, the test results are given both for regulatory and optimising control under typical ore concentrate plant disturbances.
T H E PLANT AND ITS SIMULATION Figure 1 shows a concentration plant with three grinding sections and a single flotation section. The ore, fresh from a storage pile, is water treated in the mill forming a slurry. The discharge from each mill is stored in a sump, to which water is added with the slurry then pumped on to a battery of classifiers. A fraction of the classifier discharge is fed back to the mill while the remainder continues on its path to the flotation plant, where the high copper content particles are separated from the rest. The basic grinding control consists of: a) fresh ore PI control manipulating three belt speed, b) water flow PI control manipulating the aperture of the valves, and c) PI control of the sump pump.
GrindingSection 1 Rougher Flotation and Concentrate
Crushing ore
ore
Grinding Section 2
FirstCleanning Tails
ore
Grinding Sectio~ 3
Fig. 1 Diagram of the concentration plant. The phenomenological model for each grinding section consists of: the mill model, the sump model and the pump-cyclone circuit model [7]. The mill model comprises ten non-linear differential equations, providing mass balances for each class of particle size. The sump model is a mass balance for class of ore and a general balance for the flow of stored pulp. Finally, the pump--cyclone circuit model is based on the relationship developed by Lynch and Rao. The flotation section includes both rougher and first cleaning flotation each of which is made up of a series of cells. The basic flotation control consists of PI controls of the level and ratio control of reactives respect to the total fresh ore feed. The flotation process is modelled using flow balances and relationships that consider two phases (slurry and foam) and two classes of ore (rich and poor) [8]. The main relationships are mass balance equations, by ore type, for each of the phases that provide five differential equations per cell. As the plant only consists of two cells, the dynamic flotation model comprises ten non-linear differential equations. The concentrator plant simulator also includes, for each grinding section, an ore feed controller, a water/ore ratio controller at the mill feed, a water flowrate controller to the sump and a sump level controller determining the pump velocity. It also considers local cell level and flotation reactant flow controllers. Level references will be fixed while the references of reactant flows shall be proportional to the amount of o r e processed, The simulator was programmed in MATLAB/SIMULINK, using a third-order Runge Kutta integration routine.
Supervisionand economicoptimalcontrolof mineralprocessingplants
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OBJECTIVE FUNCTION The objective function, F, represents the profits, in US$/hr, of a grinding-flotation plant and can be formulated as [9]: F = I - %,i. - %,,-
with I the fraction of the revenues from the sale of fine copper produced, where fine copper refers to the amount of copper, in ton/hr, present in the concentrate ore flow of the flotation plant, that will be finally marketed, Cgrin the variable grinding costs, Cltot the variable notation costs, and Cfzx the plant's fixed costs. Income during the time span (to, ty), is given by: t:
1=
fKPycF/~(t+x:) at
(2)
to
where K is a factor indicating the percentage of revenues drawn from concentration of the ore, Pfc is the price per tonne of fine copper, Ffc is the fine copper in the concentrate ore flow, and xf is the mineral processing delay in the grinding-flotation plant. The fine copper flow, Ffc, is expressed as a function of the recovery in smelting, Rsme, the feed grade gf and the tailing grade gt [10]: = R
(3)
in which T(t) represents the fresh feed rate to the grinding, in ton/hr, and, Xg is the mineral processing delay in the grinding plant. Therefore, given a constant feed grade, income takes the form: If
(4) t0
where
f~ = KPIcR,,.,gl
(5)
~2 = g-PfcRsme
(6)
Only the costs associated with the steel and the energy consumed are considered in regard to the grinding costs. It is also assumed, according with industrial plant operation experience, that the variable consumption of steel is proportional to the energy used. Therefore, tf
Csri. = ~TW(t + "~g)dt
(7)
Io
where
= g, Rs,]+ + e .
(8)
with Pst the cost per tonne of steel used, Rst/w the rate of steel used per unit of energy consumed by the grinding plant, Pw the cost of electricity per kW and W the power demand of the mills. The costs due to the steel corrosion, that are independent of the energy consumption, are considered in the fixed costs, Cfvc, of equation (1).
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In determining an expression for the variable cost of flotation, only the most significant cost is taken into account, that is that of the reactants: the lime, the collector, and the foaming agent. These reactants are administered according to feed weight, and, as such the flotation cost is given by: If
C$o, = ~dpZ(t) at
(9)
where to
(lo)
= P,,.R,,. + Poo,Rco, +
with Plim, Pcol and Pfoam the cost of the reactants (lime, collector, and foaming agent) and Rlira, RcoI and efoam their respective addition rates. On using (4), (7) and (9) it is possible to evaluate (1), starting with the parameters given and the following variables: feed weight, tailing grade and the power demand. Finally, transforming the objective function in terms of the discrete time variable k and expressing it in its predictive form, the function that should be maximised by the economic optimal control strategy is achieved:
(11)
F=~" k=l
~,-d~-~2~,(k o + k + K / ) r(k o + k ) - y~.,~(k o +k+Ks) -C),, i=1
where ~, y wi are the predictions of the tailing grade and the power consumed in each of the grinding sections, k0 is the current time, ~Cgand Kf are the mineral processing times Xg and xf expressed in terms of the number of samples, Nf is the prediction horizon and N s is the number of grinding sections.
OPERATING CONSTRAINTS The operating constraints are limits of the measured and manipulated variables that are placed to ensure the non-occurrence of undesired effects. In a plant such as the one under study, the group of operating constraints having to be considered include restrictions per grinding section and also global restrictions for both the grinding and flotation plant. These are specified as follows: Constraint R 1. Lower limits for grinding power demand, to avoid overfilling, and upper limits to avoid motor overload. Constraint R 2. Maximum and minimum limits on the sump, to avoid spillage or blockage of the pipes to the classifier. Constraint R 3. Upper limits on the percentage of pulp solids in the classifier feed to avoid blockages. Constraint R 4. Upper limit on the particle size of the grinding's product, measured as a percentage of particles over 100 mesh, in order to avoid classifier blockage. Constraint R 5. Upper and lower limits on each section's feed rate, due to the operating range of the feeders. Constraint R 6. Upper and lower limits on the rate of addition of water to the sump due to pipe limitations. The six constraints are crucial as the infringement of any of the conditions would lead to a sequence of operations that would significantly reduce the plant performance. These constraints should be considered for Nf samples, as follows.
Supervision and economic optimal control of mineral processing plants
631
Constraint RI:
1
Wlupp <- ~.pp [,
k=-l,2,..,Ns (12)
~,o.j L~Cko + k / ko)J A
where W.(k o +k/ko) is the prediction of the power demand for section i, set at instant kO. Constraint R2:
~,o./ _
k=-l,2,..,N I
(13)
L3,o.j LZ~Cko+k/ko)j LZ~.p.j where
L(k o+k/ko) is the prediction of the sump level for section i, set at instant ko.
Constraint R3:
k=l,2,...,Nf
(14)
D3(ko + k / ko)J LD3.ppj with 19i(k o +k/ko) being the prediction of the percentage of solids of section i, set at instant ko. Constraint R4:
~=,o. <_la~C~o+~/~o)l<_ ~...,
a3,o.] L@k,,+k/ko)j
with
k=l,2,...,Nf
(15)
a~...j
Gi(k o+k/ko) the prediction of the particle size for section i, set at instant kO.
Constraint R5:
k-=l,2,...,Nf
A~,o.j L~3Cko+k/ko)j with Ai(ko+/ak o)
LA3.ppj
the prediction of water flow used in section i, set at instant/%
(16)
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C. Mufioz and A. Cipriano
Constraint R6:
L,,,,,p 1., k0)/ <. IT2up L,ow L Cko+k/ o)J r ,o.
l :ko + /
k=-l,2,...,Nf
(17)
with T(k o +k[ko) the prediction of the feed rate sent to section i, as set at instant ko.. It is necessary to impose the following constraints upon the flotation plant: Constraint R 7. Upper limits on the pulp flows to the flotation plant in order to avoid flotation cell overflow. Constraint R 8. Lower and tipper limit on the percentage of solids to the flotation plant, that enables the plant to operate within a pre-determined density range and avoid sanding problems in flotation cells. Constraint R 9. Lower limit on the concentrate grade arising from an operating requirement of the smelting plant. Constraint RIO. Upper limit on the tailings grade in order to comply with the requirements of the tails treatment plant. In order to satisfy restriction R 7 one needs to restrict the flow of pulp to the flotation section. It is, however, necessary to estimate the flow as the large quantity of bubbles make measurement impossible. Given that the dynamics of the flotation plant are much slower than that of the grinding it is reasonable to estimate this flow Q through the following relationship:
Q(k)
A(k + r~g) + T(k + r.g) 5,,
(18)
where A is the total water added, T is the total tonnage of mineral fed to the grinding and 8m is the density of the mineral. A relationship estimating the percentage of solids in the slurry is determined the same way: S(k) =
T(k + Ks) T(k + ~¢g)+ 8wA(k + r.g)
(19)
where 5 w is the density of the water. While the restriction on the percentage of solids in the flotation slurry
S>Slo w is non-linear, it can be transformed into a linear restriction:
1 -SiowT(k +Ks) > S(k) = 8.,A(k + Ks)
(20)
Using the relationships (18) and (20), constraints R 7 to RI0 are established for each sampling within the interval defined as the current time and the prediction horizon N2 as follows: Constraint R7:
k=l,2,...,N; i=I
(21)
Constraint R8: Ns
1-S,ow _, ,(ko +k/ko) i=l
N.~
>
J,(ko + k / k 0 ) , k=-l,2,..,Nf i=1
(22)
Supervision and economicoptimalcontrolof mineralprocessingplants
633
Constraint R9: ~, ( k 0 + k / k 0) 5.-"g, ,pp, with
k =-1,2,...,Nf
(23)
~i(ko+k/ko) the prediction of the tailings grade set at instant ko.
Constraint R10:
gc,o,, < L(ko + k / ko), k=-l,2,...,N where
(24)
~,c(ko+k/ko) is the prediction of the concentrate grade set at instant k0.
CONTROL STRATEGY It is possible to build several strategies based on the previous analysis which satisfy pre-determined requirements. Two strategies follow that are compared in terms of the amount of computing effort required.
Centralised control strategy As has just been outlined before, in order to optimise the objective function keeping the whole plant operating well it is necessary to impose constraints on certain variables. The most direct way of satisfying these requirements is to design an optimiser subject to the constraints that calculates optimal set-points for the final controllels of each section (tonnage, pump speed, and water). This optimiser, which we call centralised, has to maximise the objective function (11) while satisfying constraints R 1 to R10 under both steady and dynamic conditions. Appropriate proce,;s control requires the strategy to determine a new set of set-points for the local controllers with much shorter responses than the system's most significant time constants [11]. Considering that the power responds to changes in the flow of water to the tank at a time constant of 5 to 7 minutes it would be desirable to take references from the local controllers at least once a minute. It is also important that the optimiser is able to solve the optimal control problem such that the constraints are satisfied throughout the prediction horizon. The prediction horizon should be fixed so that the process operates a great deal of the time in a steady state. As the effect of manipulated variables such as tonnage is only noted at times of over three hours in the flotation section, the prediction horizon should be of at least that order of magnitude. How, therefore, should six specific constraints for each of the grinding sections (R 1 to Rt) and four global plant constraints be satisfied while considering that in this case the circuit contains three grinding sections, bringing the total number of constraints to consider up to 22, which has to multiplied by 180 (3 hours at an optimisation interval of one minute), meaning that the optimal control problem has to analyse 3,960 constrains every iteration. In order to calculate the control horizon we need to bear in mind th~Ltthis has to be the least dynamic component of the process, which in this case is the tailings grade faced with loading changes that take approximately one hour. As the optimiser's sampling time is one minute and there are nine references to determine in the optimisation (tonnage, water and pump speed for each of the three grinding sections) it raises the number of optimisation variables to 540. Nowadays there exist computers available with enough speed rates to achieve this requirement. However, trying to group these tasks in a single algorithm elevates its complexity. Troubleshooting identification turns on a hard task, malting it inappropriate for industrial use. In other hand, the centralization of the all tasks makes the strategy less robustness, and any failure of the algorithm would leave the optmisation control strategy ineffectual.
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C. Mufiozand A. Cipriano
Decentralised control strategy
Figure 2 shows an alternative strategy, which incorporates a regulatory control stage for each grinding section whose objective is to ensure that each section's critical variables are held at values fixed by hierarchical level above, that act upon references provided by local controllers for each of the grinding sections. As these variables need to satisfy both range and change constraints, the best way of implementing the multivariable controller is through the use of predictive control [12].
Economic [ Decentralized ¢ Optimisation
I
I
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IW3' D3*G3* [ Multivariable T
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Fig.2 Hierarchical control strategy. The optimisation stage provides each of the grinding sections with references concerning the amounts of regulation, as shown in Figure 2. This disaggregation is based on the different response times of the grinding and flotation plant as well as differences in sensor sampling times associated with each plant. It is reasonable to assume that, when a change in the input variables is produced, the flotation plant responds more slowly than the grinding sections, in such a way that, while the flotation plant is still responding, the grinding variables have reached its steady state. Therefore, it is natural to set a one minute sampling time for the grinding plant regulators, which is consistent with commonly used data acquisition systems, and an optimisation period of twelve minutes. That is reasonable considering the times required for the analysis of the flotation grade. The objective of the upper hierarchical level is to maximise the objective function, but without allowing the infringement of the plant's global constraints (constraints R 7 to RIO). As the control horizon is one hour and the prediction horizon is three hours upon considering an updating time of twelve minutes, the optimal control problem takes on 285 constraints and 30 optimisation variables, and demands significantly less computing effort than the centralised system. Regulation in the decentralised control strategy
As the pump speed directly affects the sump level, while its effect on other significant variables is only transitory, the use of a PI regulatory controller is recommended for the pump speed to control the sump level. This controller contributes to the avoidance of R 2 constraint infringements since its manipulated variable, the pump speed, generally exhibits a great degree of variability. The remaining significant variables, power, percentage of solids and particle size need to be controlled in order to satisfy operating
Supervision and economicoptimalcontrolof mineralprocessingplants
635
restrictions through the manipulation of the tonnage and sump water references. Here, operating constraints R I, R 3 to R 6 are expressed as: Range and change constraints on the manipulated variables; tonnage and sump water. Range constraints on the controlled variables; power, particle size and percentage of solids. Within the framework of the imposed restrictions the grinding process exhibits a near linear behaviour and a reduced sensitivity to disturbances. This makes it sufficient, for regulation purposes, to use linear models that are robust against disturbances, and is the reasing behind proposing the algorithm used which also includes possibilities for parameter adaptation. MPCT, a set of routines for developing predictive control strategies in the MATLAB environment [13], was used for testing arLd evaluating regulatory control via simulation. As MPCT does not determine optimal control in situations where regulation of the problem presents no feasible solution, the method is complemented with a set of rules that gradually, relax the constraints upon the controlled variables in a hierarchical manner. The highest priority is assigned to the power component as oversupply or mill overfilling are critical events. A lower priority is assigned to the particle size, while the lowest is assigned to the percentage of solids.
Decentralised control strategy optimisation As we mentioned before, the response time of the grinding plant is of the same order of magnitude that the flotation sampling period, which has been taken as the time interval for the optimal control iteration. Therefore, for purposes of optimisation, in each new optimization sample, the grinding plant can be considered to be in a steady state. This means that the optimising control only requires steady power, percentage of solids and particle size models for each of the grinding sections. For the evaluation of the objective and the dynamic monitoring of the flotation constraints, R 9 and RIO, dynamic tailings and concentrate grades models should be used. As these are non-linear relationships, it is proposed to use Hammerstein models [14] that include both a non-linear static component and a linear dynamic component in identifying the flotation relationships.
Models The linear multiv~riable dynamic model, of each mill, relates mill power consumption, W, percentage of solids, D, and size ore distributon, G, with fresch feed ore, T, and surnp water adition, A. The model is formulated as:
-zX ( k )AD,(k) AG,(k)
= M...,(z
_, ,[ A T j ( k - 1 ) )[A,4,(k-
-1
1)/_]
(25)
where matrix Mmi includes transfer function components in Z-1. For the tailings and concentrate grades multivarible, non-linear Harmnerstein-type models are used. The models relates the grades gc and g~ with the total tonnage, T, and a static non-linear function of the average particle size of product, v. The model has the following structure:
Ck)l rB,,,cz-,) B,,oCz-l)]rrCk-U
BoGCz-')JLv(k-1)]
(26)
in which Agt, Agc, Bt,T, Bt,G, Be.l; Be,G are polynomials of z-~, and v is given by:
v(k) = f (G(k))
(27)
636
C. Mufioz and A. Cipriano
The total tonnage T and the average particle size of product G are: Ns
Ns
T(k)=ZT~(k) and G(k)= ZG'(k)Tj(k) i=l i-~l
Predictors at K steps
(28)
Ns
~ Tj(k) i=1
Expressions need to be developed for the predictions for power, percentage of solids and the flow of water consumed in each of the grinding sections and for the tailings and concentrate grades in the flotation plant. It is then possible to evaluate the objective function and constraints predictively, and later formulate the optimal control algorithm. The former expressions are derived from equations (25) to (28).
The optimisation problem We can formulate the objective function (11), starting from the predictions for power and the tailings grade, in terms of the optimisation variables, which are the future particle size and tonnage reference values in each of the sections. The numeric solution of the optimisation problem (11) with constraints also requires the consideration of constraints R 1 and R 3 to R10 in terms of the optimisation variables.
Calculation of the mill regulator references Once the optimal controls {G i, Ti} have been determined, it is necessary to obtain the new power and percentage of solids references, using the static grinding models. These references remain constant during the optimisation interval.
Optimisation algorithm It is possible to construct a flow diagram of the optimisation algorithm (Figure 3), from the previous analysis, essentially revealing four tasks: 1. 2. 3. 4.
Obtain future tonnage and particle size values that optimise the objective function (11) while taking the constraints into account. With the particle size and tonnage values calculated at instant k+l, find the corresponding regulatory control set-points using the static grinding models. The values obtained are applied as the references for the grinding's multivariable regulators. Wait until time period Atopt has elapsed before undertaking the next optimisation.
Solution of the optimisation problem AG*,AT* =arg(max {F(AG,AT) }) subject to RI and R3 to RI0
I
AG*,AT*
Calculation of the optimal references W*, D* and G , using the grinding static models ,
~ W*, D* y G* Transfer of IW', D* and G* as references for regulatory control
÷ Waiting At~wtime required for an optimisation cycle
I Fig.3 Flow chart of the optimiser.
Supervision and economic optimal control of mineral processing plants
6 3 7
SIMULATION RUNS System response without disturbances Figures 4 to 7 show the evolution of the main variables and economic indicators for the plant over time (PR-1 test). Initially, the plant is operated using a regulatory control strategy for 190 minutes, at which point the optimising control strategy is activated. 250
Fresh ore feed
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.... ~ .........
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50 100 Partk:le ore size
150
200
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3(~
350
50 % sc4ids
100
150
200
250
300
350
100
150
200
250
300
350
I
I
50
I
Minutes
Fig.4 Section 1 variables under regulatory and optimising control.
25O
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2OO 150 100
50 25O
i~,
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,
.
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.
.
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.
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.
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100
150
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3O0
350
100
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Mil power
.
.
.
1800
1600
Partide ore size
2O
15 35
~
ids
3O 50
M~es
Fig.5 Section 2 variables under regulatory and optimising control.
638
C. Muflozand A. Cipriano 250
Fresh ore feed
50 100 sump water
150
200
200
300
350
250
00
100
100
200
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300
350
~
':
!
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[ 1600
25
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.........
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......
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00 I00 Particle oce size
100
i
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a~
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Fig.6 Section 3 variables under regulatory and optimising control. Concentrate ~j~de 02
i
i
o. . . . - x I[]50L Z5
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3
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i
i.
i..........
i .
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150
200
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Fig.7 Grades and economic indicators under regulatory and optimising control. In the three grinding sections, the strategy holds the controlled variables (power, particle size and percentage of solids) at around the pre-determined reference values, see Figures 4 to 6, for which both the tonnage and the water flowrate for each section are modified independently. Due to this, the economic profits also presents variabilities (see Figure 6). No constraints are activated during the test while both the manipulated and controlled variables exhibit very little variation.
Supervision and economicoptimalcontrolof mineralprocessingplants
639
From the moment the optimiser is activated, at minute 190, all of the process's variables produced high variability, although none exceeds the constraints. Figures 4 to 6 show the optimal control altering the tonnage in each section so that one of the tonnages is always at its maximum permitted value, another at an intermediate point with the third at a lower value. A similar situation occurrs with both the particle size and the percentage of solids. This all produces significant changes in the dynamic of the plant, however, as average particle size values and values for the total power and tonnage are maintained the plant's grades appears almost unaltered. The enormous ch~mges in all of the process variables upon activating the optimal control strategy has a period equal to that of the optimisation time. This is due to the fact that the objective function does not have a term weighting the control energy spent.explicitly incorporated, resulting in the plant responding in a very similar fashion to a "dead-beat" type regulator [15]. A term in the objective function, representing the control energy, could be incorporated to improve plant response. This, however, would significantly undermine the objective function with it thereby losing its "economic profits" connotation. An alternative is to impose additional constraints on the variation of optimisation variables, forcing the controller to perform slower optimisation, solving the problem of extreme variability. The additional constrains fix the solution to a one local optimum, avoiding jumps between diferent solutions. This method is called "improved optimising control". Figures 8 and 9 show the results obtained from section 1, the respective grades and economic indicators when xunning test PR-1, and considering improved optimising control. It is possible to see the appreciable reduction in the variation, in the Figures, providing a better response while achieving similar costs and profits to those obtained with the original version of the optimiser.
250
Fresh ore feed
! 20O
!
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!
!
~-~• ..........:i......... -:--~-~--~ - -: : : :
!
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......... :: .... '.
'50 .... i ......... i ........ i ......... i ......... i .......... i.......... i .... lOO 250
i I 50 100 Sump water
2® '5011~ 10o
25
......
i 150
i 200
~i......... ~i ......... ~i
i 250
I 300
i , 350
i ..... i:-- i...i
50 100 Mill power
150
200
250
300
350
50 11111 Particle ore size
150
200
250
300
350
50 % solids
100
150
200
250
300
350
511
100
150
200
250
300
350
15 35
Minutes
Fig.8 Section 1 variables under regulatory and improved optimising control.
System response with disturbances Test PR-2 is developed in order to prove that the strategy works well under disturbances. This test includes the following events:
640
c. Mufiozand A. Cipriano The regulatory control strategy is activated after 10 minutes. After 70 minutes the value of parameter Q is changed from 105 to 95, indicating that the mineral is harder. After 130 minutes the value of parameter Q is readjusted from 95 to 105. The optimising control strategy effects after 190 minutes. After 250 minutes the value of parameter Q is changed from 105 to 95 After 310 minutes the value of parameter Q is readjusted from 95 to 105 The test ends after 370 minutes had elapsed.
In this description Q is a parameter of the mill model representing the hardness characteristic of the mineral [7].
Concentrate grade
02 o.ls
i
i
;
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x 1CRY" 2.5
1.52
450
5500
i . . . . .
100
:
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i . . . . . J.......... i.......... j... "
:
200
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300
350
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ti
. ;. . . . . . . . 50 100 Grinding costs
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i
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i i
i
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300
350
i
;
i
250
300
350
250
300
350
250
300
Income=
450O 40OO 3000
1500
I
I 200 Minutes
Fig.9 Grades and economic indicators under regulatory and improved optimising control. Figure 10 shows the evolution of the main variables of section 1 as the test developed, while Figure 11 shows the grades and economic indicators over the same time period. As soon as the disturbance is issued, generated by the increase in mineral hardness, the levels fell, although by a lesser amount than that without using optimal control. While regulatory control responds to disturbances by reducing the feed to the grinding sections and increasing the water supply in order to maintain the particle size constant, the optimising control strategy reduces the total tonnage processed very little accepting a slightly higher particle size which in turn produces a small impact on the process relationships and an almost insignificant reduction in the profits. The most significant effect of increasing the mineral hardness under optimised control conditions is the increase in average power consumed to almost the maximum level permitted. Table 1 presents the maximum profits under ideal circumstances, calculated under steady state conditions, for both tests PR-1 and PR-2. It also shows data for both tests drawn from regulatory and optimising control strategies. In the PR-1 test the regulatory control strategy only reaches average profits of $ 2,262 per hour against a maximum of $ 2,550 per hour. The profits achieved with the optimising control strategy are much closer to the ideal level, and even more so when considering the alternative that took into account changes in the optimisation variables. Optimising control also exhibits much better behaviour when the plant is submitted to disturbances in mineral hardness (test PR-2).
Supervision and economic optimal control of mineral processing plants
250
Fresh ore feed
~50100[ ' " ' ......... ~ - " : i ":. . . . . . . . . . " 50 250
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641
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Particle o¢e size
20 15 35
50 % solids
100
150
50
100
150
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Fig. 10
Section 1 variables under regulatory and improved optimising control upon mineral hardness disturbances.
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Profits
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0 50
~ . . . . .5. . . . . 0:" . . . . . .0. . . ; .~. . . . . . . .' ~'. . ~. . . . . . . . l(X]
150
'200
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'. . . . . . . . . . 300
~'.... 350
Minutes
Fig. 1 1
Grades and e c o n o m i c indicators under regulatory and improved optimising control upon changes in mineral hardness.
642
C. Mufiozand A. Cipriano TABLE 1 Profits evaluation (S/hr.)
PR- 1 PR-2
Ideal 2,550 2,500
Regulatory control Optimising control 2,262 2,494 2,187 . --
Improved optimising control 2,530 2,477
CONCLUSIONS This work evaluates the application of an optimising economic control technique on a simulated mineral grinding-flotation plant. It has exhibited good performance in keeping economic variables at higher levels than those obtained using only regulatory control. This is due to compensation in changes to the profits with greater regulated variable flexibility while maintaining the variables within operating ranges as defined by technical constraints. The strategy employs predictive control principles based on linear models for the regulation and non-linear models for the optimisation. The objective function used for the regulatory control of each grinding section is a predictive controller standard-type that weights error minimisation between the outputs and the references, as in control energy. The optimising control strategy only considering economic aspects exhibits improved indicators, but generates erratic responses. In order to solve this problem further restrictions were added to optimisation variable changes which smooth the evolution of the variables greatly, but without significantly affecting the economic indicators. These positive results are due to the fact that the optimising control strategy maintains the feed at its greatest value by using the water flowrate to the holding sump to compensate for disturbances in mineral hardness. Only when this is no longer possible is the feedrate reduced in order to avoid infringing the operating constraints.
ACKNOWLEDGEMENTS The authors are grateful for FONDECYT grant 1960394 and Mr. Alex Crawford for the English translation of this paper.
REFERENCES .
2. 3.
.
5.
.
Hulbert, D. et al., Multivariable Control of an Industrial Grinding Circuit. In Proceedings of the 3rd IFAC Symposium on Automation in Mining, Minerals and Metal Processing, 1980, 311-322. Hodouin, D., Najim K., Adaptive Control in Mineral Processing. In Canadian Mineral Processing Division of CIM, 1985, 85, N ° 965, 70-78. Gonzfilez, G., Barrera R., Cartes J., Larenas, G., Cipriano A. Feddersen A. and Zamora C., A Multivariable Dynamic Suboptimal Control Strategy for Mineral Processes with Uncertainties in Model and Inputs. In Proceedings of the 6th IFAC Symposium on Automation in Mining, Minerals and Metal Processing, 1989, 78-83. Manrique, R., Fuzzy Expert Control of Flotation Plants. MSc. thesis, Catholic University of Chile, 1996. Zavala, E., P~rez, R., Mufioz, C. and Cipriano, A., Heuristic and Model Predictive Control Strategies for a Simulated Flotation Circuit. In Proceedings of the 8th IFAC International Symposium on Automation in Mining, Minerals and Metal Processing, 1995, 59--65. Cipriano A. and Manrique, R., Comparison by Simulation of Fuzzy Control Strategies for Mineral Flotation Plants. In Proceedings of the 7 ° Latinoamerican Congress of Automatic Control, 1996, 25-31.
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10. 11. 12. 13. 14. 15.
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Cipriano, A. and Mufioz, C., (1995). A Dynamic Low Cost Simulator for Grinding-flotation Plants. Proceedings of the 4th IFA C International Symposium on Low Cost Automation, LCA'95, Buenos Aires, September 13-15, 294-299. P&ez-Cotrea, R., Gonz~lez, G., Casali, A., Cipriano, A., Barrera, R. and Zavala, E., (1998). Dynamic Modelling and Advanced Multivariable Control of Conventional Flotation Circuits. Minerals Engineering, 11(4), 333-346. Domic, T., Technical-economic Optimisation of a Mineral Flotation Plant. In Minerales, 1980, 35(151), 13-22. Wills, B., (1988). Mineral Processing Technology, 1988, Pergamon Press. Bordons, C., Generalized Predictive Control and Applications. Ph.D. thesis, University of Sevilla, 1994. Camacho E. and Bordons, C., Model Predictive Control in Process Industry, 1995, SpringerVerlag. Morari,/vI. and Ricker, N., Model Predictive Control Toolbox for Use with MATLAB, 1994, The Mathworks, Inc. Haber, R. and Unbehauen, H., StructureIdentification of Non-linear Dynamic Systems--a Survey on InpuffOutput Approaches. In Automatica, 1990, 26(4), 651-678. Isermann, R., Digital Control Systems, 1981, Springer-Verlag.
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