886 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
is used. Additionally, to ensure control actions that respect
the process dynamics and are smooth over the prediction
horizon, Eq. (20) also imposes a penalty for abrupt changes
in manipulated variables when comparing the values oper-
ating pressure at the current time (i) and the previous time
(i–1), with its final form then presented as:
f Q
Q Q
BSA
BSA BSA
U
U U
p
p p
obj
Ref
Ref
Ref
Prod Ref
U
i
i i
p
m
m m
2 2
1
1
2 2
m
i-1
i i-1 C C
=
-
+e
-
+
-
+
-
-
-
e
c d
o
m
o
n
(20)
where
U
C is the penalty coefficient for the roll peripheral
velocity and given by 3×106, while
p
m
C is the penalty coef-
ficient for the operating pressure and given by 20×106.
To define the optimal trajectory for both controlled
and manipulated variables, the NMPC structure applies
a constrained non–linear optimization method named
Sequential Quadratic Programming—SQP (Wilson, 1963)
as the optimization algorithm, in which the gradient and
Hessian of the objective function (Eq. (20)) are calculated
using the centered finite differences method. Application
of the SQP relied on a stipulated tolerance for the objective
function as 10–12, whereas constraints followed the opera-
tional limits of the manipulated variables as presented in
Table 1. Considering the limitation of the HPGR inves-
tigated in the present work when operating at high roller
roll peripheral velocities (Section “Industrial HPGR”), the
maximum value allowed for the machine throughput was
set as 750 t/h.
For tunning the NMPC, an optimal set of values for
the control horizon (CH) and prediction horizon (PH) are
presented in the first section of “Results and discussions”.
Case Study Simulations
Two case study simulations are performed to investigate the
NMPC flexibility when dealing with different control strat-
egies and feed BSA variabilities. Case study #1 proposes a
setpoint of 1,850 cm2/g for the product BSA and 600 t/h
throughput, which are the usual target values required in
operation. A second case study (#2) proposes new setpoints
for the product BSA and throughput as 1,950 cm2/g and
700 t/h, respectively. This second scenario is designed to
understand the NMPC behavior when there is a need to
increase plant production and improve product quality.
Both case studies simulate a time interval of 180 min
with important changes in the HPGR feed BSA over time.
To emulate the progression of the HPGR feed BSA between
two states for these case studies, it is important to recognize
that changes will follow the dynamics imposed by the ball
milling and classification stages located upstream in the cir-
cuit. As such the present work assumes that the progression
of the HPGR feed BSA is described as:
exp^-fth@ BSA BSA BSA
Feed I I =+--^th ^BSAF h61 (21)
where f vis a fitting parameter, and BSAI and BSAF are,
respectively, the initial and final BSA of the feed between
states. Eq. (21) is fitted based on previous results presented
elsewhere from simulations of a ball milling stage (Carvalho
and Tavares, 2009) with optimal value of f as 0.09. Eq. (21)
accounts for changes in the feed BSA from when the initial
level of BSA (BSAI) is either lower or higher than the final
level of BSA (BSAF). Figure 2a then presents the evolution
of the HPGR feed BSA from 1,500 cm2/g to 1,600 cm2/g
(red line) and 1,600 cm2/g to 1,500 cm2/g (blue line) in the
time window of 50 min, which demonstrates that the BSA
of the HPGR feed stabilizes after this period. Figure 2b, on
the other hand, presents the variation of the HPGR feed
BSA in the entire period assessed in both case studies ana-
lyzed, where BSA vary in the range from 1,500 cm2/g to
1,650 cm2/g.
RESULTS AND DISCUSSIONS
Results presented are solely based on predictions made with
the Modified Torres and Casali model (Section “Modeling
background”). Although results from the present work have
not yet been validated with experimental data, validation
of the model that served as the basis for it for pressing iron
ore concentrates in industrial HPGRs using both steady-
state (Campos et al., 2021) and pseudo-dynamic (Campos
et al., 2023b) simulations put this modeling approach in a
reliable position to assess the operation in the present study.
Prediction and Control Horizons
Analysis of the convergence of product BSA (a) and
throughput (b) is presented in Figure 3 over 100 seconds
of operation, comparing different prediction and control
horizons. A careful analysis of the results, when compared
to the process dynamics presented for similar operations
elsewhere (Rodriguez et al., 2022), allows us to conclude
that a prediction horizon (PH) of 15 s and a control hori-
zon (CH) of 5 s proved more suitable for tuning the con-
vergence dynamics of the process. Performance variables,
as shown, reached steady state between 30 and 60 seconds,
with the convergence dynamics of throughput slightly lon-
ger than that of product BSA. Similar convergences for both
product BSA and throughput are reasonable assumptions
given the short prediction horizon adopted. Experimental
results confirming these trends in HPGR dynamics for
pressing iron ore concentrates, however, are still missing.
is used. Additionally, to ensure control actions that respect
the process dynamics and are smooth over the prediction
horizon, Eq. (20) also imposes a penalty for abrupt changes
in manipulated variables when comparing the values oper-
ating pressure at the current time (i) and the previous time
(i–1), with its final form then presented as:
f Q
Q Q
BSA
BSA BSA
U
U U
p
p p
obj
Ref
Ref
Ref
Prod Ref
U
i
i i
p
m
m m
2 2
1
1
2 2
m
i-1
i i-1 C C
=
-
+e
-
+
-
+
-
-
-
e
c d
o
m
o
n
(20)
where
U
C is the penalty coefficient for the roll peripheral
velocity and given by 3×106, while
p
m
C is the penalty coef-
ficient for the operating pressure and given by 20×106.
To define the optimal trajectory for both controlled
and manipulated variables, the NMPC structure applies
a constrained non–linear optimization method named
Sequential Quadratic Programming—SQP (Wilson, 1963)
as the optimization algorithm, in which the gradient and
Hessian of the objective function (Eq. (20)) are calculated
using the centered finite differences method. Application
of the SQP relied on a stipulated tolerance for the objective
function as 10–12, whereas constraints followed the opera-
tional limits of the manipulated variables as presented in
Table 1. Considering the limitation of the HPGR inves-
tigated in the present work when operating at high roller
roll peripheral velocities (Section “Industrial HPGR”), the
maximum value allowed for the machine throughput was
set as 750 t/h.
For tunning the NMPC, an optimal set of values for
the control horizon (CH) and prediction horizon (PH) are
presented in the first section of “Results and discussions”.
Case Study Simulations
Two case study simulations are performed to investigate the
NMPC flexibility when dealing with different control strat-
egies and feed BSA variabilities. Case study #1 proposes a
setpoint of 1,850 cm2/g for the product BSA and 600 t/h
throughput, which are the usual target values required in
operation. A second case study (#2) proposes new setpoints
for the product BSA and throughput as 1,950 cm2/g and
700 t/h, respectively. This second scenario is designed to
understand the NMPC behavior when there is a need to
increase plant production and improve product quality.
Both case studies simulate a time interval of 180 min
with important changes in the HPGR feed BSA over time.
To emulate the progression of the HPGR feed BSA between
two states for these case studies, it is important to recognize
that changes will follow the dynamics imposed by the ball
milling and classification stages located upstream in the cir-
cuit. As such the present work assumes that the progression
of the HPGR feed BSA is described as:
exp^-fth@ BSA BSA BSA
Feed I I =+--^th ^BSAF h61 (21)
where f vis a fitting parameter, and BSAI and BSAF are,
respectively, the initial and final BSA of the feed between
states. Eq. (21) is fitted based on previous results presented
elsewhere from simulations of a ball milling stage (Carvalho
and Tavares, 2009) with optimal value of f as 0.09. Eq. (21)
accounts for changes in the feed BSA from when the initial
level of BSA (BSAI) is either lower or higher than the final
level of BSA (BSAF). Figure 2a then presents the evolution
of the HPGR feed BSA from 1,500 cm2/g to 1,600 cm2/g
(red line) and 1,600 cm2/g to 1,500 cm2/g (blue line) in the
time window of 50 min, which demonstrates that the BSA
of the HPGR feed stabilizes after this period. Figure 2b, on
the other hand, presents the variation of the HPGR feed
BSA in the entire period assessed in both case studies ana-
lyzed, where BSA vary in the range from 1,500 cm2/g to
1,650 cm2/g.
RESULTS AND DISCUSSIONS
Results presented are solely based on predictions made with
the Modified Torres and Casali model (Section “Modeling
background”). Although results from the present work have
not yet been validated with experimental data, validation
of the model that served as the basis for it for pressing iron
ore concentrates in industrial HPGRs using both steady-
state (Campos et al., 2021) and pseudo-dynamic (Campos
et al., 2023b) simulations put this modeling approach in a
reliable position to assess the operation in the present study.
Prediction and Control Horizons
Analysis of the convergence of product BSA (a) and
throughput (b) is presented in Figure 3 over 100 seconds
of operation, comparing different prediction and control
horizons. A careful analysis of the results, when compared
to the process dynamics presented for similar operations
elsewhere (Rodriguez et al., 2022), allows us to conclude
that a prediction horizon (PH) of 15 s and a control hori-
zon (CH) of 5 s proved more suitable for tuning the con-
vergence dynamics of the process. Performance variables,
as shown, reached steady state between 30 and 60 seconds,
with the convergence dynamics of throughput slightly lon-
ger than that of product BSA. Similar convergences for both
product BSA and throughput are reasonable assumptions
given the short prediction horizon adopted. Experimental
results confirming these trends in HPGR dynamics for
pressing iron ore concentrates, however, are still missing.