882 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
which is considered to provide high reduction ratio during
breakage (OCS, 2019). This strategy sets the HPGR being
controlled solely based on torque variations, which is lim-
ited to changes in the operational strategy when dealing
with important fluctuations in feed size distribution and/or
material competence (Campos et al., 2023).
Beyond the main strategies and studies discussed
above, model predictive control algorithms (MPC) applied
to HPGRs have risen as important alternatives. However,
their application has been barely investigated in the lit-
erature. Johansson and Evertsson (2018) proposed to use
a sequence of steady-state process conditions to describe
the time-dependent operation of the HPGR. Among the
main equations they used, some empirical relationships
were presented to predict the roll dynamics and the particle
bed compaction behavior. The model was validated within
a narrow range of operating conditions and demonstrated
relatively good ability to describe the HPGR performance
at an industrial scale. On the other hand, Vyhmeister et
al. (2019) proposed a model predictive control based on
the Torres and Casali model (Torres &Casali, 2009).
They designed and implemented a multivariable control-
ler in an HPGR dynamic model where the MPC consid-
ered the specific energy consumption and the product size
distribution as the two main controlled variables, whereas
roll peripheral velocity and operating pressure as the only
manipulated variables. They partially validated the model
for a pilot-scale HPGR. Nevertheless, although the authors
recognized the well-known relationship between operating
pressure and operating gap, they did not consider the latter
as a manipulated variable and set its value to 24 mm. This
simplified assumption when dealing with the significant
changes in operating pressures creates a bias in the results.
Using a hybrid model with the population balance equation
and information provided by DEM simulations, Herbst et
al. (2011) implemented a control loop for tunning the hopper
level in an HPGR operation in which the roll peripheral veloc-
ity was the only manipulated variable. The simplicity of the
control loop limited the model application and did not make
validation possible (Herbst et al., 2011). Recent works by
the authors applied the Modified Torres and Casali HPGR
model (Campos et al., 2019 2021) as a real-time digital
assistant to optimize an industrial HPGR pressing iron ore
concentrates (Campos et al., 2023a), but use of the approach
in a control loop is still missing. Additionally, despite the
several applications discussed above, the assumption of lin-
ear MPCs and the lack of proper description of the HPGR
dynamics are limiting the application of those models as
proper MPC tools for industrial HPGRs.
The present work then applies the Modified Torres
and Casali model as part of a non-linear model predictive
control (NMPC) algorithm to an industrial HPGR press-
ing iron ore concentrates. The model is assumed as valid
to describe the plant performance and investigation on the
NMPC is demonstrated from proposed case study simula-
tions to emulate a realistic scenario.
MODELING BACKGROUND
Among the main phenomenological mathematical models
that are able to describe the HPGR performance (Morrell
et al., 1997 Torres and Casali, 2009 Dundar et al., 2013),
a modified version of the model proposed by Torres and
Casali (2009), named Modified Torres and Casali model
(Campos et al., 2019 2021), has been successfully applied
by the authors to describe the performance of industrial
HPGRs pressing fine iron ore concentrates consider-
ing both steady-state (Campos et al., 2021) and pseudo-
dynamic (Campos et al., 2023a) approaches. This modeling
approach calculates the HPGR power based on the torque
for both rollers multiplied by the angular roll velocity as:
sina P U 2F 2 m
ip la
=k (1)
where κ is a fitting parameter that allows adjusting the esti-
mate of the nip angle (Campos et al., 2019), U is the roll
peripheral velocity, αip is the calculated nip angle calculated
(Torres and Casali, 2009) and Fm is the compressive force
(Torres and Casali, 2009):
F p D L 2 m m =(2)
where pm is the hydraulic pressure, D is the roll diameter
and L is the roll length.
The HPGR throughput is given as:
Q U L| 100
100
g g g t d =-a k (3)
where Ug is the material velocity, χg is the working gap, ρg is
the flake density and δ is the proportion of material ejected
by the edge of the rolls given by:
lna D Umax
U g {
d |
=-y
x a k k (4)
where Umax is the maximum roll velocity allowed for
the machine and φ, υ and τ are fitting parameters. The
material velocity is estimated as:
U
Ut
g
g g
a c t |
|
=(5)
where
a t is the bulk density,
c |is the critical size given
b0 y cos D^1
c g ip ||la =+-h(Torres and Casali, 2009
Campos et al., 2019).
which is considered to provide high reduction ratio during
breakage (OCS, 2019). This strategy sets the HPGR being
controlled solely based on torque variations, which is lim-
ited to changes in the operational strategy when dealing
with important fluctuations in feed size distribution and/or
material competence (Campos et al., 2023).
Beyond the main strategies and studies discussed
above, model predictive control algorithms (MPC) applied
to HPGRs have risen as important alternatives. However,
their application has been barely investigated in the lit-
erature. Johansson and Evertsson (2018) proposed to use
a sequence of steady-state process conditions to describe
the time-dependent operation of the HPGR. Among the
main equations they used, some empirical relationships
were presented to predict the roll dynamics and the particle
bed compaction behavior. The model was validated within
a narrow range of operating conditions and demonstrated
relatively good ability to describe the HPGR performance
at an industrial scale. On the other hand, Vyhmeister et
al. (2019) proposed a model predictive control based on
the Torres and Casali model (Torres &Casali, 2009).
They designed and implemented a multivariable control-
ler in an HPGR dynamic model where the MPC consid-
ered the specific energy consumption and the product size
distribution as the two main controlled variables, whereas
roll peripheral velocity and operating pressure as the only
manipulated variables. They partially validated the model
for a pilot-scale HPGR. Nevertheless, although the authors
recognized the well-known relationship between operating
pressure and operating gap, they did not consider the latter
as a manipulated variable and set its value to 24 mm. This
simplified assumption when dealing with the significant
changes in operating pressures creates a bias in the results.
Using a hybrid model with the population balance equation
and information provided by DEM simulations, Herbst et
al. (2011) implemented a control loop for tunning the hopper
level in an HPGR operation in which the roll peripheral veloc-
ity was the only manipulated variable. The simplicity of the
control loop limited the model application and did not make
validation possible (Herbst et al., 2011). Recent works by
the authors applied the Modified Torres and Casali HPGR
model (Campos et al., 2019 2021) as a real-time digital
assistant to optimize an industrial HPGR pressing iron ore
concentrates (Campos et al., 2023a), but use of the approach
in a control loop is still missing. Additionally, despite the
several applications discussed above, the assumption of lin-
ear MPCs and the lack of proper description of the HPGR
dynamics are limiting the application of those models as
proper MPC tools for industrial HPGRs.
The present work then applies the Modified Torres
and Casali model as part of a non-linear model predictive
control (NMPC) algorithm to an industrial HPGR press-
ing iron ore concentrates. The model is assumed as valid
to describe the plant performance and investigation on the
NMPC is demonstrated from proposed case study simula-
tions to emulate a realistic scenario.
MODELING BACKGROUND
Among the main phenomenological mathematical models
that are able to describe the HPGR performance (Morrell
et al., 1997 Torres and Casali, 2009 Dundar et al., 2013),
a modified version of the model proposed by Torres and
Casali (2009), named Modified Torres and Casali model
(Campos et al., 2019 2021), has been successfully applied
by the authors to describe the performance of industrial
HPGRs pressing fine iron ore concentrates consider-
ing both steady-state (Campos et al., 2021) and pseudo-
dynamic (Campos et al., 2023a) approaches. This modeling
approach calculates the HPGR power based on the torque
for both rollers multiplied by the angular roll velocity as:
sina P U 2F 2 m
ip la
=k (1)
where κ is a fitting parameter that allows adjusting the esti-
mate of the nip angle (Campos et al., 2019), U is the roll
peripheral velocity, αip is the calculated nip angle calculated
(Torres and Casali, 2009) and Fm is the compressive force
(Torres and Casali, 2009):
F p D L 2 m m =(2)
where pm is the hydraulic pressure, D is the roll diameter
and L is the roll length.
The HPGR throughput is given as:
Q U L| 100
100
g g g t d =-a k (3)
where Ug is the material velocity, χg is the working gap, ρg is
the flake density and δ is the proportion of material ejected
by the edge of the rolls given by:
lna D Umax
U g {
d |
=-y
x a k k (4)
where Umax is the maximum roll velocity allowed for
the machine and φ, υ and τ are fitting parameters. The
material velocity is estimated as:
U
Ut
g
g g
a c t |
|
=(5)
where
a t is the bulk density,
c |is the critical size given
b0 y cos D^1
c g ip ||la =+-h(Torres and Casali, 2009
Campos et al., 2019).