5.1. Process Description
The Williams–Otto reactor case study [
39] is a benchmark case study widely used to analyze the performance of RTO methods. Reactants A and B react in a CSTR to produce products E and P, and a side product G. While the reaction in the plant follows a three-step reaction mechanism, the nominal model considers a two-step reaction mechanism, ignoring the formation of an intermediate component C, leading to a structural plant-model mismatch. The reaction mechanisms for the plant and in the nominal model are:
The goal is to maximize the profit function
, where
are the measured steady-state concentrations of products
, and
is the summation of the feed flowrates of reactants A and B, i.e.,
.
and the reaction temperature
are the manipulated variables (
) with a range
and
. The steady-state plant model (
) and the nominal model (
) are:
and
where
is the concentration of component
i and
2105 kg is the mass of the reaction mixture in the reactor.
represent the reaction rates in the plant, and in the nominal model. The reaction rates and their kinetic parameters are defined as:
For the purpose of illustration of the performance of the proposed proactive perturbation scheme, we consider that and are and 3 .
5.2. Simulation Results
The modifier adaptation scheme MAWQA with GMA [
26] is used in the simulation study to compare the methods
active perturbation around the current input,
active perturbation around an estimate of the next input, and the proposed
proactive perturbation scheme using the tuning parameters listed in
Table 1. In MAWQA with GMA, the plant gradients are estimated from past inputs and steady-state measurements using QA. All schemes are initialized at
. For the simulation study, the measured plant profit is corrupted by Gaussian noise with zero mean and
standard deviation.
Figure 7 illustrates the evolution of the inputs (scaled) and the plant profit function from
MAWQA with GMA,
MAWQA with GMA with active perturbation schemes APCI and APENI, and
MAWQA with GMA with the proposed proactive perturbation scheme. Descriptions of all markers in the figure are given in
Table 2. The axis for the value of the profit function (
) can be found on the right-side of the plot and the axis for the inputs (scaled) to the plant is on the left. Each vertical grid line in the plot represents the completion of an iteration, which includes all steps from giving input to a process to computing a new input by solving a modifier-adaptation problem. For MAWQA with GMA and MAWQA with GMA with active perturbation schemes APCI and APENI, after applying
([
,
]) at
, two input steps (perturbations [
,
]) with a scaled step length
are performed to compute the plant gradients using finite differences. Each input is applied to the plant until it reaches a steady-state, i.e., for
. The steady-state plant measurement (
) for each applied input are obtained after
from applying an input. Upon computing the plant gradients using the steady-state plant measurements, the modifier adaptation problem in (4) is solved to compute the the input
. As the condition for cardinality is not met, i.e.,
,
additional perturbations ([
,
]) are performed after
([
,
]) to compute the plant gradients. Similar to the 0th iteration, the MA problem in (4) is solved in the
iteration to compute
. From here on, the cardinality condition is always satisfied. Therefore, QA is used for gradient approximation for all further iterations. In MAWQA with GMA, additional input perturbations are made only if the criticality-check criterion is not satisfied, for example in the
iteration after applying
to the plant.
In MAWQA with GMA with active perturbation schemes APCI and APENI, additional perturbations are made once the cardinality condition is satisfied, i.e., from the 2nd iteration onwards, and if there are no past inputs lying closer than to the scaled planned perturbation inputs. In MAWQA with GMA with APCI, the additional perturbations are stopped from the 11th iteration, as from , there are no past inputs lying farther than from all the planned perturbation inputs. In MAWQA with GMA with APENI, the perturbation inputs in kth iteration are chosen around an estimate of iteration input for th iteration. From the iteration onwards, all the planned perturbation inputs are not farther than from earlier inputs, so all active perturbations are stopped.
In the new scheme, MAWQA with GMA with proactive perturbation, after applying the input
([
,
]) to the plant, as
and
, two input perturbations ([
,
])
are performed to compute the plant gradients using finite differences. All necessary steady-state measurements for gradient correction, i.e.,
and two additional perturbations
are available after
24
. During the waiting period, i.e., from
9
to
24
,
additional input perturbations
are performed to gain additional plant information. As the input perturbations when
are considered as inputs from MA (according to the flowsheet in
Figure 6), among the 5 additional perturbations,
are renamed as
. At 24
, steady-state measurements for
and
are available. Therefore, an MA problem is formulated and solved to compute the next input
. For the next iteration, as
and
, FD is no more used for gradient approximation. However, there are not enough measurements for QA, i.e.,
, after applying
. As
, input perturbations
and
are added to smoothly switch from using FD for gradient approximation to QA. After
,
and a new steady-state measurement is always available every
thereby overcoming
and
. From here on, a new iteration input is computed every
taking into account the latest measurement information available.
Although all the schemes converged to an input in the neighborhood of the plant optimum, the MAWQA with GMA scheme took the most time, 375
and 20 iterations to converge. The time of convergence is indicated with a red-dashed vertical line in
Figure 7. The MAWQA with GMA, APCI, and APENI gained additional plant information from suboptimal input perturbations, and therefore converged in 303, 249
and
iterations. Finally, MAWQA with GMA and proactive perturbation scheme converged in only 123
with 31 iterations. It gained more information per time from additional perturbations than that of the earlier proposed active perturbation schemes and also by using the measurement information immediately after it is available.
The mean profit function for 100 simulations for all schemes is shown in
Figure 8. The mean and standard deviation of the profit function did not vary significantly upon convergence. The mean and standard deviation of the times of convergence for all modifier adaptation schemes is also shown. In the proactive perturbation scheme, the profit function converges earlier to the plant optimum, followed by the active perturbations schemes APCI, APENI, and the standard modifier adaptation scheme. The mean time of convergence for the MAWQA with GMA if
, shown in green, is longer than the mean time of convergence of the proposed proactive perturbation scheme as the proposed scheme also overcomes
in addition to
. This illustrates that the proposed scheme overcame the effect of time delay
in this case.
The Friedman ranking test [
40] was performed for a ranking comparison of the proposed proactive perturbation scheme with standard MAWQA, and the APCI and APENI schemes based on the time of convergence for 100 realizations of the measurement noise. The Friedman test is a nonparametric statistical test to recognize performance differences among different methods across multiple test samples. For each realization of the measurement noise, the methods are ranked based on their time of convergence. The lowest mean rank for a method implies its consistently superior performance, and vice versa for the highest mean rank. The mean rank of the standard scheme with no additional perturbations is
, for APCI it is
, for APENI
, and the proposed proactive perturbation scheme has a mean rank of
. Thus, the proposed scheme clearly showed a superior performance when compared to that of the other methods.