(284a) Combination of Stochastic NMPC/DRTO and Dynamic (re-)Evaluation of the Description of Model Uncertainty in the Optimization of Batch Campaigns

Robust dynamic optimization/adaptive control has been proposed to overcome a significant drawback of
conventional dynamic optimization (DRTO) and optimal control (NMPC), namely,
the requirement for accurate process models. One of the existing types of
robust NMPC/DRTO is stochastic NMPC/DRTO (Rossi et al., 2016). This approach relies
on stochastic optimization techniques and approximates the uncertainty in the
process model using an appropriate set of uncertain parameters (SUP) and their
joint probability distribution function (PDFUP). Although this type of strategy
is applicable to a large variety of systems, we will consider the case of batch
processes.

It is well-known that the
performance of stochastic NMPC/DRTO primarily depends on the selection of the
SUP and on the estimate of the PDFUP. The latter can be updated periodically based
on the measurements acquired in successful batch cycles, using, for instance,
Monte-Carlo approaches or Bayesian frameworks (Mockus
et al., 2015). However, the PDFUP mainly affects the performance of the
stochastic NMPC/DRTO algorithm and not its robustness. The selection of the SUP
is the more crucial and challenging aspect because it may strongly affect both
the robustness and the performance of the stochastic NMPC/DRTO method.
Moreover, in order to meet real time computational constraints, the SUP can
only include a limited number N_p (N_p ≤ 4, 5) of
uncertain parameters.

Typically, the SUP is identified statically using
sensitivity analysis (Saltelli et al., 2005) or based
on empirical knowledge of the batch process. However, the use of a fixed SUP is
often sub-optimal because the importance of any given uncertain parameter can
be expected to vary over time, even over a single batch cycle. This is because
the relevance of the uncertain parameters may depend on both their stochastic
properties (PDFUP) and the operating conditions of the batch process. While the
PDFUP does not significantly change over time in a single batch cycle, the same
does not apply to the operating conditions of the process, which are
continuously modified to compensate for process disturbances and to improve
process performance.

Therefore, we identify the SUP in real-time, using a three-stage,
hierarchical procedure (Figure 1). First, we employ conventional methods based
on sensitivity analysis and/or empirical knowledge of the batch process to
construct a set of N_p^tot
parameters called SUP^tot (N_p^tot ≤ 10, 15) (Phase I). This
set includes those parameters, which have the most impact on the model
predictions over a wide range of operating conditions. Secondly, after every
batch cycle, we employ the algorithm called PDFE&U (Probability
Distribution Function Estimation & Update), presented at the 2016 AIChE Annual Meeting, to re-estimate in parallel the joint
probability distributions of all the subsets of N_p uncertain
parameters (SUPs) included in SUP^tot
(Phase II). In this phase, we also define proper indicators of both the
statistical dispersion of all the uncertain parameters and their correlation. Note
that the application of parallel computing allows us to carry out this phase in
the idle time between two consecutive batch cycles. Finally, after every
control action (specifically, in the idle time between two consecutive actions),
we apply the novel algorithm called RTSASUP (Real-Time Selection of the Active
Set of Uncertain Parameters) to select the optimal SUP (SUP^opt)
(Phase III). This newly estimated set of uncertain parameters is used, in the
stochastic NMPC/DRTO engine, to compute the control action after the next one (in
other words, the SUP^opt identified at the
end of control action n is used to
compute control action n+2).

Figure  SEQ Figure \* ARABIC 1: Architecture of the proposed strategy for
the online identification of the optimal uncertainty set.

Note that RTSASUP chooses SUP^opt
as the set of N_p uncertain parameters that have the most influence
on the predictions of the process model, show the largest statistical
dispersion and exhibit the smallest correlation among them. In order to do this,
the algorithm measures the sensitivity of the process model to the uncertain
parameters by solving a series of decoupled small-scale dynamic optimization problems
in parallel, and then combines this latter information with the indicators of
the statistical dispersion and the correlation of the uncertain parameters,
estimated in Phase II. The use of parallel computing allows RTSASUP to be
efficient enough to run in the idle time between two consecutive control
actions (approximately a minute or two of clock time).

We demonstrate the proposed approach to the online identification
of the optimal SUP (SUP^opt) using a batch
version of the Tennessee Eastman challenge. The case study confirms that the
online re-estimation of the SUP carried out via RTSASUP benefits both the
performance and the robustness insured by stochastic NMPC/DRTO frameworks. Such
efficiency improvements appear to be more pronounced in the presence of process
perturbations, i.e. when the operating conditions of the process experience
larger variations in single batch cycles.

References:

L.
Mockus, J.J. Peterson, J.M. Lainez,
G.V. Reklaitis, 2015, Batch-to-batch variation: a key
component for modeling chemical manufacturing processes, Organic Process
Research & Development, 19, 908-914.

F.
Rossi, G. Reklaitis, F. Manenti, G. Buzzi-Ferraris, 2016, Multi-scenario robust online
optimization and control of fed-batch systems via dynamic model-based scenario
selection, AIChE Journal, 62, 3264-3284.

A. Saltelli, M. Ratto, S. Tarantola, F. Campolongo, 2005, Sensitivity analysis for chemical models,
Chemical Reviews, 105, 2811-2828.