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

- Conference: AIChE Annual Meeting
- Year: 2017
- Proceeding: 2017 AIChE Annual Meeting
- Group: Computing and Systems Technology Division
- Session:
- Time:
Tuesday, October 31, 2017 - 8:00am-8:19am

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.