(434g) Parameter Estimation for Real-Time Optimization Under Model Uncertainty and Measurement Noise

The increasing competitiveness of processing industries has prompted the emergence of model-based steady-state optimization to gain an economic edge. Accordingly, real-time optimization (RTO) has emerged as a commonly used method to optimize process systems. While high-fidelity models are typically used for RTO, they generally do not capture the full range of phenomena that a process may exhibit, thus resulting in model uncertainty. This uncertainty is typically addressed through the so-called two-step scheme [1].

The two-step RTO scheme pairs a parameter estimation (PE) step with the economic optimization (RTO) step. Typically, the PE step uses time-averaged steady-state plant measurements and a least-squares formulation to estimate uncertain model parameters. These updated parameters are provided to the RTO such that it can compute plant set points. In turn, the set points are provided to a control layer, which aims to drive the plant towards the RTO-dictated steady state. This set point update process is repeated at set intervals (i.e., once every RTO “period”). The typical two-step RTO approach works well if the measurement noise is relatively low such that the PE problem will result in accurate/precise estimates over time. However, this is not always the case as measurement error owed to instrumentation is unavoidable and can result in set point suboptimality and large variation in the RTO decisions [2]. Through large noisy variations in steady-state measurements, the PE problem can become ill-conditioned, causing the propagation of measurement noise to the parameter estimates [2]. This can have the unwanted effect of producing erroneous set points and putting undue burden on the control layer through frequently varying set points.

In this work, we present a low-variance (lv) PE approach that is suited to noisy measurement environments. The approach proposed herein works in a twofold manner. In the first step, the information content (IC) metric [3] is used to quantify the PE problem’s variability by assessing PE sample standard deviations under different measurement assumptions. The IC for different measurements is acquired by performing “challenger” PE problems and comparing them to “benchmark” PE problems, whereby neither problems’ solution is implemented in the RTO. The challenger problem is a version of the benchmark problem with a measurement omitted; this way the effect of the inclusion of a given measurement can be assessed using IC. Depending on the IC of a given measurement, it can be removed from the PE formulation, and the challenger problem becomes the benchmark problem. This procedure is repeated for various measurements such that the best subset is chosen for the actual PE problem (i.e., the one given to the RTO and implemented in the plant). In the second step, parameter bounds are introduced to filter poor estimates and avoid large suboptimalities in the RTO. These bounds are computed using the sample standard deviations and updated upon changes in operating conditions; the computation of these bounds is enabled by parameter data collected during the IC procedure. The standard deviation of the best benchmark problem is used as a parameter filter; henceforth, estimates outside those bounds are not provided to the plant. Together, these two novelties result in more precise and accurate RTO decisions when compared to the standard two-step RTO scheme.

A detailed algorithm describing the procedure outlined above is presented in this work. The benefit of this scheme with respect to previously proposed approaches [2,4] is practicality, as it does not require additional decision layers (e.g., state estimation or sensitivity estimation) and is relatively straightforward to implement (i.e., uses the existing process layers in a novel way). The lv-PE scheme has been tested against the traditional PE scheme in the Williams-Otto plant [5] and a post-combustion carbon capture plant [6], which have significantly different timescales and model sizes. The benefits of the scheme manifest in lower process costs/higher profits over time, including up to ~90% improvement in some cases. This study shows that the proposed scheme is both beneficial (in terms of cost improvements) and practical (in terms of implementation), warranting further investigation of its application on industrial systems.

References

[1] Darby ML, Nikolaou M, Jones J, Nicholson D. RTO: An overview and assessment of current practice. J. Process Control. 2011; 21(6): 874–884. https://doi.org/10.1016/j.jprocont.2011.03.009.

[2] Bhat SA, Saraf DN. Steady-State Identification, Gross Error Detection, and Data Reconciliation for Industrial Process Units. Ind. Eng. Chem. Res. 2004; 43: 4323–4336. https://doi.org/10.1021/ie030563u.

[3] Vrugt JA, Bouten W, Weerts AH. Information Content of Data for Identifying Soil Hydraulic Parameters from Outflow Experiments. Soil Sci. Soc. Am. J. 2001; 65(1): 19–27. https://doi.org/10.2136/sssaj2001.65119x.

[4] Miletic IP, Marlin TE. On-line Statistical Results Analysis in Real-Time Operations Optimization. Ind. Eng. Chem. Res. 1998; 37(9): 3670–3684. https://doi.org/10.1021/ie9707376.

[5] Williams TJ, Otto RE. A generalized chemical process model for the investigation of computer control. IEEE Trans. Commun. 1960; 79(5): 458–473. 10.1109/TCE.1960.6367296.

[6] Patrón GD, Ricardez-Sandoval L. An integrated real-time optimization, control, and estimation scheme for post-combustion CO₂ capture. Appl. Energy. 2022; 308: 118302. https://doi.org/10.1016/j.apenergy.2021.118302.