(295f) Development of Mass and Energy Constrained Estimator Algorithms for Process Systems

State and parameter estimation of non-linear dynamic process systems using dynamic models can be very helpful for process monitoring and other applications. Neither the dynamic models nor the measurement data of an operating plant are complete and accurate; however, both of them can be utilized in an optimal quantitative estimation algorithm to yield an adaptive, real-time, and accurate condition monitoring framework. The use of Bayesian estimators based on Kalman filtering techniques like linear KF, extended Kalman filter (EKF), unscented Kalman filter (UKF) or optimization-based state estimators have been extensively studied in literature ^[1]. The models utilized in these studies are based on either data-driven or simple dynamic models based on ODEs or PDEs for the estimation ^[2]. However, these algorithms may provide inaccurate or un-physical results. For example, the estimates may fail to exactly satisfy mass and atom conservation in a reactive system or may fail to exactly satisfy energy conservation in a heat exchanger or boiler. Although, using a physics-based model in the prediction step of an estimator may satisfy these constraints. However, the update step of the estimator incorporating the real-time measurements may lead to violations of mass and energy balances. This issue of constrained state estimation that satisfies mass and energy balance has been an active area of research with a generic structure only available in the literature for imposing the constraints on the estimator ^[3]. In this work we have developed a constrained state estimation approach for use in Kalman and similar filtering algorithms that can exactly satisfy the mass and energy balances while keeping computation time low for online implementation. The approach can not only be applied to systems given by ODEs but also those given by differential-algebraic equations (DAEs).

Most of the work in the area of constrained estimation can be broadly classified into two types- one using Bayesian estimators or their modified forms with recursive and constraint-handling capabilities for online implementation and another based on optimization-based methods^[4]. For linear systems subject to linear or non-linear constraints, modifications to the standard KF have been proposed using ‘’ad hoc” clipping techniques where unconstrained state estimates obtained are projected onto constraint space^[5]. To address the constraint estimation issue in non-linear systems, a recursive optimization-based technique called recursive non-linear dynamic data reconciliation (RNDDR) for non-linear systems governed by ODEs has been utilized ^[6]. A generic constrained optimization problem is proposed in this approach to update the unconstrained state estimate of the EKF update step using equality or inequality constraints. An extension of this RNDRR approach to DAE systems following the same optimization- based methodology handling constraints was also investigated ^[7].However, using a generic optimization framework for handling mass and energy constraints for distributed DAE models can be very computationally expensive. Furthermore, there are several difficulties in straightforward application of mass and energy balances as equality constraints during a posteriori estimate of states. For nonlinear filters, while constraints can be approximately satisfied by methods such as projection-based approaches or measurement augmentation, exact satisfaction will require solving an optimization problem with the DAE model as constraint (i.e., similar to particle filtering approaches) leading to a highly computationally expensive problem. Second, since measurements of spatial accumulation of mass and energy are not available in general, and available information about accumulation depends on the spatial discretization while developing the DAE model, and there is often associated transport delay, a straightforward application of mass and energy balance as an equality constraint is not meaningful.

In this work, we have developed three approaches for satisfying mass and energy balances as well as any other equality constraint that must be satisfied. The first approach modifies computation of posterior error covariance matrix and Kalman gain to satisfy mass and energy balance. The second approach considered a simplified optimization problem supplementing the existing a posteriori update step. This approach will yield a correction to the a posteriori state estimate for exactly satisfying the equality constraints. The third approach is based on post-processing of filter a posterior estimate to exactly satisfy the equality constraints. Performances of the proposed algorithms have been validated by applying them to a reactive Van de Vusse reactor system and to the superheater section of an operating power plant that is given by a 3-D distributed DAE model.

References

[1] J. Zhao et al., “Roles of dynamic state estimation in power system modeling, monitoring and operation,” IEEE Trans. Power Syst., vol. 36, no. 3, pp. 2462–2472, 2021, doi: 10.1109/TPWRS.2020.3028047.

[2] J. Zhao et al., “Power System Dynamic State Estimation: Motivations, Definitions, Methodologies, and Future Work,” IEEE Trans. Power Syst., vol. 34, no. 4, pp. 3188–3198, 2019, doi: 10.1109/TPWRS.2019.2894769.

[3] N. Amor, G. Rasool, and N. C. Bouaynaya, “Constrained State Estimation - A Review,” no. 3, pp. 1–11, 2018, [Online]. Available: http://arxiv.org/abs/1807.03463

[4] S. C. Patwardhan, S. Narasimhan, P. Jagadeesan, B. Gopaluni, and S. L. Shah, “Nonlinear Bayesian state estimation: A review of recent developments,” Control Eng. Pract., vol. 20, no. 10, pp. 933–953, 2012, doi: https://doi.org/10.1016/j.conengprac.2012.04.003.

[5] S. J. Julier and J. J. LaViolar, “On Kalman filtering with nonlinear equality constraints,” IEEE Trans. Signal Process., vol. 55, no. 6 II, pp. 2774–2784, 2007, doi: 10.1109/TSP.2007.893949.

[6] P. Vachhani, R. Rengaswamy, V. Gangwal, and S. Narasimhan, “Recursive estimation in constrained nonlinear dynamical systems,” AIChE J., vol. 51, no. 3, pp. 946–959, 2005, doi: 10.1002/aic.10355.

[7] R. Kumar Mandela, R. Rengaswamy, S. Narasimhan, and L. N. Sridhar, “Recursive state estimation techniques for nonlinear differential algebraic systems,” Chem. Eng. Sci., vol. 65, no. 16, pp. 4548–4556, 2010, doi: 10.1016/j.ces.2010.04.020.