(17c) Nonlinear State and Parameter Estimation for Differential-Algebraic Equation Systems with Uncertain Algebraic Equations and Unknown Noise Covariance

Joint state and parameter estimation in operating plants can be very valuable for process monitoring and control. However, many industrial processes are highly nonlinear and they are given by differential-algebraic equation (DAE) systems with both exact and uncertain algebraic equations. Furthermore, noise covariance in the industrial data is unknown. Joint state and parameter estimation of nonlinear DAE systems using the industrial data of unknown noise covariance remains a challenging task.

In the existing literature on the EKF for non-linear DAEs, the algebraic equations are typically considered to be noise-free thus the algebraic equations can be differentiated converting the system to an implicit ODE. However in real life, there are systems that have both exact and uncertain algebraic equations. For such systems, the white noise term in the uncertain algebraic equations cannot be differentiated since the derivative of white noise is not well-defined. A modified extended Kalman Filter (EKF) that can handle uncertainties in both differential and algebraic equations along with exact algebraic equations and uncertain measurement noise characteristics is developed. In this approach, the error covariance matrix is split into separate square blocks for differential and algebraic variables^[1]. By writing the algebraic equations in terms of the differential variables, the updates in the error covariance matrix are computed by the linear non-linear transformation of the error covariance of the differential equations. The algorithm also accounts for unknown covariance of measurement noise for certain algebraic and state variables.

The estimation approach is applied to an industrial superheater/reheater system as part of a natural gas combined cycle plant. Due to very high temperature in these sections, measurements in these systems are rather limited thus an estimator is very useful for this system. For industrial application, computational cost of the estimation framework also needs to be reasonably low. In the past, approximate partial differential equations (PDE) models have been used for superheater state estimation using Kalman Filter and Unscented Kalman Filter^[2][3]. Fouling detection in boiler and reheater using EKF has also been studied using lumped parameter models^[4][5]. However, those techniques cannot be used for systems with uncertain algebraic equations and with unknown covariance of the measurement noise.

A dynamic 2-D first-principles DAE model is developed for the superheater/reheater. In addition to the states, parameters corresponding to the heat transfer coefficients are also estimated using the developed EKF algorithm for DAE systems. The data available from an operating power plant for about a year are used in developing and validating the estimation framework. The framework leads to fast estimation of the key state variables such as the main and reheat steam temperatures and the unmeasured variables such as the tube temperatures.

References

1. Mobed, S. Munusamy, D. Bhattacharyya, and R. Rengaswamy, “State and parameter estimation in distributed constrained systems. 1. Extended Kalman filtering of a special class of differential-algebraic equation systems,” Ind. Eng. Chem. Res., vol. 56, no. 1, pp. 206–215, 2017.
2. L. Lo and Y. Rathamarit, "State estimation of a boiler model using the unscented Kalman filter," in IET Generation, Transmission & Distribution, vol. 2, no. 6, pp. 917-931, November 2008.
3. Sato, M. Nomura, H. Matsumoto, and M. Lioka, “Steam Temperature Prediction Control for Thermal Power Plant,” IEEE Power Eng. Rev., vol. PER-4, no. 9, p. 30, 1984.
4. Prem, K. Ayyagari, R. Singh, P. Purakasthya, and P. Deeskow, “Estimation of soot deposition in the boilers of coal fired power plant,” no. 1, pp. 1–4, 2014.
5. K. Sivathanu and S. Subramanian, “Extended Kalman filter for fouling detection in thermal power plant reheater,” Control Eng. Pract., vol. 73, pp. 91–99, Apr. 2018.