Multivariable optimal control of an industrial nonlinear boiler–turbine unit

Abstract

Performance control of a boiler–turbine unit is of great importance due to demands for the economical operations of power plants and environmental awareness. In this paper, an optimal control strategy is designed to achieve the desired performance of a boiler–turbine unit. A multivariable nonlinear model of a utility boiler–turbine unit is considered. By manipulation of valves position for the fuel, steam and feed-water flows; output variables including the drum pressure, electric output and fluid density (and consequently drum water level) are controlled. Performance measure of the problem is defined such that the control efforts are minimized while the tracking objectives are obtained. In development of the optimal control strategy, the “variation of extremal” approach is used as an effective tool to handle the nonlinear uncertain problems. Tracking performance of the system is investigated and compared for three cases; tracking from a specific operating point to another ‘near’, ‘far’ and ‘farther’ operating point (depending on the distance between the operating points, the qualitative phrases ‘near’, ‘far’ and farther’ are used). According to the results obtained, more control efforts are required for the tracking of farther operating points (generally). Also, it is investigated that the designed optimal controller guarantees the robust performance of the system in the presence of model parametric uncertainties.

Appendix

Using Eq. (6), coefficients of state and co-state influence matrices (${\tilde{\mathbf{P}}}_{{\mathbf{x}}}$ and ${\tilde{\mathbf{P}}}_{{\mathbf{p}}}$) in Eq. (12) are evaluated as:

$$ \begin{aligned} \left[ {\frac{{\partial^{2} {\mathbf{H}}}}{\partial P\partial X}} \right] & = \left[ {\begin{array}{lll} {0.0020\,x_{1}^{{{1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0pt} 8}}} \,\psi_{1} + 0.0018\,x_{1}^{{{9 \mathord{\left/ {\vphantom {9 8}} \right. \kern-0pt} 8}}} \,\psi_{2} } \hfill & 0 \hfill & 0 \hfill \\ { - 0.0821\,x_{1}^{{{1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0pt} 8}}} \,\psi_{1} - 0.073\,x_{1}^{{{9 \mathord{\left/ {\vphantom {9 8}} \right. \kern-0pt} 8}}} \,\psi_{2} - 0.018\,x_{1}^{{{1 \mathord{\left/ {\vphantom {1 8}} \right. \kern-0pt} 8}}} } \hfill & { - 0.1} \hfill & 0 \hfill \\ {0.01294\,(\psi_{1} + x_{1} \,\psi_{2} ) + 0.00223} \hfill & 0 \hfill & 0 \hfill \\ \end{array} } \right] = \left[ {\frac{{\partial^{2} {\mathbf{H}}}}{\partial X\partial P}} \right]^{T} \\ \left[ {\frac{{\partial^{2} {\mathbf{H}}}}{{\partial P^{2} }}} \right] & = - \left[ {\begin{array}{lll} {\frac{0.405}{{r_{1} }} + \frac{{1.62 \times 10^{ - 6} }}{{r_{2} }}x_{1}^{{{9 \mathord{\left/ {\vphantom {9 4}} \right. \kern-0pt} 4}}} + \frac{0.01125}{{r_{3} }}} \hfill & { - \frac{{6.57 \times 10^{ - 5} }}{{r_{2} }}x_{1}^{{{9 \mathord{\left/ {\vphantom {9 4}} \right. \kern-0pt} 4}}} } \hfill & {\frac{{1.16 \times 10^{ - 5} }}{{r_{2} }}x_{1}^{{{{17} \mathord{\left/ {\vphantom {{17} 8}} \right. \kern-0pt} 8}}} - \frac{0.1244}{{r_{3} }}} \hfill \\ {sym.} \hfill & {\frac{0.00266}{{r_{2} }}x_{1}^{{{9 \mathord{\left/ {\vphantom {9 4}} \right. \kern-0pt} 4}}} } \hfill & { - \frac{{4.72 \times 10^{ - 4} }}{{r_{2} }}x_{1}^{{{{17} \mathord{\left/ {\vphantom {{17} 8}} \right. \kern-0pt} 8}}} } \hfill \\ {sym.} \hfill & {sym.} \hfill & {\frac{{8.372 \times 10^{ - 5} }}{{r_{2} }}x_{1}^{2} + \frac{1.376}{{r_{3} }}} \hfill \\ \end{array} } \right]_{\text{symmetric}} \\ \left[ {\frac{{\partial^{2} {\mathbf{H}}}}{{\partial X^{2} }}} \right] & = \left[ {\begin{array}{lll} {2\,(q_{1} - \,r_{2} \psi_{2}^{2} ) + \psi_{1} \,\psi_{3} - 0.00225\,p_{2} \,x_{1}^{{ - {7 \mathord{\left/ {\vphantom {7 8}} \right. \kern-0pt} 8}}} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {2\,q_{2} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {2\,q_{3} } \hfill \\ \end{array} } \right] \\ \end{aligned} $$

(16)

where,

$$ \psi_{3} = \left( {2.531 \times 10^{ - 4} p_{1} - 0.01026p_{2} } \right)x_{1}^{ - 7/78.8} $$

(17)

and ψ ₁, ψ ₂ are given by Eqs. (9) and (10).