Energy-based control of a distributed solar collector field (Q1614366)

The dynamics of the oil temperature \(T(t,x)\) along a tube of a distributed solar collector field are described by the energy balance given by the linear first-order PDE \[ A_0 \frac{\partial T(t,x)}{\partial t}+ q(t) \frac{\partial T(t,x)}{\partial x}= \frac{\eta_0 G_0}{c_0 \rho_0} \mu (x) I(t) \] with boundary condition \(T(t,0) = T_{in} (t)\), where \(A_0\) is the tube inner cross-sectional area in \(m^2\), \(q(t)\) is the oil pump volumetric flow rate, \(\eta_0\) is the collector optical efficiency, \(c_0\) is the specific oil heat capacity in \(J/K kg\), \(G_0\) is the collector aperture in \(m\), \(\rho_0\) is the oil mass density in \(kg/m^3\), \(\mu(x)\) is the tube/collector characteristic function of solar exposure, and \(I(t)\) is the solar radiation for \(0 \leq x \leq L\) and \(t \geq 0\) with tube length \(L > 0\) in \(m\). This model represents a simplified version as described in [\textit{Klein} et al. (1974); \textit{A. Orbach}, \textit{C. Rorres} and \textit{R. Fischl}, Automatica 17, 535-539 (1981; Zbl 0474.49007); \textit{L. Carotenuto}, \textit{M. LaCava} and \textit{R. Raiconi} [Int. J. Syst. Sci. 16, 885-900 (1985; Zbl 0568.93053); and \textit{E. Comacho} and \textit{M. Berenguel}, Int. J. Adapt. Control Signal Process. 11, 311-325 (1997; Zbl 0900.93233)]. Here the heat losses and the conductivity of the tube are neglected. The objective is to control the temperature \(T(t,L)\) to its specific setpoint depending on the control input of the oil pump flow rate \(q(t)\) with \( 0 < q_{\min} \leq q(t) \leq q_{\max} \) (\(q_{\min}\) safety limit, \(q_{\max}\) pump capacity limit). For this purpose, define the internal energy \[ U(t)= \int^L_0 c_0 T(t,x) \rho_0 A_0 dx . \] First, results on the convergence of \(U(t)\) and \(T(t,x)\) to \(U^*\) and \(T^*(x)\), respectively, as time \(t\) tends to \(+\infty\) are shown using the Laplace transform of the differential equation for \(U(t)\) and the Lyapunov functional \[ V(t)= \frac{1}{2} \int^L_0 [T(t,x)-T^*(x)]^2 dx . \] Second, the authors discuss implementation issues of the controller. Third, experimental results follow, indicating that the performance of the energy-based controller is similar to that of the best model-based controllers and better than some fine-tuning controllers. Thus, the energy-based controller for the internal energy of the solar power plant provides a conceptually simple, robust and highly performing nonlinear controller whose convergence is mathematically justified.