# Identification and control of non-linear solar collector plants with varying delay

Solar power is becoming increasingly important due to the increasing problems with greenhouse gases in the atmosphere. Solar energy can be collected in many different ways including

- solar cells, directly converting solar radiation to electricity.
- fluid filled solar collectors, where the solar radiation heats the energy transporting fluid (e.g. water).
- mirrors, where the solar radiation is focused on a storage tank, e.g. for generation of steam.

The present project is focused on identification and control of a solar collector plant that heats water, to be used for cooling or heating of buildings. A special purpose is to identify non-linear models of the plant, to use the so obtained models for control and to evaluate gains. The project is performed within the EC HYCON project, contract number FP6 - IST - 511368.

The solar air conditioning plant is located in Seville, Spain, and is used to acclimate the Laboratories of the System Engineering and Automation Department of the University of Seville. The plant can operate for cooling in the summer and heating in winter. It consists of a solar field that produces hot water which feeds an absorption machine that generates chilled water and injects it into the air conditioning system, achieving a cooling power of 35 kW, in the refrigeration case. In the heating case the hot water is fed directly to the air distribution system.

To provide the necessary models, algorithms from a parallel project at the department (Recursive identification of non-linear systems using ordinary differential equation models) have been successfully applied to identify a nonlinear model of the solar collector part of the plant. The input signal is the flow of water through the solar collectors and the output is the water temperature at the outlet of the solar collectors. The two measurable disturbances are the solar radiation and the water temperature at the inlet of the solar collectors. The plant has two severe complications.

- The transport delay associated with the water flow result in very long time delays affecting the input signal and the measurable disturbances. The delays are different.
- The transport delays are time varying and heavily dependent on the flow.

New delay compensation schemes have been developed for the above reasons. After compensation the signals can be successfully run through the identification algorithms of the parallel project. One result is displayed in Figure 1. As can be seen the identified non-linear model is far more accurate than the linear one. The mean square error of the non-linear model is a factor of 10 (!) less than the mean square error of the best linear model found.

*Figure 1: Validation of a non-linear (dashed), and a linear (dash-dotted) model of the output temperature from the solar collectors. The solid line is the measured output.*

The accuracy of the non-linear model is particularly important when feedforward control is to be used. The objective in the project has been to develop a controller for control of the outlet temperature, using the solar radiation and input temperature for feedforward. Flow dependent sampling is used to compensate for the input signal dependence of the time delays. This embeds the controller in an environment where the time delays are approximately constant. A nonlinear optimal feedforward controller was then developed based on the model obtained from identification. The optimal controller penalizes the control error and the derivative of the input signal. This avoids offset and secures accurate control over the widest possible range. The feedforward controller is applied in an MPC re-computational setting. As reported in the paper 1 and the thesis 2, simulations with measured disturbances show that the controller performs well also during partly cloudy weather, this being a particularly difficult scenario. This is displayed in Figure 2, which shows a comparison between the performance of the controller (solid) and cases where constant control is applied. Note that the controller does not include any feedback or constraint handling. The reason is a desire to highlight achievable feedforward gains.

*Figure 2: Feedforward control performance, using measured disturbances and the identified model. Controller performance (solid) is displayed together with constant control cases (dashed). The reader is refered to the paper 1 and the thesis 2 for more results and a more detailed description of the control problem.*

The work with the controller has lead to two additional findings of general interest in MPC based feedforward control.

- The conventional choice of controller time horizon may lead to inaccurate and biased feedforward control/too slow integrating control, when running open loop optimal controllers in a re-computational MPC setting. A cure for this problem has been found and is reported in the thesis 2 and the paper 3.
- A formal result on the equivalence between two formulations of the optimal control problem has been proved for SISO non-linear systems with long time delays. The result implies large reductions of the computational complexity when the open-loop optimal control sequence is found from the Euler-Lagrange equations. Also this result is reported in the paper 3.
- The formal result on the equivalence between two formulations of the optimal control problem has been extended to the MIMO case in the paper 1 and the thesis 2.

### References

1. L. Brus, T. Wigren and D. Zambrano, "MPC of a nonlinear solar collector plant with varying delays", IET J. Contr. Theory and Applications, vol. 4, no. 8, pp. 1421-1435, 2010.

2. L. Brus, "Nonlinear identification and control with solar energy applications," Ph.D. dissertation, Department of Information Technology, Uppsala University, Uppsala, Sweden, April 25, 2008.

3. T. Wigren and L. Brus, "Criteria and time horizon in feedforward MPC for non-linear systems with time delays", Proc. 7th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2007 , Pretoria, South Africa, pp. 782-787, August 22-24, 2007. DOI 10.3182/20070822-3-ZA-2920.00129.