The main control loop in a distributed collector solar field like the one of Fig. 1.4 aims at driving the temperature of the fluid leaving the collector loops to a desired value, by manipulating its flow. As explained in Sect. 1.2.1 the manipulated variable is the command of the pump/valve system that drives fluid flow. The process output can be either the average of the temperature of the fluid at the output of the collector loops or its maximum. Since in practical terms there is little difference in controlling these two variables, the average fluid temperature is used.
The main disturbances to the system are:
• Changes in incoming solar radiation due to passing clouds and moisture scattered in the atmosphere;
• Changes in the temperature of the fluid at the input of the pipe;
• Changes in ambient temperature;
• Disturbances due to various factors such as dust deposition on the mirrors and wind affecting the shape of the mirrors.
The first three types of disturbances in the above list can be measured and used as feedforward signals. In particular, feedforward from solar radiation is quite important
to improve the performance. The other two disturbances are less important. It should be remarked that the measure of incoming solar radiation may present difficulties due to the large size of the field that may lead to situations in which the collectors are not uniformly illuminated.
Figure 1.14 shows two examples of daily records of solar radiation. Due to the Sun’s apparent movement, the intensity of radiation arriving at a given point on Earth has a sinusoidal evolution with the hour of the day. As is well known, its peak value depends on the period of the year, for details see for example, Tiwari (2002) or any other reference on solar energy. Superimposed on this deterministic variation, there are two main stochastic fading effects. One is associated with moisture scattered in the atmosphere. By absorbing radiation in a random way, it causes fast small amplitude changes of the intensity of the radiation arriving to the collector field. The fluctuations on the record on the left of Fig.1.14 are predominantly due to moisture. The other factor that reduces radiation intensity are passing clouds. The record on the right of Fig. 1.14 shows three groups of clouds passing that drastically reduce the arriving solar radiation intensity, sometime to values close to just 100 Wm-2.
It should be remarked that, specially at the beginning and at the end of the day, the radiation intensity varies such as to resemble a ramp function. Hence, basic system theory arguments allow us to conclude that if the controller contains just one integrator a steady state error will result. This will be made apparent later in some experimental results.
Given the above definition of manipulated, process and disturbance variables, the block diagram of a distributed collector solar field controller is as in Fig. 1.13. In many cases, feedforward terms from ambient temperature and from inlet fluid temperature are not used. It is remarked that the inlet fluid temperature is affected by the load on the system, that is, by the amount of energy per unit time extracted from the fluid by the secondary circuit connected to the equipment that consumes power.
An alternative structure uses two feedback loops connected in cascade as shown in Fig. 1.15 (for the sake of simplicity the block diagram in this figure omits the sensors and the fluid flow control system). The inner loop (associated with controller Ci in Fig. 1.15) controls the average temperature T of the fluid leaving the collector loops and includes the feedforward actions as explained above. Its manipulated variable is the fluid flow command. The outer loop (associated with controller C2) uses as manipulated variable, the temperature set-point of the inner-loop (denoted u2 in Fig. 1.15) and as the process variable to be controlled, the fluid temperature at the inlet to the storage tank. In-between the end of the collector loops and the storage tank there is a pipe that collects heated fluid from the collector loops and transports it to the storage tank.
The motivation for using cascade control is common to what is found in other processes (Stephanopoulos 1984). In addition to decoupling the design of controllers for different parts of the plant, it reduces the effect of disturbances affecting the inner loop, thereby decreasing their effect in the outer-loop and improving tracking performance.
The dynamics of the relation between the average temperature at the output of the collector loops and the temperature at the storage tank inlet is mainly a delay that varies with the average value of fluid flow. This means that, if this outer loop is to
Fig. 1.16 і of a distributed collector field controlled by a PID, demonstrating non-linear
be closed with a controller that is robust with respect to changes in plant delay. A suitable controller is provided by some Model Predictive Control algorithms. More details on adaptive cascade control solutions will be provided in Chap. 3.
In general, unless explicitly stated otherwise, when we consider the control of the field, we refer to the situation of Fig. 1.13 and not to cascade control. Furthermore, we refer to the average of the temperatures of the fluid leaving the collector loops simply as “the temperature”.
A detailed discussion of the dominant dynamics of distributed collector solar fields is made in Chap. 2. Here, based on experimental data as well as simulations, we just point out a number of characteristic features of distributed collector solar fields that motivate the use of adaptive control techniques and identify some major difficulties facing controller design.
The first point to note is that the system is nonlinear. The nonlinearity is shown by the simulation results of Fig. 1.16. These records concern the situation in which the temperature is controlled with a fixed gain PID that is well tuned for a working point close to 200 °C. With this controller, the reference is varied between 180 and 280 °C in five steps of 20 °C each. As can be seen, at lower temperatures the response is sluggish, but, as the working point increases, there is more and more oscillation. Around 280 °C there is a constant amplitude oscillation. The linear constant parameter controller was not able to adequately control the system over the whole operating range due to the inherent system nonlinearity.
In qualitative terms, this can be easily understood. At working points corresponding to higher temperatures, the average fluid flow is smaller because the fluid requires more time to absorb the energy required to reach a higher temperature. Therefore, the system transport delay increases and the stability margin of the controlled linear approximation decreases causing the closed-loop response damping to decrease.
It should be noted that the situation is complex because the operating point depends on the fluid flow which is the manipulated variable. Actually, as will be shown in Chap. 2 there is a product of the manipulated variable by the state and hence this type of system is called “bilinear” (Elliot 2009).
This discussion leads us to the following conclusions concerning distributed collector solar fields:
• For a controller to be effective over a wide range of reference values it must take into account plant nonlinearity and compensate it;
• Adaptive controllers relying on linear control laws can compensate nonlinear effects only up to some extent. These controllers are able to adapt to the working point and yield a good response for small changes of the reference around it, but the performance degrades when the aim is to track large and fast reference changes. Chapters 5 and 6 consider nonlinear controllers able to cope with this problem. In particular, Chap. 5 will present experimental results on the response to a 50 °C sudden variation of the reference with virtually no overshoot.
Another important issue is uncertainty. In distributed collector solar fields uncertainty comes from several sources. These include parametric uncertainty and unmodeled terms.
Issues affecting parametric uncertainty are manifold: When identifying a model, parameter estimates become known only up to a certain precision. This is common to all modeling processes. More specific to distributed collector solar fields are the processes that lead to parameters change. One is aging: With the passage of time the working fluid used to capture solar energy may change its thermal characteristics. Another is dust deposition/removal, by factors such as wind or rain, over the surface of concentrating mirrors that changes its reflectivity coefficient.
Apparently equal systems may actually present different behavior when subjected to the same stimulus. An example is provided by Fig. 1.17 that shows the temperature records at the outlet of two different collector loops, made at the same time, with the same fluid flow and under the same solar radiation incidence. Nevertheless, the records are not only different, but the maximum is attained in each of them at different time instants.
Unmodeled terms have also other sources. A major example concerns the fact that the plant is infinite dimensional but that controller design usually rely on finite dimensional models. Furthermore, several subsystems, such as energy accumulation in the glass covering the pipe conveying the heating medium, or the dynamics of the flow controller, are usually neglected and contribute to uncertainty in plant model knowledge. Another example is provided by the difference between the fluid flow command and the actual fluid flow (Fig. 1.18). The temperature controller manipulates the fluid flow command but the fluid flow control loop adds a delay of 15 s which is significant (in the computer control experiments described in this book the sampling time is 15 s).
The presence of uncertainty factors motivate the use of adaptive control. Based on current plant data, the adaptation mechanism corrects the controller gains so as to optimize their match to plant dynamics. A word of caution should however be said: Adaptation corresponds to a highly nonlinear feedback that may destabilize the plant if proper care is not taken to produce an adequate design (Ioannou and Fidan 2006). This said, adaptive control algorithms are powerful tools that enhance the performance of distributed collector solar fields.