1. Introduction[1]

Nowadays, solar energy is exploited for various purposes such as refrigeration, detoxification, water sweetening, etc. Solar energy is the most plentiful sustainable source of energy that sends more than 150,000 T.W of energy to the earth. The atmosphere reflected half of this energy, and the earth’s surface absorbs the other half. Only a little fraction of solar energy will be enough to meet the world's energy demand [1].

There are several ways, directly such as photovoltaic systems or indirectly, to convert solar energy into electrical energy. Indirect use of solar thermal energy is more for homemade, industrial, and power plant uses. Electric power generators are used in all power plants, such as hydropower plants, steam power plants, and gas power plants. A generator is a device that converts mechanical energy into electrical power. In solar power plants, the main task of the solar system (that's called collector) is to produce the steam needed to rotate the turbines. The collector is a structure on which the mirrors are placed and focuses the sun's radiation on the center of the fluid carrier tube. The power generation cycle of a solar collector plant is illustrated in Fig. 1.

 

 

Fig. 1. Electricity generation cycle in solar collector power plant

 

The solar collector field has a tracking system that follows the sun from the east to the west [2]. Fluids used in solar collectors may be air, water, oil, or other organic solvents. Solar collectors have different types such as Fresnel lens concentrating collectors, compound parabolic collectors, flat-plate collectors with reflectors, and parabolic trough collectors (PTC)[1]. The purpose of the control in this system is to hold the output oil temperature at a favorable level notwithstanding disturbances such as changes in radiation, changes in the reflection of mirrors due to the presence of dirt, ambient temperature changes, and Inlet oil temperature, etc. Of course, the most important factor influencing these systems is solar radiation, and, directly or indirectly, other factors can be dependent on this factor. The temperature of the outlet oil for use in steam turbines should usually be above 285 °C; also to prevent the absorption of the tube material, the discord between the inlet and outlet temperatures of the oil should not exceed 80 ° C. If the temperature difference is more significant than this, the oil may leak because the oil pressure is high [2].

Table (1): Symbols

Description (Units)	Symbol
Tube related
Oil  related
Density
Special heat capacity
Area
Diameter of mirror opening
Inner diameter of transmission
Differential tube length
Diameter of collector opening
Flow of oil
Effective level of collectors
Temperature
Environment temperature
Time (second)
Set point temperature
Solar Irradiation
Space m
Mirror optical efficiency
Global coefficient of thermal losses
Heat transfer coefficient between oil & tube

 

Two categories of control methods, basic and advanced control, are used for distributed solar collector fields. PID control, feed forward control, and cascade control are among the basic controllers used in these power plants. Proposed methods for controlling distributed solar collector fields by the year 2007 have been studied in [3]. PID control is recognized as one of the most used controllers in the industry. The use of PID control with fixed parameters was investigated in 1987 by Camacho et al [4], but they were not able to reach a good performance of the system without the additional compensator in the control loop [5]. Actually, in all experiments, a PID control combined with a feed forward controller which is used to compensate for the effect of measurable disturbances has been applied [6]. In [7, 8], both dynamic and static feed forward terms have been developed. The combination of the PID and feed forward controller is the basis of a new generation of direct steam-generating solar power plants [9, 10]. The PI controller provided in [11] presents robust behavior. In addition, this controller can be an easy-to-use controller.

By splitting the control problem into two control loops, an inner control loop (slave) to compensate for disturbances and the outer control loop (master) to control the process output, cascade control can be developed to cancel the effects of the disturbances[3].

Because of the nonlinear nature of the system and the variety of disturbances, the use of base controllers would not be appropriate for model-based approaches. Therefore most researchers have sought to control this system using more advanced controllers. Several advanced control methods, such as robust control or adaptive control, have been developed around linear control systems. Some of the other control methods such as neural network control and fuzzy control developed around artificial intelligence control. In [12] the same feed forward controller and identification mechanism were presented in a comparative predictive control. Based on a simplified transfer function of the model, an adaptive robust predictive controller is developed in [13]. In [14] by using a simplified transfer function model that part of it includes resonance features, the alternative control plan was designed. This controller has been developed in [15]. An optimal control formulation is suggested in [14, 16] for achieving maximum efficiency of electricity production in the framework of controlling distributed collector systems. In 1985-1986 an alternative method by using a second-order Lyapunov function and adapting the law of control was proposed by Camacho et al [17, 18]. In [3, 19], the response of the system is divided into the forced and free terms and a model predictive control (MPC) is presented. Therefore, the effect of disturbances is taken into account using a nonlinear model of the free response, and the optimal sequence of control actions is obtained without a need for numerical methods. Also, by using a simplified mathematical model of the plant a nonlinear model-based predictive controller (NMPC) was designed [20]. This paper estimated the parameters of the nonlinear model. In [21] an adaptive feedback linearization scheme is proposed by benefiting from Lyapunov's rule. Some of the techniques for designing feedback systems with a high degree of robustness have been proposed by Camacho et al. [20]. In [22] by the reference temperature as the only system input, a nonlinear system subjected to disturbances, is controlled as an uncertain linear system. Then a feed forward controller is used as a robust controller [3]. In [23] the PDE field description is approximated by a compact parameter model. In [24] by using a simple feed forward method an automatic control approach was developed.  In [19] various methods of advanced control including adaptive control, predictive controller, nonlinear internal controller, fuzzy control and optimal control have been investigated. In the context of these types of systems, the fuzzy control for controlling distributed solar collector field is considered from the outset. In [25], the use of supervisory fuzzy predictive control has been performed. Also a smart predictive controller besides the same mentioned idea has been used in [26, 27]. In [28] a fuzzy prediction control scheme is developed and in [29], a recursive neural network is used to obtain an overnight inverse of the system by applying fuzzy logic controller techniques.

Of course, in the most recent research, we can mention the Takagi Sugeno fuzzy model in which the nonlinear behavior of the system is identified and predicted [30]. Ref. [31] for solar trough systems proposed an observer-based model predictive control strategy, and the performance of the proposed control strategy is compared to a PID controller and a gain scheduling generalized predictive controller. A nonlinear predictive controller is also used in [32] and in [33] a solar power plant is described by partial differential equations and sliding mode control is applied to it. In [34], For solar thermal power generation system with fast time-varying, strong interference, and uncertainty characteristics, combined with compensatory the feed forward compensation to measurable disturbance, strong robustness of sliding mode control, and the advantages of predictive control to handle obvious constraints effectively, a method of feed forward sliding mode predictive controller is put forward to apply in solar thermal power generation heat collecting system. In the latest research, it is possible to provide optimal control in [35]. Increasing the electricity generated is the main objective of the above control strategies.  Also, in [36] using PI control with feed forward control and using optimal control strategy, it has been tried to keep the production of electric power at desired set point during the day with partial radiation. In [37], the output oil temperature control is presented using the filter dynamic matrix control (FDMC). In [38], a novel strategy in oil temperature control is presented using the combination of parametric uncertainty robust control and the Takagi-Sugeno fuzzy model. In [39], a multi-model active fault-tolerant controller is established through the recursive least square method. A nonlinear controller has been designed in [40] to track reference temperature and its stability has been proven by Lyapunov's case. Of course, the existence of chattering in the control signal for this system is very clear. In the latest investigates, parabolic trough collector (PTC) and linear Fresnel reflector (LFR) are comprehensively and comparatively reviewed in terms of historical background, technological features, recent advancement, economic analysis, and application areas [41] and a detailed review of the experimental and numerical works carried out on heat transfer enhancement techniques which focus on minimization of heat loss, is presented in [42]. All the disturbances, including external disturbances and internal disturbances, were lumped into one disturbance and this disturbance is estimated and rejected by an active disturbance rejection control (ADRC) [43]. In [44], a feed forward-based strategy is proposed to control the outlet temperature of the collectors of a solar plant using the defocus angle as the manipulated variable. Predicting transformer oil temperature, in [45] had been investigated to accurately get the running state of transformer.

Inspired by the aforementioned research, the adaptive nonlinear control is investigated to adjust the outlet oil temperature of the collectors in this paper. In the control design, a high-performance controller is proposed for distributed solar collector field as the smooth signal control of this controller can achieve a fast response and low chattering. Compared with the existing studies, this work has two distinct characteristics:

Contrary to the works in [5, 7-10], a strong control is presented against uncertainty and disturbance in distributed collector system utilizing by robust nonlinear control method and Lyapunov approach.

In addition to guaranteeing the robustness of the designed controller, the proposed controller has a smooth signal control that can achieve a fast response and low chattering.

In the next section, system modeling and various numerical models presented for linear parabolic solar power plants are discussed and two models are used: a) distributed parameter model, and b) concentrated parameter model. In Section 3, first, a nonlinear controller is designed and next, an adaptive nonlinear controller is designed to control the output oil temperature of this plan and its stability has also been proven. Finally, Section 4 presents the simulation results of the control method based on the parameters of the Shiraz solar power plant which shows the proper performance controller. Concluding remarks are provided in section 5.

 

2. Mathematical Modeling and System Description

Several methods for numerical modeling of distributed solar collector power plants are found in the literature which was accepted [46]. Most of these models are obtained by experiments and the behavior of the ACUREX solar collector field. The main task of modeling the ACUREX was done by Carmona [47] and introduced in 1997 by Camacho. It should be noted in the modeling of a solar collector that a part of the concentrated radiation is absorbed by the oil and the other part is lost through metal tubes and glass. Given the thermal conductivity above the metal tube, we assume that the surface temperature inside and outside them is equal. Generally, the models presented for the solar collector field can be divided into two categories: "Fundamental models" and "Data-driven models". In the fundamental model, the dynamic field equation is achieved by the dynamic equations for each path, which are as follows:

(1)

 

Where

 

(2)

 

Eq. (1) describes the energy absorbed by the oil tube, the energy lost to the environment, and finally the energy transferred to the oil. The amount of energy absorbed by the oil follows another dynamic equation:

(3)

 

Some papers have proposed an energy equilibrium model that does not consider heat loss and has been used by some researchers. This equation is described as follows [48].

(4)

 

Depending on how the equations are solved, different equations can be presented based on the obtained data. In this paper, the "centralized parameter model" is used to design the controller, and the "distributed parameter model" is used to perform the simulation study.

 

• Centralized Parameter Model

The Centralized parameter model is used to represent the total energy changes of the field using a compact equation. The values of the parameters used in this equation are calculated by the least-squares identification method in [49] and the centralized parameter model is presented as follows:

(5)

 

In this equation,  is output oil temperature,  is inlet oil temperature,  is average inlet and outlet oil temperature, is the function of temperature, and the other parameters are described in Table 1. Due to the simplicity and absence of complex parameters in this equation as well as a relatively accurate representation of the amount of system energy, this equation is more commonly used for control purposes and is less used to express the complete and accurate behavior of the system.

 

• Distributed Parameters Model

The dynamics of the distributed solar collector field is the description of the energy equilibrium by partial differential equations. Dynamic field equations are obtained by writing the dynamic equations of each path. These equations are given in equations (1) & (3).  Several methods have been proposed to solve them such as the method of line, finite difference method, partial iteration process, and so on. The general idea of most partial differential equation solving methods is that the continuous domain of the problem is discretized at first. Then the unknown function or its derivatives are approximated by a numerical technique (in fact, the difference between the different methods is due to the difference in the type of approximation used). Combining the discrete and the approximate equations converts the differential equation to a matrix equation that by solving it, the solution of differential equations is obtained.

(6)
(7)

 

In these equations, i represents the collector number used in the Distributed Solar Collector Field. It should be noted that the equations apply only to the active regions. In this equation, in addition to total system efficiency, optical efficiency is also considered. η0 as optical efficiency is one of the important parameters in this system which is a function of optical properties (mirror reflection coefficient, glass tube crossing coefficient, and receiver tube absorption coefficient) and angle of impact (the angle created between sunlight and the line perpendicular to the collector surface) [50]. The optical efficiency of the parabolic trough collector (PTC) is calculated by the following equation [51]:

(8)

 

The parameter ρ represents the reflectivity of the mirror, (τα) is the natural absorption and transfer factor, γ is the coefficient of determination, and K(θ) is the angle-modifier coefficient of radiation, which is a function of hours and days of the year. To calculate this parameter, the collector efficiency at each angle is calculated first [51, 52]:

(9)
(10)
(11)
(12)

N represents the number of days, φ is the geographical width of the solar collector field, θ_Z is the angle of the sun's peak, δ is the angle of deviation, θ is the collision angle and t is time in hours. Using the values obtained by the above equations, K(θ) is calculated by the following equation [53, 54]:

(13)

 

Also the coefficient of heat transfer is calculated by the following equation:

(14)

 

3. Control Design

Since oil temperature and pressure are the main factors in generating electricity for these types of solar power plants, an appropriate controller should be designed and used to adjust the temperature and optimize the operation of the power plant. Excessive oil temperatures can cause damage to existing equipment and hardware. On the other hand, lowering the oil temperature has a direct effect on reducing energy generation efficiency. The purpose of control in this paper is to regulate the temperature of the oil output from the solar collector despite the disturbances. For this purpose, a sliding mode controller was first designed to track the reference temperature, and then an adaptive term is added to the controller to have a robust and applicable controller in real-time conditions.

 

• Nonlinear Controllers

The purpose of the controller design in this section is to track the desired temperature. A reference temperature is considered to control the temperature of the oil under various conditions depending on factors such as the amount of sunlight, the dynamical conditions of the system, the amount of oil temperature required to generate electrical energy, and many other factors and also it is tried to keep the oil temperature within the desired range. In most of these power plants, the reference temperature is set by human operators, but in some papers, intelligent controllers such as the fuzzy controller or neural network controller have been used to calculate the reference temperature.

To design the nonlinear controller, the tracking error is first defined and the control problem is changed to a tracking problem, and by using Lyapunov's theory, control conditions and rules are calculated so that the tracking error converges to zero or at least stays in a confined area.

(15)

 

T­_des is the reference temperature, which according to the system conditions must be achievable and T is the temperature of the output oil of the system. By differentiating (15) and replacing (5) the following equation is obtained:

(16)

 

It can be further shown that by applying (17) as an input of the system (flow rate of the oil), the error is converges to zero, where k(t) is a time-varying control gain which satisfies the robustness condition which is presented in continue.

(17)

 

Proof: Lyapunov's theory is used to prove the stability of the designed controller. Therefore, a positive Lyapunov function is defined as follows.

(18)

 

Using equation (16) the following result is obtained:

(19)

 

And then by substituting the control signal presented in equation (17), equation (19) is reached as:

(20)

 

Where the functions F and G are defined as follows:

 

(21)
(22)

 

If the control gain, k(t), is given by equation (23), the robustness condition will be met.

(23)

 

Then:

 

(24)

 

Hence, the control objective has been achieved that is:

(24)

 

Due to the negative derivative of the Lyapunov's function, thus the output tracking error e(t) converges to zero.

Remark 1: It should be noted that the disturbances in this system are considered by solar radiation or are related to some of the parameters, for example (η₀) in (equation (8)) [33, 35, 38]. Therefore, if inequality (23) is satisfied for any F and G, the controller is robust against disturbances, and the convergence of tracking error is guaranteed with Lyapunov’s theorem.

Remark 2: It should be noted that equation (23) guarantees the robustness condition for the developed sliding mode controller and in practice, choosing K is difficult at all times. However, one can choose K arbitrarily large to satisfy equation (23) at all times but this can cause chattering in the input signal. This problem is discussed in the next remark. Since the objective of the designed controller is reference temperature tracking and to achieve the minimum error possible, at e(t)=0, K(t) can have its minimum value.

 

• Adaptive Nonlinear Controller

As it is explained in the previous section, calculating the control gain at all times increases the computational time and leads to further complexity of the controller. For this reason, the control gain is usually considered a constant and sufficiently large amount. However, it should also be kept in mind that by changing environmental conditions, disturbances, and even minor changes to the system such as increasing or decreasing the number of collectors, partial failure of the system and, so on, the control gain should also be changed to maintain system stability on the one hand and to prevent chattering on the other hand.

The magnitude of the created chattering depends on the control gain value. To resolve this issue, various methods are used, including continuous approximation and high-order nonlinearity, which is a method for improving nonlinear accuracy, and chattering and combinatorial methods [1]. One of the combined methods is the use of adaptive control to regulate the controller gain. Adding the adaptive part to nonlinear control in addition to helping optimum chattering, can improve the robustness against uncertainty and disturbances and also has a higher convergence rate in comparison to the conventional methods. In this paper, nonlinear control and adaptive control are combined to design a controller that performs better. In fact, the adaptive controller is used to adjust the control gain in the designed controller. Therefore, with changes to the control signal equation (17), the new control law is calculated as:

(26)

 

Where  and  is the adaptive rate. The adaptive law is chosen as follows to simplify and avoid complex calculations:

(27)

 

It is obvious that the control signal has two general changes compared to the previous controller. First, instead of control gain,  the adaptive rate,  is used which is easier to calculate. Second, a term has been added to the control law to ensure the system`s stability.

Proof: Same as in the previous section, a new Lyapunov function is defined:

 

(28)

 

In this function,  that is a positive constant is defined and  is chosen as the adaptive error as follows:

(29)

 

Here, k is assumed to be constant and arbitrarily large according to the conditions of the system. The time-derivative of the Lyapunov function can be obtained as:

(30)

 

Then using equation (13):

 

(31)

 

By substituting equation (26) into equation (31):

(32)

 

And by substituting  and  using equation (29) and equation (27), respectively:

(33)

 

And by using equations (21)- (22):

 

(34)
(35)

Since in equation (24),  and  is always positive, Therefore:

(36)

 

Therefore the output tracking error e(t) converges to zero.

In the next section, the proposed control strategies to track the reference temperature are studied for a distributed solar collector field using real data.

 

4. Simulation Results

Most of the distributed solar collector field power plants in the world have oil and steam cycles. The purpose of this paper is to control the temperature of the outlet oil in the collectors. In this section, the 250 KW power plant in Shiraz, which is a linear parabolic collector and the designed controllers have been implemented in MATLAB. The Shiraz power plant consists of 48 collectors that are in an area of 13500 m² and with 8 parallel directions and 6 collectors in each direction along the north-south U-path. The Behran heat transfer oil is used in the Shiraz solar power plant and the lowest and highest oil flow rates in the tubes are 2kg/s and 16kg/s, respectively. The oil is able to withstand heat up to 290^oC [54, 55]. The metal used for the construction of tubes is ordinary steel or carbon steel. Of course, the electrical-thermal efficiency of carbon steel is higher than that of ordinary steel [56]. In order to obtain more accurate results and evaluate the efficiency of the designed controller, the distributed parameters model which is mentioned in Section 2-2 is used. The numerical values of each of the parameters considered in the plant are given in Table 2.

Fig. 2 illustrates the effect of the control gain k in the designed controller in Section 3-1. Increased control gain makes tracking more accurate and better. In this controller, the gain control plays a decisive role in the steady state error of the system response and has almost no effect on the transient response of the system. Fig 3 shows a comparison between how the reference signal is tracked in the controller with constant k=1 (nonlinear controller (N.C.)) and the adaptive nonlinear controller (A.N.C.). As shown in the figure, both controllers track the reference temperature. Fig. 4 shows that the control signal in the controller has chattering, while the control signal in the adaptive controller is reasonably desirable. Also, the changing of adaptive gain K is shown in Fig. 5.

In real conditions, some natural factors cause the loss of solar energy and reduce the overall efficiency of the system. Some of these factors can include wreck to some mirrors and inadequate rotation to reflect sunlight, dirty mirror surfaces, dirty transmission tubes, etc. In some papers, a modification factor for the outlet oil temperature is chosen to match the simulated models to the real state. This modification factor which affects the sunlight differs depending on the environmental conditions of different days. To study the performance of the controllers, a real data simulation of the Shiraz solar power plant is compared with the results of the designed controllers. By trial and error it was founded that by choosing the modification factor of 0.35 for this day, the results of the modeling and the real state are similar.

 

Table (3 shows the condition of the power plant on this day.

 

Table (2): Parameters of the Shiraz solar collector field

Description	Value
Tube density
Special heat capacity of tube
Oil and tube area
Diameter of mirror opening
Inner diameter of transmission tube
Heat transfer coefficient between oil and tube
Differential tube length
Diameter of collector opening
Latitude of solar collectors field
Oil density
Special heat capacity of oil

 

Fig. 2. Effect of the control gain k in the nonlinear controller

 

 

 

Fig.3. Temperature tracking by nonlinear controller (N.C.) and adaptive nonlinear controller (A.N.C,)

 

 

Fig. 4. Oil flow of Shiraz power plant

 

 

Fig. 5. Variations of the controller gain  of the adaptive nonlinear controller

 

It should be noted that in the real state, the control system is regulated by an operator. Since the initial oil temperature is about 100^oC in the early hours of the day and the optimum outlet oil temperature is 265^oC, in the first time of tracking, the system is running at its minimum value. This is due to the difference between the input oil temperature and the reference temperature. An optimal curve control structure is used to solve this problem. In this way, the initial set point temperature of the system is selected 20 degrees higher than its input oil temperature, and when the temperature reaches the desired value, the desired value increases again to 20 degrees, and this process continues. Fig. 6 shows solar radiation in the city of Shiraz on November 14, 2010.

Fig 7 also shows a comparison between the controllers designed in this paper with the real state (operator-controlled). Error! Reference source not found. and Fig. 9 show the amount of oil flow in each control method and variation of control gain K in the adaptive nonlinear control scheme, respectively. As it is well known in the simulation results, the designed controllers have been able to achieve higher temperature at the output of the transmission tubes than the control mode by the operator.

In order to prove the claim that the designed controller is robust to disturbances entering the system, three different tests have been carried out. These tests depend on the intensity profile of the sunlight which is striking the solar collector. In this experiment, it was first assumed that solar radiation would be applied to the system in a normal state without any disturbance. The second test has been performed by using a varying profile with some rapid changes in the disturbances. The last test has been performed for the worst possible conditions, under unfavorable radiation. Test results are shown in Fig. 10.

As the test results show, the designed adaptive nonlinear controller is able to provide the desired response with a very smooth control signal and without chattering under different conditions.

 

 

Fig. 6. Solar radiation in the city of Shiraz

 

 

 

Fig. 7. Output oil temperature

(a)	(b)

Fig. 8. Oil flow: (a) the control signal in real state and for designed adaptive nonlinear control, (b) the control signal for designed nonlinear control

 

Fig. 9. Variations of the controller gain  of the adaptive nonlinear controller

 

(a)

 

 

(b)

 

 

(c)

Fig. 10. Investigating the robustness of the controller under different working conditions. (a) Solar radiation, (b) Output oil temperature, (c) Oil flow

 

 

 

 

Table (3) Operational Conditions of the power plant

Start time tracking	8:00
End time tracking	16:45
Oil temperature at the start time	52oC
Oil discharge at the start time
Number of circles traced	7 from 8 circles
Weather on the test day	Clear with a bit of cloud
Status of mirrors	dirty
Receiver tube status	Dirty
Correction factor in the simulation	0.35

 

5. Conclusions

In this research, it has been theoretically proved that a distributed solar collector field can be controlled by using a nonlinear controller with adaptive theory to increase the robustness and convergence speed and reduce the system chattering. First, a nonlinear controller is designed to track the set point temperature for the output oil. In this controller, there is a direct relationship between the control gain and temperature tracking performance, but an increase in control gain leads to chattering in the control signal. For this reason, an adaptive term has been added to the controller to address this problem. Finally, the simulation results using real day information at the 250 kW power plant in Shiraz showed better and more efficient performance of the proposed method.

 

[1] Submission date:15, 01, 2022

Acceptance date: 25 , 04, 2022

Corresponding author: Mahdi Khodabandeh, Department of Electrical Engineering, Hamedan University of Technology

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