Performance of Parabolic Trough Collector with Different Heat Transfer Fluids and Control Operation

Abstract

Electricity generation from solar energy has become very desirable because it is abundantly available and eco-friendly. Mathematical modeling of various components of a Solar Thermal Power plant (STP) is warranted to predict the optimal and efficient operation of the plant. The efficiency and reliability of STPs are maximized based on different operating strategies. Opting for proper Heat Transfer Fluid (HTF), which is proposed in this paper, helps in reducing operating complexity and lowering procurement cost. The Parabolic Trough Collector (PTC) is the heart of STP, where proper focusing of PTC towards solar radiation is the primary task to maximize the outlet temperature of HTF. This maximum temperature plays a major factor due to diurnal solar radiation variation, and its disturbance nature, with the frequent startup and shutdown of STP, is avoided. In this paper, the PTC component is modeled from the first principle, and, with different HTF, the performance of PTC with constant and quadratic solar disturbances is analyzed along with classical control system designs. Through this, the operator will be able to choose proper HTF and resize the plant components depending on plant location and weather conditions. Furthermore, the thermal energy is collected for therminol oil, molten salt, and water; and its performance with different inputs of solar radiation is analyzed along with closed-loop controllers. Thermal energy extracted by therminol oil, molten salt, and water with constant solar radiation results in $81.7\%,73.7\%$ and $18.7\%$, respectively.

1. Introduction

Among all the renewable energy resources available, solar energy has been of great interest to researchers and industrialists. Kalogirou [1] described the environmental problems of fossil fuels. In addition, different types of solar collectors, their construction, functionality, thermal analysis, and applications were also presented. Solar Thermal Power (STP) plants make use of solar radiation to provide heat to a thermodynamic cycle through a heat exchanger, in which two fluid streams come into thermal contact to transfer heat from primary to secondary fluid. The secondary fluid is used to drive a turbine generator to produce electricity.

A solar collector is a key element in the solar thermal energy system. It captures the incoming solar radiation and converts it into a usable form, transforming the collected heat in the collector array into electricity as efficiently as possible through increasing the upper process temperature for the conversion. The quality of components and systems in STP is a decisive criterion in achieving greater efficiencies and reducing costs. Researchers are therefore working on suitable measurement methods and devices in order to measure the quality and subsequently improve the weak points. Predominantly, therminol oil is used as Heat Transfer Fluid (HTF) in Parabolic Trough Collector (PTC), and its operating temperature of thermal oil is limited to just under 400 °C [2].

In literature, PTC consists of parabolic reflectors as concentrators, rays to be focussed on the focal line and provided with a single axis tracking mechanism with East to West or North to South tracking with reliable, commercially proven, and mature technology. Modularity, scalability and storage pave the way for large heat production, being cost effective as compared with central receiver and parabolic dish technologies. Essential qualities for a Heat Transfer Fluid (HTF) to be used as a working fluid in PTC are high thermal capacity and superior conductivity with a minimum coefficient of thermal expansion. It should be less viscous, with no corrosive behavior or toxicity, and thermally and chemically stable. The desired properties should be unaffected during the entire range of operating conditions and long durations. Apart from therminol oil, molten salt, water and nanofluids have excellent stability. Currently, there is no practical implementation studies of thermal enhancement in these nanofluids [3]. PTC with therminol oil as the heat transfer medium is among the most widely used [4]. A Linear Fresnel Reflector (LFR) type solar collector with water as HTF to generate direct steam generation, which does not require a separate Heat Exchanger (HX) block for generation of steam, is used as an alternative—whereas LFR has low efficiencies, affected by heavy disturbances such as wind velocity, cloud cover, shadowing, among others [5]. Sauceda et al. [6] showed that the thermal efficiency of PTC is 15% higher than LFR.

Camacho et al. [7] developed a PTC mathematical model using the energy balance for therminol oil as a fluid and absorber pipe energy balance alone. Powell et al. [8] developed a PTC mathematical model with an additional energy balance for a glass pipe by considering a few realistic losses compared to the model proposed by Camacho et al. [7]. Using a Powell et al. [8] dynamic model, Kannaiyan et al. [9] validated the model with real plant data from a 1MWe gurgaon plant [10]. Silva et al. [11] presented a study on HTF as a therminol oil temperature profile of PTC that was modeled using a transport equation and solution obtained using the method of characteristics. Camacho et al. [12] described several control system designs using data-driven models for PTC.

Molten salt as HTF had the benefit of lower procurement cost, and operating at a higher temperature (565 °C) [13]. Molten salt has already proved itself as a storage medium on a commercial scale [4]. Using molten salt as HTF with a hybrid adaptive-control scheme and a time-warped predictive controller, the outlet temperature of molten salt is controlled for intermittent solar radiation [13].

PTC with water as HTF is called Direct Steam Generation (DSG). The DSG is comprised of several advantages such as it is inexpensive since it eliminates extra cost of the heat exchanger and it does not use hazardous chemicals. In addition, It is non-corrosive [14]. Yan et al. [15] used PTC with water as HTF and developed a dynamic model with three differential equations for the temperature of the water, pipe, and glass profiles, which are analyzed without describing the momentum equation. Several researchers have developed a dynamic model for DSG as a set of Differential-Algebraic Equations (DAEs) [16,17,18]. In this study, there is no two-phase steam generation that occurs, pertaining to solar radiation excitation.

Chatoorgoon [14] developed stability studies for homogeneous two-phase flow, which works efficiently for both small-time and large-time steps. Odeh et al. [19] analyzed a two-phase flow model using single-phase and two-phase Heat Transfer Coefficient (HTC) correlation coefficients used in DSG.

Recent innovation on design on PTC with a double U-tube sun-tracked with HTF is modeled using an energy balance equation along a numerical scheme for computation grid length, and experimental validation is discussed in detail in [20]. Furthermore, a rotating absorber tube and its thermal variable performance are discussed in [21]. Maintaining HTF outlet temperature by regulating the mass flow rate is essential in improving plant efficiency [22]. Similarly, cyclic startup and shutdown of STP can also be regulated [23].

To avoid such cyclic startup and shutdown in STP, PTC outlet temperature is regulated through control system design. PTC dynamic modelling and control studies help to reduce the operating cost in different disturbances such as wind speed and the temperature of HTF flowing in. Solar radiation on PTC is not predictable due to cloud cover, and the temperature of HTFs coming into the PTC is not constant. By maintaining HTFs’ outlet temperature at setpoint, better performance of PTC assistance to enhance electric power generation in STP is achieved. Setpoint of HTF’s outlet temperature is maintained by manipulating the HTF’s flow rate through PTC. To maintain HTF outlet temperature of PTC at a desired setpoint, irrespective of disturbances, along with the heat gain gradient of HTF also being maintained within limits to avoid thermal losses and oil leakage, it is essential for the safe operation of STP [7,24]. Furthermore, it also helps to find the best operating strategies to produce maximum electricity generation.

In several literature works, this has been chosen as a control and manipulated variable with several controller techniques such as PID, FeedForward [25,26] feedback linearization, Fuzzy, GPC, gain scheduling [7,25], among others. Predictive functional control and PI control for PTC with therminol oil with different solar disturbance and its performance are studied [27].

Direct steam generation is obtained using a mass flow rate of water flow as manipulated variables in PTC [17]. Using a PI controller, linear transfer function steam outlet temperature is controlled [28]. A brief review on controller methods implemented on direct steam generation is discussed [29].

The aim of the present study is to construct a dynamic model of a PTC for various HTFs with therminol oil, molten salt, and water; and then analyze their performances in different scenarios, such as heat gain performance. The purpose of this paper is to fill the research gap of utilizing the performance of alternative HTFs, molten salt, and water, for PTC and to compare with therminol oil. While comparing these HTFs, transient simulation with different disturbance fluctuation is explored along with relevant classical control action. The methodology followed in this paper is discussed as follows: the design of controller within their limits and physical reliability is essential. Thus, the design of a proper control system needs four sequences of steps as follows: (1) Using a validated dynamic model of PTC [9], step input is applied on its manipulated variable; (2) Process Variable (PV) data are collected; (3) Best fit Transfer function is obtained through a MATLAB system identification toolbox; (4) Controller tuning parameters are obtained based on fitted transfer function and IMC tuning rules; (5) Solar and thermal energy extraction for different HTFs, and its controller performance is studied.

The remaining sections of this paper are organized as follows: Section 2 presents the modeling of parabolic trough collector, and the control structure design of PTC is discussed in Section 3. In Section 4, case studies of HTFs are discussed, followed by conclusions in Section 5.

2. Modeling of a Parabolic Trough Collector

Modeling of a Parabolic Trough Collector (PTC) obtained on the basis of conservation of energy in a finite volume for each of its three components, named HTF, absorber pipe, and glass envelope. The dynamics of these three components are closely related to one another [8]. The glass tube is used to reduce the convective loss from the receiver tube by transmitting 90% of incoming short wave solar radiation while not transmitting the emitted longwave radiation outward by an absorber tube [1].

2.1. PTC with HTF

The dynamic model of PTC consists of three Partial Differential Equations (PDEs), one each for HTF, absorber pipe, and glass envelope. In the PTC modeling, all the significantly dominant factors are taken into consideration. The schematic of PTC with HTF as therminol oil and molten salt is shown in Figure 1.

The detailed energy balance of PTC per unit length is presented herewith [8].

Table 1. PTC: Notation and values of parameters.

Symbol	Description	Units	Values
$A_A,A_E$	Cross-sectional area: absorber pipe and glass envelope	m2	$1.9\times {10}^{-3},2.8\times {10}^{-3}$
$C_A,C_E,C_{p,o}$	Specific heat capacity: absorber pipe, glass envelope and therminol/molten salt/water	J/kg/K	460 [8], 840 [8], Function of temperature
$h_{air},h_p$	Convective heat transfer coefficient: Air-glass envelope, absorber pipe-therminol/molten salt/water	W/(m2 °C)	25, Dittus Bolter (Equation (A1))
${\dot{m}}_o$, I	Mass flow rate of therminol/molten salt/water, Solar radiation incident on collector surface	kg/s, W/m2	3, Figure 1
$p_{A,j}$	Absorber pipe perimeter:($j=i$) inner and ($j=o$) outer	m	0.157 [8], 0.188 [8]
$T_A,T_E,T_o$	Temperature: absorber pipe, glass envelope, and therminol/molten salt/water	°C	Variable
$T_{air},T_{sky}$	Temperature: Ambient, effective sky temperature	°C	40, 40
L, Da, W	Total Length of solar collector, Diameter of absorber, Width of mirror aperture	m	500, 0.06, 5.75
${\eta }_{opt}$, ${\xi }_A$, ${\xi }_E$	Total optical efficiency, Emissivity: absorber pipe, glass envelope	–	0.4, $0.18$ [8], $0.9$ [8]
${\rho }_A,{\rho }_E,{\rho }_o$	Density: absorber pipe, glass envelope and therminol/molten salt	kg/m3	7850, 2400, Function of temperature
$\sigma$, $\upsilon$, u, K	Stefan–Boltzmann constant, Kinematic viscosity, Velocity of fluid, Thermal conductivity	W/(m2K4), m2/s, m/s, W/(mK)	$5.67\times {10}^{-8}$ [8]

Subscript ‘o’ stands for therminol oil/molten salt/water.

shows the relevant variables used in Equations ((1) to (9)) presented in this study.

Energy balance of HTF of PTC is obtained as (1):

$$\frac{\partial T_o}{\partial t}\left({\rho }_o{C}_{p,o}{A}_A\right)={\dot{m}}_o{C}_{p,o}\frac{\partial T_o}{\partial x}+h_p{p}_{Ai}(T_A-T_o)$$

(1)

Energy balance for absorber pipe of PTC is obtained as (2):

$$\begin{array}{ccc}\hfill {\rho }_A{C}_A{A}_A\frac{\partial T_A}{\partial t}& =& h_p{p}_{Ai}(T_o-T_A)\hfill \\ & & -\frac{\sigma }{\frac{1}{{\xi }_A}\frac{1-{\xi }_E}{{\xi }_E}\left(\frac{r_{Ao}}{r_{Ei}}\right)}{p}_{Ao}\left(T_A^4-T_E^4\right)+I{\eta }_{opt}W\hfill \end{array}$$

(2)

Energy balance for glass envelope is obtained as (3):

$$\begin{array}{ccc}\hfill {\rho }_E{C}_E{A}_E\frac{\partial T_E}{\partial t}& =& \frac{\sigma }{\frac{1}{{\xi }_A}\frac{1-{\xi }_E}{{\xi }_E}\left(\frac{r_{Ao}}{r_{Ei}}\right)}{p}_{Ai}\left(T_A^4-T_E^4\right)\hfill \\ & & -\sigma p_{Eo}{\xi }_E(T_E^4-T_{sky}^4)-h_{air}{p}_{Eo}(T_E-T_{air})\hfill \end{array}$$

(3)

The PDEs are converted into Ordinary Differential Equations (ODEs) using the backward finite difference method. The number of necessary grid points for the computation is obtained by its successive increment and tracking the consequent change in the temperature until the change is less than the prescribed level. Based on the analysis, a total of 15 grid points are found to be necessary for a PTC of 500 m long. The number of state variables present in PTC for therminol oil and molten salt is 45 (i.e., $15\times 3$) and derivatives of the states are integrated over time. The details of mathematical modeling of PTC with therminol oil as HTF and its discretization, grid point computation, validation with real plant data, and implementation numerical scheme are discussed in detail by Kannaiyan et al. [9].

The proposed PTC model uses therminol oil and molten salt, and the pressure loss in both cases is negligible. Therminol oil and molten salt thermodynamic properties are obtained as shown in Appendix A. However, in the case of water as HTF, the pressure loss from input to output of PTC is significant due to the occurrence of the two-phase generation. The computation variables and its schematic, and its simulation approach for PTCTHO, PTCMS and PTCWAT are shown in Figure 2 and Figure 3.

2.2. Parabolic Trough Collector with Water as HTF (PTCWAT)

Dynamic simulation of PTCWAT is obtained using Chatoorgoon’s model [14] that discusses the conservation of mass, momentum, and energy (Equations (4) to (6)) with a discretization scheme. In the energy Equation (6), term $q_w$ captures the heat energy supplied from solar energy [9]. In addition, Equations (2) to (3) are also considered, which carry HTF to obtain the temperature increase of water at a total length of PTC at 500 m. Steam generation contributes to significant pressure loss that occurs with water as HTF, and it affects the energy output of PTC. Accounting for that dynamic model of PTCWAT consists of the following equations derived from first principles as [14]:

$$\mathrm{Mass}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho u\right)}{\partial x}=0$$

(4)

$$\mathrm{Momentum}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\frac{\partial \left(\rho u\right)}{\partial t}+\frac{\partial \left(\rho u^2\right)}{\partial x}+\frac{\partial P}{\partial x}+C_k\rho u^2+\rho g=0$$

(5)

$$\mathrm{Energy}:\hspace{1em}\frac{\partial }{\partial t}\left[\left(\rho \left(h+\frac{u^2}{2}\right)\right)\right]+\frac{\partial }{\partial x}\left[\left(\rho u\left(h+\frac{u^2}{2}\right)\right)\right]+\rho ug=\frac{\partial P}{\partial t}+q_w$$

(6)

$$\mathrm{State}:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\rho =f(P,h)$$

(7)

Energy balance for PTCWAT absorber pipe is obtained with (8):

$$\begin{array}{ccc}\hfill {\rho }_A{C}_A{A}_A\frac{\partial T_A}{\partial t}& =& h_p{p}_{Ai}(T_{(w/st/2\varphi )}-T_A)-\frac{\sigma }{\frac{1}{\xi A}+\frac{1-{\xi }_E}{\xi E}\left(\frac{T_A}{T_E}\right)}{p}_{Ao}(T_A^4-T_{sky}^4)\hfill \\ & & +I_c{\eta }_{opt}W-h_{air}{p}_{Eo}(T_A-T_{air})\hfill \end{array}$$

(8)

Energy balance for PTCWAT glass envelope is obtained with (9):

$$\begin{array}{ccc}\hfill {\rho }_E{C}_E{A}_E\frac{\partial T_E}{\partial t}& =& \frac{\sigma }{\frac{1}{\xi A}+\frac{1-{\xi }_E}{\xi E}\left(\frac{T_A}{T_E}\right)}{p}_{Ai}(T_A^4-T_E^4)\hfill \\ & & -\sigma p_{Eo}{\xi }_E(T_E^4-T_{sky}^4)-h_{air}{p}_{Eo}(T_E-T_{air})\hfill \end{array}$$

(9)

PTCWAT has one mass balance, three conservation equations (water, absorber tube and glass tube) and one momentum equation. The schematic of PTC with water as HTF is shown in Figure 4. In Appendix C, the relation of solar energy received per unit volume on an absorber pipe for sectional length for PTCWAT is discussed in detail. Parameters involved in PTCWAT are listed in

Table A1. LFR: notation and values of parameters (Equations (4) to (9)).

Symbol	Description	Units	Values
$T_{(w/st/2\varphi )}$	Temperature of water/steam/ two phase mixture depending on operating condition	°C
$\rho$	Density of working fluid	kg/m3
h	Specific enthalpy of working fluid	J/kg
q	Quality of steam
u	Velocity flow velocity of working fluid	m/s
P	Pressure	bar
$C_k$	Friction constant
$q_w$	Heat per unit volume of flow (refer (A2))	W/m3
$A_A$, $A_E$	Cross-sectional area: absorber pipe and glass envelope	m2	$1.9\times {10}^{-3},2.8\times {10}^{-3}$
$C_A$, $C_E$	Specific heat capacity: absorber pipe and glass envelope	J/kg/K	460 [8], 840 [8], Function of temperature
$h_{air}$, $h_p$	Convective heat transfer coefficient air–glass envelope, HTC of working fluid	W/(m2 °C)	25, refer to section $S.2.2.2$ [19]
I	Solar radiation incident on PTC collector surface	W/m2
L	Length of PTC	m	500
$T_A$, $T_E$	Temperature: absorber pipe, and glass envelope	°C	Variable
$T_{air}$, $T_{sky}$	Temperature: Ambient, effective sky temperature	°C	25, 40
W	Width of mirror aperture	m	14
${\xi }_A$, ${\xi }_E$	Emissivity: absorber pipe and glass envelope		$0.18,0.9$
${\eta }_{opt}$	Total optical efficiency of PTC	–	$0.4$
$\sigma$	Stefan–Boltzmann constant	W/(m2K4)	$5.67\times {10}^{-8}$
g	Gravitational acceleration	m/s2	$9.8$
${\mu }_l$	Liquid dynamic viscosity	kg/(ms)
${\mu }_g$	Steam dynamic viscosity	kg/(ms)
${\mu }_{fg}$	Viscosity difference between liquid and steam	kg/(ms)
$f_f$	Pipe friction constant

for Equations from (4) to (9).

In this simulation, PTCWAT consists of ‘n’ sections from its entire length, so the number of state variables present is given by the length of pipe multiplied with sections of pipe ($500\times 50\times 5=$ 125,000). The numerical solution used for solving the PTCWAT is implicit Euler with a step size of time ($dt$) for 60 s [9]. Thermodynamic properties of water and steam are accessed from MATLAB library files Xsteam [30].

In PTC with water as HTF, the process variable is considered as enthalpy instead of the temperature of water flowing out. In the proposed case studies, the steam generation case does not occur, and it is possible, if there is an increase in optical efficiency of PTC or inlet input temperature of the water is increased, that steam generation occurs.

If direct steam generation occurs from the PTC, the temperature of the water is constant; however, the steam quality varies due to heat gain from the solar radiation. Steam quality is more at the end of exit; then, pressure loss occurs more with reference to input pressure applied at the input of PTC. This is one of the case studies discussed in this article and performed in Appendix D to realize the steam quality generation.

3. Control Structure Design for PTC

3.1. System Identification of PTC with HTFs

Development of linear system is obtained by choosing a specific operating point, exciting the Manipulated Variable (MV) through step input, and, through the response of the PTC process variable (PV), a suitable proxy model of PTC is obtained. Based on the obtained data order of the system, the nature of the transfer function is selected based on the complexity of controller design. In this study, First Order (FO) is selected based on sufficient fit percentage.

First Order (FO) Approximation

First Order (FO) is approximated with process gain ($K_p$) and time constant ($\tau$). Process gain ($K_p$) is obtained based on the ratio of change in the magnitude of output (PV) to the change in magnitude of input (MV). The time constant ($\tau$) is obtained by predicting the time taken for the output variable to reach $63\%$ of change in the Process Variable (PV) to its steady-state value. PV responds to a change in manipulated variable [31], and it is obtained as shown in (10). The parameters of the transfer function under consideration were obtained using the system identification toolbox in MATLAB 2012a; details of the best fit equation are discussed in Appendix E:

$$G\left(s\right)=\frac{K_p}{\tau s+1}$$

(10)

The initial value assigned during this FO simulation for therminol oil and molten salt is as follows: Oil inlet temperature 200 °C, the flow rate of oil 3 kg/s, the initial temperature of absorber and glass are 201 °C and 40 °C, respectively. In the case of water inlet temperature set at 125 °C and the flow rate of oil 2 kg/s, the initial temperatures of absorber and glass are 126 °C and 40 °C, respectively.

Table 2. Step test inputs for system identification of PTC for HTFs.

	Input during Step Test
	I (W/m2)	${\mathit{\eta }}_{\mathit{opt}}$	${\dot{\mathit{m}}}_{\mathit{o}}$ (Kg/s)	Manipulated Variable (MV)	Process Variable (PV)
PTCTHO	600	0.4	3 to 4	Mass flow rate of Therminol oil flow in towards PTC	Therminol oil outlet temperature
PTCMS	600	0.4	3 to 4	Mass flow rate of molten salt flow in towards PTC	Molten salt outlet temperature
PTCWAT	600	0.4	1 to 2	Mass flow rate of Water flow in towards PTC	Water outlet temperature

shows other variables during the step test. The remaining parameters are set as per the design of PTC as shown in Table 1. Figure 5 shows the step test for PTC with therminol oil and molten salt as HTF by simulating Equations (1) to (3). Figure 6 shows the step test for PTC with water as HTF by simulating Equations (4) to (9). Based on Figure 6, the transfer function is fitted as shown in

Table 3. Control parameters for different HTFs.

Parameters System	PTCTHO	PTCMS	PTCWAT
Manipulated variable (MV)	Therminol oil flow rate	Molten salt flow rate	Water flow rate
Process Variable (PV)	Therminol oil temperature flow out of PTC	Molten salt temperature flow out of PTC	Water temperature (steam enthalpy) flow out of PTC
Disturbance	Therminol oil inlet temperature, Solar radiation	Molten salt inlet temperature, Solar radiation	Water inlet temperature, Solar radiation
Approximation Model (S domain)	$\frac{-17}{360S+1}$	$\frac{-21}{961S+1}$	$\frac{-80}{360S+1}$
Controller Tuning $(K_p;T_i)$	−0.107, 247	−0.142, 720	−0.0353, 270
Fit Percentage (%)	92.37	98.86	86.37

PTCTHO = Parabolic Trough Collector with therminol oil as heat transfer fluid; PTCMS = Parabolic trough collector with molten salt as heat transfer fluid; PTCWAT = Parabolic trough collector with water as the heat transfer fluid.

.

Using those identified transfer functions, controller tuning was obtained and utilized in the dynamic model of PTC for case study simulation.

3.2. Controller Tuning

Several tuning algorithms are available in the literature. The Internal Model Control (IMC) is employed for the PTC since it explicitly uses the desired closed-loop response and the specific form of the transfer function of the system used, in order to obtain the tuning parameters [31,32]. The various tuning rules to obtain $K_c$, ${\tau }_i$, and ${\tau }_d$, depending on the form of the transfer function used for the components of the STP, are outlined in

Table 4. IMC—controller tuning rules.

${\mathit{G}}_{\mathit{P}}$	${\mathit{G}}_{\mathbf{CL}}$	${\mathit{K}}_{\mathit{C}}$	${\mathit{\tau }}_{\mathit{i}}$	${\mathit{\tau }}_{\mathit{d}}$
${\displaystyle \frac{K_P}{\tau s+1}}$ [31]	${\displaystyle \frac{\gamma s+1}{\lambda s+1}}$	${\displaystyle \frac{2\tau -\lambda }{K_P\lambda }}$	${\displaystyle \frac{2\tau \lambda -{\lambda }^2}{{\tau }_P}}$	-

.

3.2.1. Implementation of the Digital Form of PID Controller

The general continuous form of PID controller is given as shown in (11):

$$\begin{array}{c}\hfill u\left(t\right)=K_p\ast [e\left(t\right)+\frac{1}{{\tau }_i}\int e\left(t\right)dt+{\tau }_d\ast \frac{de\left(t\right)}{dt}]\end{array}$$

(11)

Implementation of a velocity form of PID controller [2] is shown in (12):

$$△u\left(k\right)=q_{o}e\left(k\right)+q_{1}e(k-1)+q_{2}e(k-2)$$

(12)

$$\begin{array}{ccc}\hfill q_0=K_p\left(1+\frac{{\tau }_d}{T_o}+\frac{T_o}{T_i}\right);& & q_1=-K_p\left(1+2\frac{{\tau }_d}{T_o}\right);q_2=K_p\left(\frac{{\tau }_d}{T_o}\right)\hfill \end{array}$$

Several tuning algorithms are available for control systems based on that FO model, and, out of those, IMC is the preferred one. The tuning values of the IMC model are shown in Table 4.

3.2.2. Static Feedforward Control of PTC

In this section, static feedforward with a feedback controller is implemented to track the setpoint and to reject the measured disturbance. A schematic of manipulated and control variables along with disturbance is shown in Figure 7. The control variable for PTC control is to control the outlet HTFs’ temperature by manipulating the HTF flow rate through PTC.

Static feedforward control of PTC is designed based on energy balance, and its result is obtained as shown in Equation (13). Then, it is combined with feedback PI control to obtain a resultant manipulated variable:

$$\begin{array}{c}\hfill {\dot{m}}_{(o,FF)}(h_o-h_{oi})=IA{\eta }_{opt}\\ \hfill {\dot{m}}_{(o,FF)}=\frac{IA{\eta }_{opt}}{(h_o-h_{oi})}\end{array}$$

(13)

where ${\dot{m}}_{(o,FF)}$, $h_o,h_{oi}$ represent mass flow rate of HTFs, enthalpy of HTF fluid out of PTC, and enthalpy of HTF fluid flow towards the PTC, respectively. The enthalpy values are obtained as discussed in detail in Appendix A.

3.2.3. Performance Metrics of System

PTC in Closed Loop (CL) performance is characterized based on a few metrics such as Integral Absolute Error (IAE), Integral Squared Error (ISE), Integral Time Absolute Error (ITAE) and Integral Time Squared Error (ITSE), as shown in

Table 5. Performance metric of the controller.

Performance Metric $({\mathit{e}}_{\mathit{r}\mathit{r}}=\mathit{S}\mathit{P}-\mathit{P}\mathit{V})$	Expression
Integral Absolute Error (IAE)	${\int }_0^{\infty }\mid e_{rr}\mid dt$
Integral Squared Error (ISE)	${\int }_0^{\infty }{e}_{rr}^{2}dt$
Integral Time Absolute Error (ITAE)	${\int }_0^{\infty }t\mid e_{rr}\mid dt$
Integral Time Squared Error (ITSE)	${\int }_0^{\infty }t{e}_{rr}^{2}dt$

. Those metrics are obtained to characterize the tracking error residual.

4. Case Study of PTC with HTFs

Performance of PTC with different HTFs such as therminol oil, molten salt and water is evaluated with the following four case studies:

• Constant solar radiation in open Loop (COL);
• Constant solar radiation in closed loop (CCL);
• Quadratic solar radiation in open loop (QOL);
• Quadratic solar radiation in closed loop operation (QCL).

The performance of the case studied is evaluated by solar energy received ($Q_{sr}$) and the heat gain on PTC ($Q_{th}$), as shown in (15):

$$Q_{sr}={\int }_{S{P}_t}I\left(t\right)WLdt$$

(14)

$$Q_{th}={\int }_{SPt}\left(h_o\left(t\right)-h_{oi}\left(t\right)\right){\dot{m}}_o\left(t\right)dt$$

(15)

where $SPt$ stands for a time period of solar radiation applied for simulation as shown in Figure 8b. $h_o$ and $h_{oi}$ represent enthalpy of HTFs flow out of PTC and enthalpy of HTFs flow in, respectively.

Solar radiation and temperature of therminol oil/molten salt flowing in towards PTC for this respective case study are shown in Figure 8a,b. Figure 9 shows that the temperature of water flowing in towards PTC is shown with similar solar radiation, which is applied as shown in Figure 8b. In the case study of quadratic solar radiation, a disturbance in solar radiation is affected at the leading edge of 2 h and trailing edge of 6 h, respectively, and the corresponding radiation decreases by $65\%$ and $68\%$, respectively. In CCL and QCL case studies, in the first one hour of operation, PTC is simulated in open-loop operation with constant values of MV, and then control action is activated after 1 h of operation. The performance metrics of the controller for PTC case studies are discussed in

Table 6. PTC Controller performance on case studies.

Controller Performance		PTCTHO	PTCMS	PTCWAT
CCL	IAE	$4.31\times {10}^4$	$2.94\times {10}^4$	$1.041\times {10}^4$
	ISE	$1.06\times {10}^6$	$3.48\times {10}^5$	$6.20\times {10}^5$
	ITSE	$1.18\times {10}^8$	$1.37\times {10}^8$	$1.27\times {10}^6$
	ITASE	$1.059\times {10}^9$	$1.032\times {10}^9$	$6.97\times {10}^7$
QCL	IAE	$7.66\times {10}^4$	$1.66\times {10}^5$	$1.31\times {10}^4$
	ISE	$1.71\times {10}^6$	$2.89\times {10}^6$	$8.56\times {10}^5$
	ITSE	$5.48\times {10}^8$	$1.65\times {10}^9$	$1.95\times {10}^6$
	ITASE	$5.52\times {10}^9$	$1.92\times {10}^{10}$	$1.10\times {10}^8$

. These metrics show the controller performance for the simulation period, and the corresponding setpoints are updated every two hours.

The Process Variable (PV) performance of HTFs for case studies is shown in Figure 10, Figure 11 and Figure 12, whereas the Manipulated Variable (MV) variation of HTFs for case studies is shown in Figure 13, Figure 14 and Figure 15.

4.1. Constant Solar Radiation with Open Loop Operation (COL)

Table 7. Control parameters performance for case studies. (Abbreviations used in this table are: THO = Therminol oil, MS = Molten salt, WAT = Water, LE = Leading edge at 2 h, TE = Trailing edge at 6 h, PV: Process variable, MV = Manipulated variable, SSPV = Steady State value of PV, SSMV = Steady State value of MV, PV_MAX = Maximum value of PV.)

		SSPV ${(}^{°}\mathbf{C})$	PVMax ${(}^{°}\mathbf{C})$	Temperature of HTF at 0, 4, 8 h ${(}^{°}\mathbf{C})$	PV Variation Due to Solar Disturbance ${(}^{°}\mathbf{C})$		MV Variation Due to Solar Disturbance (Kg/S)
					LE	TE	LE	TE
COL	THO	288	288	200, 288, 288	-	-	-	-
	MS	324	324	200, 324, 324	-	-	-	-
	WAT	114	114	35, 144, 144	-	-	-	-
CCL	THO	-	372	200, 288, 288	-	-	-	-
	MS	-	356	200, 330, 296	-	-	-	-
	WAT	-	184	35, 184, 130	-	-	-	-
QOL	THO	-	355	200, 355, 269	323 to 286	338 to 247	SSMV: 3	SSMV: 3
	MS	-	414	200, 412, 315	360 to 342	396 to 352	SSMV: 3	SSMV: 3
	WAT	-	160	35, 160, 141	150 to 146	156 to 135	SSMV: 2	SSMV: 2
QCL	THO	-	380	200, 355, 269	323 to 286	308 to 284	3 to 0.512	4 to 6.5
	MS	-	384	200, 330, 315	340 to 322	302 to 277	5.52 to 2.43	7 to 6.5
	WAT	-	192	35, 144, 128	148 to 147	138 to 128	0.5 to 0.5	6.5 to 4.5

shows the values of Steady State (SS), initial, MV, and PV variations during solar variation. Time taken for HTFs to reach SS is about 30 min, 1 h, 31 min and 36 min, respectively. For COL, the solar energy received is 19,906 MJ, and the collected thermal energy for therminol oil, molten salt, and water is 16,277, 14,678, and 3733 MJ, respectively. Its corresponding performance analysis is shown in

Table 8. PTC case studies’ performance.

		PTCTHO	PTCMS	PTCWAT
Solar energy collected $Q_{sr}$ (MJ)	COL	-	19,906	-
	QOL	26,989
	CCL	-	19,906	-
	QCL	26,989
Thermal energy extracted $Q_{th}$ (MJ)	COL	16,277	14,678	3733
	QOL	22,210	19,456	5450
	CCL	15,845	15,040	3892
	QCL	22,256	22,101	5712

PTCTHO = PTC with therminol oil; PTCMS = PTC with molten salt; PTCWAT = PTC with water.

. Therminol oil collects more energy compared to molten salt and water at an efficiency of $81\%$, whereas, for molten salt and water, it has the efficiency of $73\%$ and $18.7\%$, respectively.

4.2. Constant Solar Radiation with Closed Loop Operation (CCL)

For the proposed CCL, the solar energy received is 19,906 MJ. The collected thermal energy for therminol oil, molten salt, and water is 15,845, 15,040, and 3892 MJ, respectively, and its performance analysis is shown in Table 8. Therminol oil collects more energy compared to molten salt and water; it is at an efficiency of $79\%$. However, for molten salt and water, it has the efficiency of $75\%$ and $19\%$, respectively.

4.3. Quadratic Solar Radiation with Closed Loop Operation (QOL)

For the QOL, the solar energy received is 26,989 MJ. The collected thermal energy for therminol oil, molten salt, and water is 22,210, 19,456, and 5450 MJ, respectively, and its performance analysis is shown in Table 8. Therminol oil collects more energy compared to molten salt and water; it is at an efficiency of $82\%$, whereas, for molten salt and water, it has the efficiency of $72\%$ and $20\%$, respectively.

In Leading Edge (LE) of disturbances (2 h), the temperature varies by about 31 °C for therminol oil. Furthermore, in the case of molten salt and water, it is about 18 and 4 °C, respectively. In the Trailing Edge (TE) of disturbances (6 h), temperature varies by about 91 °C for therminol oil, and, in the case of molten salt and water, it is about 81 and 21 °C, respectively.

4.4. Quadratic Solar Radiation with Closed Loop Operation (QCL)

Solar energy received for the QCL is 26,989 MJ, and collected thermal energy for therminol oil, molten salt and water is 22,256, 22,101, and 5712 MJ, respectively. Furthermore, its performance analysis is shown in Table 8. Therminol oil collects more energy compared to molten salt and water; it is at an efficiency of $82\%$, whereas, for molten salt and water, it has the efficiency of $81\%$ and $21\%$, respectively.

In LE of disturbances (2 h), temperature varies by about 31 °C for therminol oil, and, in the case of molten salt and water, it is about 18 and 4 °C, respectively. In TE, temperature varies by about 91 °C for therminol oil. In the case of molten salt and water, it is about 81 °C and 21 °C, respectively.

5. Conclusions

In this paper, the performance of PTC with HTF of therminol oil, molten salt and water is analyzed with different case studies, and its main outcomes are summarized as follows:

• With constant solar radiation of 600 W/m² for an 8-hour duration, solar energy collected was about 19,906 MJ. Based on this condition, the open-loop performance showed higher heat gain collected by therminol oil with an improvement by 16,277 MJ, i.e., 81%, whereas molten salt and water collected with 14,678 MJ and 3733 MJ, respectively.
• In a quadratic solar radiation case study, the solar energy collected about 26,989 MJ; open-loop performance had a higher heat gain collected by therminol oil by approximately 22,210 MJ, i.e., 82%, whereas molten salt and water collected with 19,456 MJ and 5450 MJ, respectively.
• In the closed-loop performance with the designed PI controller, therminol oil performance showed better tracking and disturbance rejection. With constant solar disturbances, molten salt showed better performance, whereas, in the case of disturbance of solar radiation, the tracking and disturbance reaction of PTC with molten salt is not adequate.
• During intermittent solar radiation, temperature variation with therminol oil is affected by 91 °C, whereas molten salt and water becomes affected by 44 °C and 21 °C, respectively.
• Heat gain captured by water as HTF is less compared to therminol oil and molten salt. For four case studies, it is in a range of about $18.7$ to $21.6\%$. The oscillations of water flow rate as MV are more compared to therminol oil and molten salt.

Based on the open and closed-loop operation of PTC with different HTF, proper HTF can be selected based on cost and operational complexity. This study sets the benchmark problem for PTC in both open and closed-loop with different HTF such as therminol oil, molten salt, and water. Advance controller design technique and optimization based control with plant constraints are the future scope of this article.