Research on Multi-Model Switching Control of Linear Fresnel Heat Collecting Subsystem

Abstract

Aiming at the stochasticity, uncertainty, and strong perturbation of the linear Fresnel solar thermal power collection subsystem, this study establishes a multivariate prediction model for the linear Fresnel collector subsystem based on complex environmental characteristics and designs a PID controller and MPC controller for the tracking and control of the outlet temperature. By analyzing the heat transfer process of the collector, constructing a model in Multiphysics for three-dimensional modeling of the collector, extracting data through simulation, fuzzy clustering the data and using different clustering centers for parameter identification in order to obtain the multi-model. By using the field data from the site of Dunhuang Dacheng Linear Fresnel Molten Salt Collector Field, considering the inlet temperature, normal direct irradiance and wind speed are used as the perturbation quantities, and the flow rate of molten salt is used as the control quantity. Considering three representative weather conditions, the switching criterion of minimizing the real-time point error is adopted for switching the outlet temperature of the collector. Simulation analysis results show that under the same conditions, the tracking error of the single model is relatively large, with the output temperature error fluctuating between −100 °C and 100 °C and containing many burrs. In contrast, the output temperature error of the multi-model switching control is controlled within 50 °C, which features a smaller tracking error and a faster tracking speed compared with the single-model control. When faced with large disturbances, the multi-model MPC switching control achieves better tracking performance than the multi-model PID switching control. It tracks temperatures closer to the set value, with a faster tracking speed and more excellent anti-interference performance.

1. Introduction

The world is witnessing a disturbing acceleration in the number, speed, and scale of climate record-breaking increases. Under current policies, greenhouse gas emissions will increase by 16 percent in 2030. Today, the projected increase is 3%. However, GHG emissions must still be projected to fall by 28% in 2030 for the 2 °C pathway of the Paris Agreement and by 42% for the 1.5 °C pathway. All countries must urgently accelerate economy-wide, low-carbon transformations to achieve the long-term temperature goal of the Paris Agreement [1]. Signaling a further significant increase in support for clean energy investments in energy efficiency [2], in solar thermal power generation, the linear Fresnel solar thermal power generation has the unique advantages of flexible structural arrangement, low investment and maintenance costs, etc. It shows a very promising potential as it proceeds rapidly to commercial maturity [3].

The working principle of the linear Fresnel collector subsystem is to reflect and gather incident sunlight through primary mirrors to heat the molten salt in the collector tube and to keep the outlet temperature within a certain range by adjusting the flow rate of molten salt, so as to ensure the relative stability of power generation. A variety of control algorithms have been applied to solar collector systems by a wide range of scholars, and their control is all aimed at reducing tracking error and controlling the outlet temperature of the collector in a relatively smooth range [4,5,6,7,8,9]. However, in the face of unknown interference and natural factors, it is not easy to maintain the outlet temperature within a certain range. Therefore, the outlet temperature of molten salt is an important control target for power generation stabilization, which has certain research value and practical application needs.

With the rapid development of solar thermal power generation technology, various control algorithms have been applied to solar heat collection systems by numerous scholars. Their control purposes are all to reduce tracking errors and keep the outlet temperature of the collector within a relatively stable range. Reference [10] designed a Filtered Dynamic Matrix Control (FDMC) algorithm for solar heat collection fields. By adding a filter to the prediction error, it enhances the robustness and anti-interference capability under multiple time delays. Reference [11] proposed a hybrid logical dynamic predictive controller, which adopts a simplified lumped parameter model for prediction and simulation, thereby reducing heat losses caused by cloud shading in large-scale solar fields. Reference [12] proposes a dual-mode model predictive anti-interference controller, which consists of a feedforward compensator with an improved disturbance observer and a feedback controller based on dual-mode model predictive control. Experimental results show that in the face of disturbances, it can achieve unbiased control of the outlet temperature of the heat collection field with a small overshoot and a short adjustment time. Reference [13] designs a Dynamic Matrix Predictive Control (DMC) and incorporates a Kalman filter, thereby enabling the system to exhibit stronger stability and robustness when facing large fluctuations in solar radiation intensity. Reference [14] adopts soft sensor modeling and sliding mode predictive control methods. Based on the measured data, it controls the outlet molten salt temperature of the heat collection subsystem in the 50 MW linear Fresnel CSP (Concentrated Solar Power) plant of Gansu Dunhuang Dacheng, which improves the prediction accuracy and anti-interference ability of the heat collection subsystem. Reference [15] conducts cluster analysis on on-site measured data and establishes a predictive multi-model for the heat collection subsystem. The results of multi-model predictive control show that compared with a single model, the multi-model has the advantages of higher precision and shorter lag time. However, this multi-model does not consider wind speed as a disturbance term of the system.

Existing linear Fresnel collector subsystem outlet temperature control uses common industrial PID control algorithms. The controlled object is basically a single model, and in the face of the wind speed of the environment, it makes the establishment of the model as well as the control of the existence of large deviations difficult. In summary, this paper carries out three-dimensional modeling simulation of the vacuum collector tube, and through the extracted data set for the establishment of a multi-model, the model takes into account the time-varying wind speed conditions. Finally, this is accomplished through the design of multi-model PID and multi-model MPC controllers and through the measured data for comparative simulation analysis.

We summarize the main contributions as follows:

2. Physical Modeling of Vacuum Collector Tubes

This section presents a 3D modeling simulation of the collector tube through COMSOL Multiphysics (hereafter referred to as COMSOL). The version used in this study is 6.1, and the coupled Multiphysics simulation is performed through solid and fluid heat transfer fields and fluid fields in COMSOL. The computer hardware configuration is as follows: processor: 12th Gen Intel(R) Core(TM) i7-12700H 2.30 GHz; graphics card: RTX 3070Ti (Laptop). The manufacturer of COMSOL software version 6.1 is COMSOL AB, with its headquarters located in Stockholm, Sweden.

2.1. Operating Principle

The linear Fresnel concentrator collector system, where a fixed receiver is mounted on a series of small towers and produces a linear focal point, and long rows of flat or slightly curved mirrors moving independently on an axis reflect the sun’s rays back to the fixed receiver to heat the molten salt inside the vacuum collector tube, as shown in Figure 1. The collector tube is referenced to a 4060 mm solar medium and high-temperature vacuum collector tube, which consists of a metal inner tube and a transparent glass outer tube, and the heat loss of the collector tube is reduced by pumping the interlayer between the inner and outer tubes into a vacuum and maintaining it for a long period of time. In this case, the material of the metal tube is stainless steel 321, and the outer side is air domain. The collector tube geometrical parameters are shown in

Table 1. Collector tube geometric parameters.

Designation	Unit	Value
Inner diameter of metal collector tube	m	0.081
Outer diameter of metal collector tube	m	0.09
Glass casing inner diameter	m	0.139
Outer diameter of glass casing	m	0.145
The emissivity of metal tube surface coating	-	0.08
Glass casing emissivity	-	0.86
Collector vacuum	pa	0.001

.

The established collector model contains heat conduction, heat convection, and heat radiation in three heat transfer modes; heat conduction includes the heat transfer between the glass casing and the metal tube; due to the vacuum between the metal tube and the glass casing, the heat convection is ignored, and only the natural convection and forced convection between the outside of the glass casing and the air are considered; the heat radiation is the heat transfer between the vacuum layers and the radiation to the external environment. The energy transfer profile is shown in Figure 2.

2.2. Boundary Condition

In the process of heat collection, most of the energy is still concentrated in the lower part of the collector tube due to the great non-uniformity of the radiant heat flow on the surface of the collector tube. Therefore, the energy flow density distribution in Reference [17] is used to set up 6 groups along the circumferential angle of the collector tube, as shown in Figure 3. Energy flux density distribution: Boundary conditions for radiation from a surface to the environment are adopted in order to consider the process of radiative heat dissipation. In COMSOL, the physical field heat transfer equations are represented by Equations (1) and (2) for solids and fluids, respectively. Figure 4, which shows the energy transfer profile, indicates that the conduction heat flux and the temperature in the lower half of the collector is slightly higher than the upper half due to the influence of the non-uniform energy flow density.

$$\left\{\begin{array}{c}\rho C_{p}u\cdot \nabla T+\nabla \cdot q=Q+Q_{ted}\\ q=-k\nabla T\end{array}\right.$$

(1)

$$\left\{\begin{array}{c}\rho C_{p}u\cdot \nabla T+\nabla \cdot q=Q_P+Q_{vd}\\ \rho ={\displaystyle \frac{P_A}{R_{S}T}}\\ q=-k\nabla T\end{array}\right.$$

(2)

where $k$ is thermal conductivity; $C_p(\mathrm{J}/\mathrm{k}\mathrm{g}\cdot \mathrm{K})$ is constant pressure heat capacity; $P_A(\mathrm{p}\mathrm{a})$ is absolute pressure; $Q(\mathrm{W})$ is additional heat source items; $\rho (\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$ is densities; $q_{ted}(\mathrm{W})$ is thermoelastic damping, a heat source term due to compression or expansion of a solid; $Q_P(\mathrm{W}/{\mathrm{m}}^3)$ is the pressure function; $Q_{vd}(\mathrm{W}/{\mathrm{m}}^3)$ is viscous dissipation, $q(\mathrm{W}/{\mathrm{m}}^2)$ is the conducted heat flux vector; $R_s(\mathrm{J}/\mathrm{k}\mathrm{g}\cdot \mathrm{K})$ is the gas constant; and $T(°\mathrm{C})$ is temperature.

The heat transfer medium is 60% NaNO3 and 40% KNO3 ratio of molten salt. Molten salt generally begins to decompose and produce gas at more than 600 °C and begins to become viscous at 240 °C, crystals are precipitated at 238 °C, and it begins to solidify at 220 °C, so the appropriate temperature range is between 290 and 580 °C [18]. Whereas the physical parameters of the molten salt are strongly affected by temperature variations, the values of the physical parameters were fitted to the following Equations (3)–(6) using the nonlinear least squares method [19].

$$\rho =2090-0.636T_f$$

(3)

$$C=1447.5+0.1718T_f$$

(4)

$$\lambda =0.442+1.954\times {10}^{-4}{T}_f$$

(5)

$$\mu =22.714-0.12T_f+2.281\times {10}^{-4}{T}_f^2-1.474\times {10}^{-7}{T}_f^3$$

(6)

During the flow of molten salt, considering the hydraulic modeling requirements of the circulating pipelines of the collector system, the adjustable range of the molten salt flow rate in the collector branch is 0.5 to 2.5 m/s. The nature of the flow of the molten salt depends on the Reynolds number, as shown in Equation (7), which is specifically defined by the following criteria: $\mathrm{Re}<2300$ for laminar flow, $2300\le \mathrm{Re}\le 10000$ for a transition state, and $\mathrm{Re}>10000$ for turbulence [20].

$$\mathrm{Re}={\displaystyle \frac{{\rho }_{f}vd}{\mu }}$$

(7)

where $d(\mathrm{m})$ is the metal tube inner diameter and $v(\mathrm{m}/\mathrm{s})$ is the medium flow rate.

The physical field is set to laminar flow, where the velocity distribution of the fluid in the cross-section no longer changes due to the long pipe, i.e., a fully developed flow with a pressure inlet and velocity outlet. Gravity is neglected, and there is no slip wall. The steady state equations for laminar flow are given in Equation (8).

$$\left\{\begin{array}{c}\rho (u_2\cdot \nabla )u_2=\nabla \cdot [-p_{2}I+K]+F\\ \rho \nabla \cdot u_2=0\\ K=\mu (\nabla u_2+{(\nabla u_2)}^T)\end{array}\right.$$

(8)

where $K(\mathrm{N}/{\mathrm{m}}^2)$ is the viscous stress tensor, $I(\mathrm{N}/{\mathrm{m}}^2)$ is the unit tensor, and $F(\mathrm{N}/{\mathrm{m}}^3)$ is the volumetric force.

The external forced convection condition is used to consider the introduced wind speed. In the feasibility study report of the thermal power plant [21], the collectors are arranged in the north–south direction. Their wind rose map indicates that the wind direction is mostly from west to east, which corresponds to the arrangement of the collector field, so it is the inlet-to-outlet direction; the wind speed is set by the common wind speeds in the feasibility study report. Equation (9) represents the physical field equation of convective heat flux.

$$\begin{array}{c}-n\cdot q=q_0\\ h=\left\{\begin{array}{c}2{\displaystyle \frac{K0.3387{\mathrm{P}}_r^{1/3}{\mathrm{Re}}_L^{1/2}}{L{(1+{(\frac{0.0468}{\mathrm{Pr}})}^{2/3})}^{1/4}}}\cdots if{\mathrm{Re}}_L\le 5\cdot {10}^5\\ 2{\displaystyle \frac{K}{L}}{\mathrm{Pr}}^{1/3}(0.037{\mathrm{Re}}_L^{4/5}-871)\cdots if{\mathrm{Re}}_L>5\cdot {10}^5\end{array}\right.\\ q_0=h(T_{ext}-T)\end{array}$$

(9)

where $L(\mathrm{m})$ is length, $T_{exe}(°\mathrm{C})$ is the outside temperature, $h(\mathrm{W}/{\mathrm{m}}^2\cdot \mathrm{K})$ is the heat transfer coefficient, and $\mathrm{Pr}$ is the Prandtl Number.

2.3. Mesh Generation

A user-controlled mesh was selected for the object geometry in COMSOL. The mesh quality map is shown in Figure 5, indicating that the quality of the constructed mesh is better overall.

In the steady state study calculations, COMSOL selects the solution method by default based on the characteristics of the model. Therefore, the relative tolerance defaults to 0.001 and is automatically linear. The separation solver–direct solver is selected, which has the advantage of being robust and general and more robust than the iterative solver. The PARDISO solver is selected among the direct solvers, which is relatively faster.

The simulation analysis was carried out by COMSOL, varying different parameters for the simulation, and 300 sets of data were extracted from it to provide data support for the establishment of the multi-model.

3. The Establishment of Multivariate Model

In order to build a multi-model to reflect the multiple complexities of the heat-trapping process, 300 data sets were clustered to obtain the clustering centers, and the parameters were identified for each class of data with different clustering centers.

3.1. Cluster Analysis

The fuzzy C-mean clustering algorithm is based on initializing the number of clusters and the cluster center and continuously updating the degree of affiliation and the cluster center to minimize the objective criterion until the cluster center is no longer changing or the difference between the objective function values of the two iterations is within the permissible range [22]. The steps are as follows:

Step 1: Determine the number of categories $C$ and the fuzzy weight index $m$, and initialize the clustering center $\nu$.

Step 2: The fuzzy affiliation matrix is calculated through the following Equation (10):

$$u_{ij}=\left\{\begin{array}{cc}{\left[{\displaystyle \sum _{k=1}^c\frac{{‖x_i-{\upsilon }_j‖}^{\frac{2}{m-1}}}{{‖x_i-{\upsilon }_k‖}^{\frac{2}{m-1}}}}\right]}^{-1}& ‖x_i-{\upsilon }_k‖\ne 0\\ 1& ‖x_i-{\upsilon }_k‖=0 \mathrm{and} \mathrm{k}=j\\ 0& ‖x_i-{\upsilon }_k‖=0 \mathrm{and} \mathrm{k}\ne j\end{array}\right.$$

(10)

where $u_{ij}$ is the fuzzy affiliation of individual $x_i$ to class $j$, and ${\nu }_j$ is the cluster center of class $j$.

Step 3: Use the following Equation (11) to calculate the clustering center:

$${\upsilon }_j={\displaystyle \frac{{\displaystyle \sum _{i=1}^n{u}_{ij}^m{x}_i}}{{\displaystyle \sum _{i=1}^n{u}_{ij}^m}}}$$

(11)

Step 4: Calculate the clustering target value from Equation (12), and determine whether the target value is satisfied. If it is satisfied, the clustering ends; if not, return to Step 2.

$$J={\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^c{(u_{ij})}^m‖x_i-{\upsilon }_j‖}}$$

(12)

Here, the advantages and disadvantages of categorization are evaluated by choosing the DB effectiveness index [23]. A smaller value of its indicator indicates better clustering effectiveness, and the DB indicator is defined as shown in Equation (13).

$$D{B}^{(-)}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{R}_i}$$

(13)

where $R_i$ is the similarity between categories i and j.

The 300 sets of data $M(T_{in},u_i,I,T_{out},v)$ were classified and analyzed by fuzzy clustering, which, according to the DB indicator, showed the best clustering effect when $C=6$, and

Table 2. Cluster results.

C	2	3	4	5	6	7	8	9
DB	0.6845	0.8686	0.7688	0.7050	0.6018	0.8741	0.6859	0.7872

represents its clustering results.

The clustering centers clustered into 6 classes are as follows:

$$\begin{array}{l}{\upsilon }_1=\left(\begin{array}{ccccc}301.1243& 0.6003& 680.2298& 442.0567& 2.6127\end{array}\right)\\ {\upsilon }_2=\left(\begin{array}{ccccc}299.6899& 0.6145& 811.4918& 502.8205& 3.1902\end{array}\right)\\ {\upsilon }_3=\left(\begin{array}{ccccc}301.7597& 0.6345& 832.6394& 427.4059& 4.8611\end{array}\right)\\ {\upsilon }_4=\left(\begin{array}{ccccc}296.1373& 0.6185& 886.3975& 527.1712& 3.7107\end{array}\right)\\ {\upsilon }_5=\left(\begin{array}{ccccc}297.8536& 0.6278& 759.2058& 440.2099& 1.9754\end{array}\right)\\ {\upsilon }_6=\left(\begin{array}{ccccc}294.4886& 0.6302& 932.0549& 484.5639& 4.6277\end{array}\right)\end{array}$$

3.2. Parameter Recognition

A multivariate model of the collector subsystem was developed using the forgotten factor recursive least squares method based on the data classification results. The inlet temperature, normal direct irradiance, molten salt flow rate, and wind speed are inputs, and the outlet temperature is the output. In order to better adapt the model to changes in the current data and to increase the impact of new data, the forgetting factor recursive least squares method is used. The parameter $\theta (k)$ is identified using a controlled autoregression model in Equation (14) below.

$$Y(k+1)={\phi }^T(k)\theta (k)$$

(14)

where, $Y(k+1)={[y_1(k+1)\dots y_6(k+1)]}^{\mathrm{T}}$

$$\theta =\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}\\ a_{21}& a_{22}& a_{23}& a_{24}& a_{25}\\ a_{31}& a_{32}& a_{33}& a_{34}& a_{35}\\ a_{41}& a_{42}& a_{43}& a_{44}& a_{45}\\ a_{51}& a_{52}& a_{53}& a_{54}& a_{55}\\ a_{61}& a_{62}& a_{63}& a_{64}& a_{65}\end{array}\right], {\phi }^T(k)=\left[\begin{array}{ccccc}{T}_{in1}(k)& u_1(k)& I_1(k)& -y_1(k)& v_1(k)\\ T_{in2}(k)& u_2(k)& I_2(k)& -y_2(k)& v_2(k)\\ T_{in3}(k)& u_3(k)& I_3(k)& -y_3(k)& v_3(k)\\ T_{in4}(k)& u_4(k)& I_4(k)& -y_4(k)& v_4(k)\\ T_{in5}(k)& u_5(k)& I_5(k)& -y_5(k)& v_5(k)\\ T_{in6}(k)& u_6(k)& I_6(k)& -y_6(k)& v_6(k)\end{array}\right]$$

The initial values are zero vectors or sufficiently small positive real vectors. Then, the multi-model equation can be obtained as shown in Equation (15). $y_i(k+1)$ is the initial predicted value at time k + 1.

$$\left\{\begin{array}{c}{y}_1(k+1)=0.1577T_{in1}(k)+0.1826u_1(k)+0.3383I_1(k)+0.2219y_1(k)+0.0013v_1(k)\\ y_2(k+1)=0.1240T_{in2}(k)+0.1535u_2(k)+0.3514I_2(k)+0.2041y_2(k)+0.0023v_2(k)\\ y_3(k+1)=0.1195T_{in3}(k)+0.1492u_3(k)+0.3212I_3(k)+0.1679y_3(k)+0.0020v_3(k)\\ y_4(k+1)=0.1227T_{in4}(k)+0.1545u_4(k)+0.3583I_4(k)+0.2124y_4(k)+0.0021v_4(k)\\ y_5(k+1)=0.1433T_{in5}(k)+0.1787u_5(k)+0.3398I_5(k)+0.2135y_5(k)+0.0018v_5(k)\\ y_6(k+1)=0.1081T_{in6}(k)+0.1373u_6(k)+0.3424I_6(k)+0.1802y_6(k)+0.0032v_6(k)\end{array}\right.$$

(15)

where $T_{ini}(k)(°\mathrm{C})$ is inlet temperature, $u_i(k)(\mathrm{m}/\mathrm{s})$ is molten salt flow rate, $y_i(k)(°\mathrm{C})$ is outlet temperature, and $v_i(k)(\mathrm{m}/\mathrm{s})$ is wind velocity.

Considering the overall length of the pipeline, the molten salt needs to collect and exchange heat in a longer pipeline, so among the factors, the change in normal direct irradiance has the greatest effect on the outlet temperature. The comparison between the predicted values and the real values is shown in Figure 6. Except for some deviations in the prediction effect of individual data points, the overall error between the overall predicted values and the real values is small, indicating that the prediction effect of the model is generally good.

The model is validated by using the data of the inlet and outlet temperature measurement points of the collector circuit of the commissioned Lanzhou Dacheng Thermal Power Generation Collector Field on 17 March 2023 and combining them with the weather conditions measured by the power plant on that day.

Each collector loop consists of 12 solar collector assemblies, and the length of a single loop is 1200 m. The effective collector area of each collector assembly is 1445 m², and the effective collector area of a single loop is 17,340 m².

The results are shown in

Table 3. Comparison table of results.

NO.	Inlet Temperature/°C	Molten Salt Velocity/m·s−1	Solar Radiation/W·m−2	Wind Speed/m·s−1	Model Exit Temperature/°C	Actual Exit Temperature/°C	Relative Error/%
1	300	0.61	688	2.5	444.72	441.1	0.8
2	297	0.62	853	2	495.41	497.6	0.44
3	303	0.63	815	4.5	426.2	427	0.19
4	296	0.63	896	2	528.74	526.8	0.37
5	298	0.62	732	2	456.71	454.6	0.46
6	290	0.61	913	8.5	480.8	480.5	0.06

for comparison with the real data. Among them, the relative errors of each group are within a reasonable range, confirming that the established predictive multi-model is reliable and can be used in the subsequent control system. The data in Table 3 are the cluster centers obtained after clustering. All data are divided into six categories, with these six data groups as the centers after clustering, and the clustering results are shown in Figure 6.

4. Multi-Model Switching Control

4.1. Design of Multi-Model PID Controller

The block diagram of the multi-model PID switching control designed in this section is shown in Figure 7. With the set value as the input and the outlet temperature as the output, the flow rate of molten salt is regulated by controlling the flow rate of molten salt, the control output is carried out separately for multiple sub-models, the error between the output of each sub-model and the set value is calculated, the smallest error is selected online as the final output, and the model of that output is selected at the same time, so as to complete the switching output control of the multi-models.

In this case, the control law of the PID controller is shown in Equation (16) below.

$$u(t)=k_p+k_i{\displaystyle \underset{0}{\overset{t}{\int }}e(t)dt+k_d}{\displaystyle \frac{de(t)}{dt}}$$

(16)

By discretizing it, the discretized PID control law expressed in the following Equation (17) can be obtained.

$$u(k)=k_{p}e(k)+k_i{\displaystyle \sum _{i=0}^{k}e(i)+k_d}(e(k)-e(k-1))$$

(17)

4.2. Design of Multi-Model MPC Controller

A model predictive controller is selected and designed to control the outlet temperature output based on the characteristics of the collector subsystem. Model predictive control is a model-based control algorithm. Compared with PID control, it has a certain degree of future predictability, and it applies the principle of online rolling optimization, which has certain advantages for the uncertainty caused by time-varying, disturbances, etc. In addition to this, it is a kind of optimal process technology due to its use of multistep prognostics, which can effectively solve the problem of time-delay processes.

The multi-model MPC switching control diagram designed in this section is shown in Figure 8. The reference set value is used as the input, the outlet temperature is controlled by controlling the flow rate of molten salt, and the outlet temperature of the molten salt is used as the output, where the output of each sub-model is used to perform an error operation, with the set value and the smallest output error value selected online for the selection on the model and switched to the output of the selected model.

By truncating the model when the step response essentially reaches a stable value within an appropriate sampling period, the response sequence is obtained for a limited period of time:

$$\alpha ={[a_1,a_2,a_3,\cdots a_N]}^T$$

(18)

where $N$ is the modeling time domain.

When there is a control increment $\Delta u(k)$ for the control quantity at moment $k$, then the predicted value at moment $k+i$ is

$$\widetilde{y_1}(k+i|k)=\widetilde{y_0}(k+i|k)+a_i\Delta u(k),i=1,2,\cdots N$$

(19)

where ${\tilde{y}}_0(k+i|k)$ is the predicted initial value.

The output values for future moments are

$$\widetilde{y_M}(k+i|k)=\widetilde{y_0}(k+i|k)+\underset{j=1}{\overset{\min (M,i)}{\Sigma }}{a}_{i-j+1}\Delta u(k+j-1),i=1,2,\cdots N$$

(20)

where $M$ is the control time domain.

Rewrite the above equation in vector form:

$$\widetilde{y_{PM}}(k)=\widetilde{y_{P0}}(k)+A\Delta u_M(k)$$

(21)

The vector consisting of the control increment changes is

$$\Delta u(k)={[\begin{array}{ccc}\Delta u(k)& \cdots & \Delta u(k+M-1)\end{array}]}^T$$

(22)

The optimal sequence of control increments is obtained by minimizing the performance index function. In order to keep the predictions of the MPC controller from diverging from the actual output, a feedback correction of the prediction model is required with an output error of

$$e(k+1)=y(k+1)-\widetilde{y_1}(k+i|k)$$

(23)

The corrected output is

$$\begin{array}{l}\widetilde{y_{cor}}(k+1)=\widetilde{y_{1N}}(k)+he(k+1)\\ \widetilde{y_{cor}}(k+1)=\left[\begin{array}{l}\widetilde{y_{cor}}(k+1|k+1)\\ \\ \widetilde{y_{cor}}(k+N)|k+1)\end{array}\right]\end{array}$$

(24)

where $h={[h_1,h_2,\cdots h_N]}^T$,$\widetilde{y_{cor}}(k+1)=\left[\begin{array}{c}\widetilde{y_{cor}}(k+1|k+1)\\ \begin{array}{cc}⋮& \end{array}\\ \widetilde{y_{cor}}(k+N)|k+1)\end{array}\right]$ where $h$ is the calibration vector and ${\tilde{y}}_{N1}(k)$ is the initial predicted value at moment $k+1$.

4.3. Performance Indicators

We want to be as close as possible to a given value for the output, and the control increments do not vary too drastically, so the performance metrics of the MPC are defined through Equation (25).

$$J={[Y(k+1)-Y_r(k+1)]}^{T}Q[Y(k+1)-Y_r(k+1)]+\Delta U^T(k)R\Delta U(k)$$

(25)

where $Y(k+1)={[y(k+1),y(k+2),\cdots y(k+p)]}^T,\Delta U(k)={[\Delta u(k),\Delta u(k+1),\cdots \Delta u(k+m-1)]}^T$, $Q$ is the output error control matrix, $R$ is the weighted volume control matrix, and $P$ is the predictive time domain.

Due to the complexity of the system operation and the actual physical constraints, there is the following Equation (26) for the local constraints:

$$\left\{\begin{array}{l}{u}_{\min }\le u(k+j-1)\le u_{\max },j=1,2,\cdots M\\ y_{\min }\le y(k+j)\le y_{\max },j=1,2,\cdots P\\ \Delta u_{\min }\le \Delta u(k+j-1)\le \Delta u_{\max },j=1,2,\cdots P\end{array}\right.$$

(26)

4.4. Switching Guidelines

Six models are established according to the system operation status. The control accuracy of model predictive control (MPC) is related to the parameter accuracy of the established predictive models. Based on the criterion that the error between the expected value and the current output is minimized, the predictive model is selected to implement model predictive control. The output value obtained from each controller is compared with the set value, and the minimum error is selected online as the output at that moment, defining its error criterion as the real-time point error, as shown in Equation (27).

$$j_{ei}=|y_i-y_{ri}|$$

(27)

4.5. Reference Track

Before the outlet temperature of the molten salt linear Fresnel collector system is put into closed-loop control, the collector system is in the preheat boot stage. The outlet temperature of the corresponding collector branch is 350 °C, and it is kept in steady cycle operation. For the setting of the reference trajectory, the outlet temperature is set to start at 350 °C, and finally, the outlet temperature criterion of 550 °C is reached by two rises.

5. Simulation and Analysis

Data from the Dacheng linear Fresnel power system, which has been put into grid-connected power generation in western Gansu, China, are taken for simulation and analysis under various weather conditions.

Scene 1: sunny weather conditions.

The solar radiation, inlet temperature, and wind speed variations on the day are shown in Figure 9a–c.

The simulation results of single-model MPC, multi-model PID switching control, and multi-model MPC switching control are shown in Figure 10, where (a), (b), (c), and (d) indicate the outlet temperature, output error, control increment, and model selection, respectively.

From the simulation results, it can be seen that the simulation results for scene 1 are generally better. The control error of the switching MPC can be seen to be smaller than that of the single-model MPC as well as the PID switching control through Figure 10a,b, and its tracking speed is faster.

In order to further reflect the simulation results obtained by various algorithms, the results are quantified in terms of the amount of overshoot, response time, average error, and root mean square error (RMSE) as shown in

Table 4. Performance comparison of each algorithm under sunny weather conditions.

Scope and Algorithm	Overshooting (%)		Response Time (min)		Average Error (-)	RMSE (-)
Setpoint change	First time (510 °C)	Second time (550 °C)	First time (510 °C)	Second time (550 °C)	Overall situation	Overall situation
Multi-model PID	0.5119	0.6657	3	3	3.3395	27.0819
Single-model MPC	0.4178	0.5339	1.5	1.5	0.8964	9.2410
Multi-model MPC	0.1701	0.8572	1.5	1.5	0.4181	4.5227

, where the average error as well as the RMSE is the global average, and the amount of overshoot as well as the response time are given by the first and second set point rise, respectively.

RMSE is shown in Equation (26) and is expressed as the deviation between the true value and the observed value.

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _1^n{(\widehat{y_i}-y_i)}^2}}$$

(28)

where $n$ is the sample size, ${\hat{y}}_i$ is the observed estimates, and $y_i$ is the actual real value.

As can be seen from Table 4, the response time when using PID is relatively slow for MPC, and its average error and RMSE are higher than that of single-model MPC and multi-model MPC. In addition to that, the average error of multi-model MPC is only 0.4181, the RMSE is 4.5227, that of single-model MPC is 0.8964, the RMSE is 9.2410, and the multi-model MPC performs better than single-model MPC. Multi-model MPC performs better than single-model MPC, which is due to the fact that the multi-model switching control can compare the current output errors of each model in real time and select the smallest error as the final output, so its tracking error is smaller and its anti-interference is relatively high.

Scene 2: cloudy weather conditions.

The solar radiation, inlet temperature, and wind speed variations on the day are shown in Figure 11a–c. Among them, it can be seen from Figure 11b that due to the receipt of cloud cover, solar radiation shows a drastic decrease and large volatility in some periods.

From Figure 11, it can be seen that the overall irradiation was better from 11:00 to 14:00 on that day. Due to the cloud cover, the solar normal direct irradiance showed a large fluctuation after 14:00 and lasted for a longer period of time. After 16:00, the inlet temperature showed a large fluctuating situation, and there was a tendency for the wind speed to increase during that time, when the control of the molten salt flow rate enabled the inlet temperature to undergo a small increase in the flow rate of molten salt.

For this, multi-model PID switching control as well as MPC switching control are performed, and the simulation results are shown in Figure 12, where (a), (b), (c), and (d) indicate the outlet temperature, output error, control increment, and model selection, respectively.

For the model predictive control of multi-model switching systems, during the model switching process, it is only necessary to simply compare the current output with the expected value, so the calculation load is not large. Generally, the shortest time for model switching is about 10 min.

From the simulation results of cloudy weather, it can be seen that due to the cloud cover, the solar normal direct irradiance appears to be more unstable, which leads to a larger fluctuation in the outlet temperature, but the error of the multi-model MPC is much smaller than that of the multi-model PID control. In addition, due to the more drastic changes in irradiation, the control increment and model selection reflect greater complexity.

In order to further represent the simulation results obtained by various algorithms, this is quantified in the form of

Table 5. Performance comparison of each algorithm under cloudy weather conditions.

Scope and Algorithm	Overshooting (%)		Response Time (min)		Average Error (-)	RMSE (-)
Setpoint change	First time (510 °C)	Second time (550 °C)	First time (510 °C)	Second time (550 °C)	Overall situation	Overall situation
Multi-model PID	1.7444	1.1328	12	4.5	5.8838	33.9793
Multi-model MPC	0.3175	1.0403	1.5	1.5	0.6389	4.8475

for cloudy weather.

As can be seen from Table 5, the response time of PID is slower than that of MPC, and the average error of multi-model MPC is 0.6389 with an RMSE of 4.8475, while the average error of multi-model PID is 5.8838 with an RMSE of 33.9793, which is much higher than that of multi-model MPC, and there exists a large instability of the output in the face of a larger perturbation that lasts for a long time, but the MPC shows more excellent anti-disturbance performance relative to PID.

Scene 3: windy weather conditions.

The variations in solar radiation, inlet temperature, and wind speed are shown in Figure 13a–c. From Figure 13c, it can be seen that the wind speed reaches high-velocity values in a short period of time.

For this, multi-model PID switching control and multi-model MPC switching control are performed, and the simulation results are shown in Figure 14, where (a), (b), (c), and (d) represent the outlet temperature, output error, control increment, and model selection, respectively.

From the simulation results, it can be seen that for scene 3, the simulation results are generally better, and the multi-model MPC switching control error is smaller than the multi-model PID switching control. Its tracking speed is faster in the face of the short time increase in wind speed.

In order to further represent the simulation results obtained by various algorithms, this is quantified in the form of

Table 6. Performance comparison of each algorithm under windy weather conditions.

Scope and Algorithm	Overshooting (%)		Response Time (min)		Average Error (-)	RMSE (-)
Setpoint change	First time (510 °C)	Second time (550 °C)	First time (510 °C)	Second time (550 °C)	Overall situation	Overall situation
Multi-model PID	0.3983	0.6183	6	2	3.5053	29.9970
Multi-model MPC	3.6781 × 10−13	0.0843	1.5	2	0.5144	4.5612

for windy weather.

As can be seen from Table 6, the simulation results in windy weather conditions are generally better, the response time of the multi-model PID is slower than that of the MPC, the overshoot of the multi-model MPC is generally much smaller than that of the multi-model PID, and the average error of the multi-model MPC is 0.5144 with an RMSE of 4.5612, while that of the multi-model PID is 3.5053 with an RMSE of 29.9970, which is much higher than the multi-model MPC, indicating that the multi-model MPC has a higher tracking accuracy compared with the multi-model PID when facing a sudden change in wind speed. The average error of multi-model MPC is 0.5144 and RMSE is 4.5612, and the average error of multi-model PID is 3.5053 and RMSE is 29.9970, which is much higher than that of multi-model MPC, which indicates that multi-model MPC has higher tracking accuracy than multi-model PID when facing the situation of sudden change of wind speed.

6. Conclusions

This thesis establishes a three-dimensional collector model through COMSOL software and proposes to include wind speed as a disturbance variable according to the environmental factors. It simulates and extracts data sets by constructing simulations in Multiphysics, establishes a multi-model by using fuzzy clustering as well as parameter identification, and verifies it through the data of the collector field that has already been put into use. Finally, it is controlled by designing PID and MPC controllers for three typical weather conditions by switching between multiple models. In terms of multi-model switching control, the following conclusions can be obtained:

(1)

According to the structure and working principle of the linear Fresnel collector subsystem, a three-dimensional model of the vacuum collector tube in line with the site conditions was established by COMSOL with some specification data from the site. The inlet temperature, normal direct irradiance, wind speed, and the flow rate of the molten salts were listed as the variables, and a Multiphysics simulation of the established model was constructed to simulate and analyze the model and extract the data, which provided data support for the clustering analysis as well as the establishment of the multi-models.

(2)

We performed fuzzy clustering of the data set through DB evaluation indexes, the identification of the parameters of each type of data through the recursive least squares method of the forgetting factor to obtain the predictive mathematical multi-model reflecting the collector subsystem under a variety of circumstances, and verified it using the data of the collector field which has been put into use, indicating that the multi-model established in this paper has a better prediction effect.

(3)

According to the absolute minimum of real-time point error for its output switching selection, through the design of PID and MPC controllers and simulation analysis for three typical weather conditions, the results show that the designed controller basically meets the tracking requirements of the outlet temperature, multi-model MPC switching control of the tracking error is smaller than multi-model PID switching control, and its tracking speed is relatively the fastest.