Yu Qiu

Ming-Jia Li

Kun Wang

Zhan-Bin Liu

Xiao-Dai Xue

Aiming strategy optimization for uniform flux distribution in the receiver of a linear Fresnel solar reflector using a multi-objective genetic algorithm

Keywords: Linear Fresnel solar reflector, Aiming strategy optimization, Flux distribution non-uniformity, Optical loss, Monte Carlo ray tracing, Multi-objective genetic algorithm

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Aiming strategy optimization for uniform flux distribution in the receiver of a linear Fresnel solar reflector using a multi-objective genetic algorithm

Introduction

The combustion of fossil fuels has resulted in not only the tight global energy supply but also serious global environmental issues like air pollution and global warming [START_REF] Mcglade | The geographical distribution of fossil fuels unused when limiting global warming to 2 degrees C[END_REF][START_REF] Li | A hybrid model for explaining the short-term dynamics of energy efficiency of China's thermal power plants[END_REF][START_REF] Yan | Handbook of clean energy systems Volume 1: Renewable Energy[END_REF][START_REF] Li | Review of methodologies and polices for evaluation of energy efficiency in high energy-consuming industry[END_REF][START_REF] Bellos | Multi-criteria evaluation of parabolic trough collector with internally finned absorbers[END_REF][START_REF] Li | Gas-side fouling, erosion and corrosion of heat exchangers for middle/low temperature waste heat utilization: A review on simulation and experiment[END_REF][START_REF] Li | Modeling a hybrid methodology for evaluating and forecasting regional energy efficiency in China[END_REF]. For solving these problems, renewable energy technologies [START_REF] Trancik | Renewable energy: Back the renewables boom[END_REF][START_REF] Du | Convective heat transfer of molten salt in the shell-and-tube heat exchanger with segmental baffles[END_REF], including Concentrating Solar Power (CSP) [START_REF] Du | Optical and radiative properties analysis and optimization study of the gradually-varied volumetric solar receiver[END_REF][START_REF] Li | Prospective fully-coupled multi-level analytical methodology for concentrated solar power plants: General modelling[END_REF],

wind power [START_REF] Castellani | Analyzing wind turbine directional behavior: SCADA data mining techniques for efficiency and power assessment[END_REF], photovoltaic [START_REF] Zhou | Multi-scale investigation on the absorbed irradiance distribution of the nanostructured front surface of the concentrated PV-TE device by a MC-FDTD coupled method[END_REF][START_REF] Liang | A detailed study on phonon transport in thin silicon membranes with phononic crystal nanostructures[END_REF], hydrogen energy [START_REF] Li | A passive anion-exchange membrane direct ethanol fuel cell stack and its applications[END_REF][START_REF] El-Melih | Reformation of hydrogen sulfide to hydrogen in the presence of xylene[END_REF] and fuel cell technology [START_REF] Li | A liquid-electrolyte-free anion-exchange membrane direct formate-peroxide fuel cell[END_REF][START_REF] Huang | Cations Diffusion in Nafion117 Membrane of Microbial fuel cells[END_REF][START_REF] Li | A Sodium-Ion-Conducting Direct Formate Fuel Cell: Generating Electricity and Producing Base[END_REF], etc., are considered to be highly competitive candidates [START_REF] Hoffert | Advanced technology paths to global climate stability energy for a greenhouse planet[END_REF]. Among these candidates, the CSP technology is considered as a promising option [START_REF] He | Non-uniform characteristics of solar flux distribution in the concentrating solar power systems and its corresponding solutions: A review[END_REF][START_REF] Weinstein | Concentrating Solar Power[END_REF][START_REF] Wang | A novel integrated simulation approach couples MCRT and Gebhart methods to simulate solar radiation transfer in a solar power tower system with a cavity receiver[END_REF] which mainly includes Linear Fresnel Reflector (LFR) [START_REF] Abbas | A comprehensive optical characterization of linear Fresnel collectors by means of an analytic study[END_REF], parabolic trough collector [START_REF] Wang | A design method and numerical study for a new type parabolic trough solar collector with uniform solar flux distribution[END_REF][START_REF] Cheng | A detailed nonuniform thermal model of a parabolic trough solar receiver with two halves and two inactive ends[END_REF][START_REF] Zheng | Numerical optimization of catalyst configurations in a solar parabolic trough receiver-reactor with non-uniform heat flux[END_REF][START_REF] Wang | Heat transfer performance enhancement and thermal strain restrain of tube receiver for parabolic trough solar collector by using asymmetric outward convex corrugated tube[END_REF][START_REF] Zheng | Thermal analysis of a solar parabolic trough receiver tube with porous insert optimized by coupling genetic algorithm and CFD[END_REF][START_REF] Li | Prospective fully-coupled multi-level analytical methodology for concentrated solar power plants: Applications[END_REF][START_REF] Wang | Progress in concentrated solar power technology with parabolic trough collector system: A comprehensive review[END_REF], solar power tower [START_REF] Zheng | Thermal analysis of solar central receiver tube with porous inserts and non-uniform heat flux[END_REF][START_REF] Du | Analysis of thermal stress and fatigue fracture for the solar tower molten salt receiver[END_REF][START_REF] Besarati | Optimal heliostat aiming strategy for uniform distribution of heat flux on the receiver of a solar power tower plant[END_REF][START_REF] Wang | Thermodynamic analysis and optimization of a molten salt solar power tower integrated with a recompression supercritical CO2 Brayton cycle based on integrated modeling[END_REF][START_REF] Wang | Integration between supercritical CO 2 Brayton cycles and molten salt solar power towers: A review and a comprehensive comparison of different cycle layouts[END_REF][START_REF] Zhu | Thermodynamic analysis and comparison for different direct-heated supercritical CO2 Brayton cycles integrated into a solar thermal power tower system[END_REF], and parabolic dish collector [START_REF] Hafez | Solar parabolic dish Stirling engine system design, simulation, and thermal analysis[END_REF][START_REF] Cui | Study on combined heat loss of a dish receiver with quartz glass cover[END_REF][START_REF] Wang | Transient thermal performance response characteristics of porous-medium receiver heated by multi-dish concentrator[END_REF].

The LFR technology is a solar power technology which has attracted more attention in recent years. In an LFR, solar radiation is concentrated onto a fixed receiver by several linear mirrors, where the receiver is usually a Single-Tube Receiver with a Secondary Collector (STRSC) [START_REF] Balaji | Optical modelling and performance analysis of a solar LFR receiver system with parabolic and involute secondary reflectors[END_REF] or a Multi-Tube Cavity Receiver (MTCR) [START_REF] Sahoo | Steady state hydrothermal analysis of the absorber tubes used in Linear Fresnel Reflector solar thermal system[END_REF]. The optical performance of an LFR which determines the input power of the system is important for the system design, performance optimization and safe operation. Hence, many studies have focused on this topic nowadays.

On the one hand, some studies on LFRs using STRSCs have been carried out. Häberle et al. [START_REF] Häberle | The Solarmundo Project. Advanced technology for solar thermal power generation[END_REF] evaluated the optical performance of the first LFR plant Solarmundo which adopts a secondary Compound Parabolic Collector (CPC). It is found that the optical efficiency of 61% is achieved at normal incidence, and the circumferential flux on the absorber tube is non-uniform. Qiu and He et al. [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF] also studied the optical characteristics of an LFR using an STRSC with a CPC by a self-developed Monte Carlo ray tracing (MCRT) model. It is found that the circumferential flux can be homogenized by the CPC with 34% of the total power on the top half of the tube. Similar non-uniform fluxes have also been obtained by Balaji et al. [START_REF] Balaji | Optical modelling and performance analysis of a solar LFR receiver system with parabolic and involute secondary reflectors[END_REF] and Craig et al. [START_REF] Craig | Finite-volume ray tracing using Computational Fluid Dynamics in linear focus CSP applications[END_REF]. Moreover, Grena and Tarquini [START_REF] Grena | Solar linear Fresnel collector using molten nitrates as heat transfer fluid[END_REF] designed a new double-wing secondary collector. The simulated result shows that the circumferential flux on tube is non-uniform as well. Around 37% of the total power is on the top half.

On the other hand, some studies on LFRs using MTCRs have also been conducted. Mills and Morrison [START_REF] Mills | Compact linear Fresnel reflector solar thermal powerplants[END_REF] proposed a novel design called compact linear Fresnel reflector which offers two alternative receivers to each mirror. Optical study depicts that a non-uniform flux on the receiver is observed. He et al. [START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF] and Moghimi et al. [START_REF] Moghimi | A novel computational approach to combine the optical and thermal modelling of Linear Fresnel Collectors using the finite volume method[END_REF] computed the fluxes on MTCRs in LFRs adopting the MCRT method and finite volume method, respectively. Both studies found that the rays almost just shine on the lower half of the tubes, which results in the non-uniform flux on each tube. It was also found that the flux non-uniformity among the tubes is significant. Abbas et al. [START_REF] Abbas | Solar radiation concentration features in Linear Fresnel Reflector arrays[END_REF][START_REF] Abbas | Parabolic trough collector or linear Fresnel collector? A comparison of optical features including thermal quality based on commercial solutions[END_REF][START_REF] Montes | A comparative analysis of configurations of linear Fresnel collectors for concentrating solar power[END_REF] and Bellos et al. [START_REF] Bellos | Experimental and numerical investigation of a linear Fresnel solar collector with flat plate receiver[END_REF] investigated the concentration features of flat receivers by the MCRT method and the Solidworks, respectively, where the flat receiver can be seen as the aperture of an MTCR or an STRSC. It was found that the flux on the flat receiver is significantly non-uniform with a hot region at the center.

To sum up, it can be concluded that non-uniform circumferential flux appears on each tube in both the STRSC and the MTCR. As a result, this flux will result in the non-uniform temperature on every tube, and it may lead to some negative effects. The main negative effect in the STRSC with an evacuated tube is the bending of the absorber tube [START_REF] Li | Thermal load and bending analysis of heat collection element of direct-steam-generation parabolic-trough solar power plant[END_REF]. This bending can be quite large, and the deflection at the middle part of the absorber tube can be 10 mm after the receiver is used for one year in a trough system [START_REF] Lu | Experimental studies on nonuniform heat transfer and deformation performances for trough solar receiver[END_REF]. This large deformation can lead to the vacuum failure and the glass envelop fracture [START_REF] Kutscher | Line-focus solar power plant cost reduction plan (Milestone Report)[END_REF][START_REF] Qiu | Thermal performance analysis of a parabolic trough solar collector using supercritical CO2 as heat transfer fluid under non-uniform solar flux[END_REF]. Data from the SEGS plant indicate that 3.37% of the total field receivers failed per year. In the MTCR, ordinary tubes without glass envelop are employed. The tubes can cope with non-uniform temperatures, because it is no need to worry about the glass breakage. As a result, the uniformity of the circumferential flux is not as important for MTCR as it is with STRSC.

It is also found that the flux among tubes in the MTCR is quite non-uniform, which may result in the unstable operation due to the uneven heating of the fluid among the tubes when parallel flow is employed. Moreover, it should also be noted that the local high temperature can accelerate the degradation of coating when there is no vaccum and cause the decomposition of the heat transfer fluid [START_REF] Häberle | The Solarmundo Project. Advanced technology for solar thermal power generation[END_REF][START_REF] Dey | Heat transfer aspects of an elevated linear absorber[END_REF][START_REF] Eck | Thermal Load of Direct Steam-Generating Absorber Tubes with Large Diameter in Horizontal Linear Fresnel Collectors[END_REF]].

To overcome these negative effects, it is necessary to improve the flux uniformity in the receiver. Review of above mentioned literatures indicates that few studies have focused on this topic.

To provide better studies for solving these problems, this work focuses on developing an optimization approach to reducing the flux non-uniformity in LFRs. The main contributions are summarized as:

(1) The originality of this work is that a multi-objective optimization approach combining Monte Carlo ray tracing (MCRT) and the genetic algorithm (GA) was developed for homogenizing the flux distributions in LFRs. LFRs with two typical receivers were studied to illustrate the application of the approach in practice, including a multi-tube cavity receiver and a single-tube receiver with a secondary collector.

(2) Optimization results with different typical conditions indicate that flux distributions in two typical receivers can be homogenized efficaciously in the whole range of the incident angle with a small drop of efficiency.

Physical model and aiming strategies

Physical model

The LFR primary mirror field with 25 cylindrical mirrors is taken as the concentrator [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF][START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF]. A Multi-Tube Cavity Receiver (MTCR) [START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF] and a Single-Tube Receiver with a Secondary Collector (STRSC) [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF] are considered as the receiver module, respectively. The configuration of the LFR is illustrated in Fig. 1 and Fig. 2. The geometric and optical parameters are shown in Table 1. Several Cartesian coordinate systems are established in Fig. 1 and Fig. 2 for describing the model. XgYgZg is the ground system, where the southern end of the field's middle line is the origin (G), and Xg, Yg, and Zg point to the south, east, and zenith, respectively. XmYmZm is the mirror system, where the southern end of each mirror's middle line is the origin (M). Xm points towards the south, and Zm is normal to the mirror's tangent plane at M and points to the upside of the mirror. Ym is normal to XmZm plane. XrYrZr is the receiver system. The intersection of Zg axis and the plane zg=Ht in XgYgZg is the origin (O), where Ht is the height of the axes of the tubes in XgYgZg. Xr, Yr, Zr point to the south, east, and zenith, respectively. The mirror field is symmetrical about its middle line. Two mirrors at symmetric positions have the same the dimensions. The mirrors are numbered M1 to M25 from the west to the east as shown in Fig. 1 and Fig. 3. The mirrors with non-constant curvatures were designed for concentrating as much power as possible, and the curvature radiuses Rm for M13~ M25 are given in Eq.( 1) [START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF]. The MTCR which includes eight absorber tubes is covered by a glass plate at the bottom. It is covered by aluminum walls and asbestos on the top as shown in Fig. 2(a). The STRSC includes a CPC and an evacuated tube as shown in Fig. 2

i i i Ri i i i                   (1)
                                            ( 3
)
where i is the ordinal number of the mirrors; other geometric parameters are given in Table 1.

One-line strategy and GA optimization strategy

The aiming line (Xr, yaim, zaim) in XrYrZr for a mirror is a line which is on the aiming plane of zr= zaim in XrYrZr and hit by the rays reflected on the center line of the mirror as shown in Fig. 1 and Fig. 3, where yaim and zaim are the y value and z value of the aiming line in XrYrZr. The aiming plane in the MTCR is at the axes of the tubes, where zaim=0. The aiming plane in the STRSC is at the aperture of the CPC, where zaim= -70 mm.

A basic aiming strategy for LFRs is the one-line aiming strategy (S1) as illustrated in Fig. 3

Mathematical model

The aiming strategy was optimized by combining an MCRT model and a multi-objective GA.

The flowchart of the optimization is shown in Fig. 4. In the optimization, the MCRT model is used to predict the flux distribution in the receiver and calculate the optical efficiency. The GA is adopted to optimize the flux distribution.

Due to the structural characteristics of the LFR, it is known that the longitudinal components of the incident solar rays have no effect on the flux non-uniformity in the receiver. Therefore, only the aiming strategies at different values of the transversal incidence angle (αs,T), which is the angle between the projection of an incident ray on YgZg plane and Yg as shown in Fig. 1, are needed to be optimized. In current optimization, the sun rays are assumed from the east when the solar azimuth (Αs) is 90°, and αs,T is equal to the solar altitude (αs).

For each receiver, two objective functions are optimized, including a non-uniformity index of the flux distribution and the optical loss (ηloss). The non-uniformity index (fST) of the STRSC is defined as the relative standard deviation of values of the flux in the circumferential elements on the absorber tube as given in Eq.( 4). The non-uniformity index (fMT) for the MTCR is defined as the relative standard deviation of values of power absorbed on all tubes as given in Eq.( 5). ηloss is defined in Eq.( 6). The optical efficiency (ηopt) is the ratio of the power absorbed by the tube and the maximum power that can be accepted by the mirrors.

      e e l l e l 2 ST l le 1 1 -1 =, i i n n n q i q q f n i q q         (4)       t t t t t t 2 MT t tt 1 1 -1 =, i i n n n E i E E f n i E E         (5) loss opt opt m m m 1 , 100% Q DNI L W n         ( 6
)
where ne is the number of the circumferential elements on the tube in STRSC; nt is the number of the tubes; ql(i) is local flux in ith element; Et(i) is the power absorbed by ith tube; Q and DNI are the power absorbed by the tube and the direct normal irradiance of the sun, respectively.

The detailed model is introduced in the following sections.

Optical model based on Monte Carlo ray tracing

The solar ray transfer from the sun to the receiver is simulated by a three-dimensional optical model developed based on the Monte Carlo ray tracing (MCRT), which is an optical simulation method that has been widely used in CSPs [START_REF] He | A MCRT and FVM coupled simulation method for energy conversion process in parabolic trough solar collector[END_REF][START_REF] Cheng | Comparative and sensitive analysis for parabolic trough solar collectors with a detailed Monte Carlo ray-tracing optical model[END_REF][START_REF] Qiu | A comprehensive model for analysis of real-time optical performance of a solar power tower with a multi-tube cavity receiver[END_REF]. The code was programmed in FORTRAN, and the flowchart is illustrated on the left side in Fig. [START_REF] Li | Review of methodologies and polices for evaluation of energy efficiency in high energy-consuming industry[END_REF]. The details of the model which can be found in

Refs. [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF] and [START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF] are omitted here, and some important information is introduced as follows.

The tracking angle of a mirror (θm) is defined as the angle between the positive directions of Ym and Yg. In the northern hemisphere, the tracking angle θm for a western mirror is given in Eq.( 7). θm at other conditions can be obtained in the similar way [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF].

m r,T s,T 0.5 ( )

      (7) 11 st s,T r,T s aim m tan =tan , =tan sin H A y y          ( 8
)
where ym is the y value of the mirror's center line in XgYgZg; αs and Αs are the solar altitude and azimuth, which can be obtained when the local latitude φ, real solar time ts and the ordinal of a day in a year are provided [START_REF] Qiu | A comprehensive model for optical and thermal characterization of a linear Fresnel solar reflector with a trapezoidal cavity receiver[END_REF]; αs,T is the transversal incidence angle which is the angle between the projection of an incident ray on YgZg plane and Yg as shown in Fig. 1; αr,T is the angle between the projection of the reflected ray on YgZg plane and Yg.

The solar rays are initialized randomly and uniformly on the mirrors. The sun shape error is considered as a pillbox with an angular radius of 4.65 mrad [START_REF] Qiu | Study on optical and thermal performance of a linear Fresnel solar reflector using molten salt as HTF with MCRT and FVM methods[END_REF]. The real-time DNI of the annual solar radiation is calculated by a clear sky radiation model [START_REF] Xu | The development of a software for solar radiation and its verification by the measurement results on the spot[END_REF], and the power of each ray (e) is calculated by Eq.( 9). m m m cos ray 1

( ) / i n e DNI L W i n     (9)
where ηcos(i) is the cosine efficiency of the ith mirror; nray is the total number of the traced rays.

Quadrilateral meshes are generated on all surfaces in the MTCR and the STRSC to count the rays absorbed on these surfaces. The sketches of the meshes are illustrated in Fig. 5. In all computations processes, the mesh systems with 60 and 68 circumferential elements on each absorber tube are used in the MTCR and the STRSC, respectively. In the optimization process, only one lengthwise element for each tube in the two receivers is adopted to reduce the amount of calculation.

In the flux computation process, 200 lengthwise elements for each tube in the two receivers are used for revealing the detailed fluxes.

If a ray is absorbed, it will be counted in the number of the rays absorbed in the local element (nray,e) where the ray is absorbed. The local flux in each element (ql) will be calculated after the tracing of the last ray by Eq. [START_REF] Du | Optical and radiative properties analysis and optimization study of the gradually-varied volumetric solar receiver[END_REF]. The Local Concentration Ratio (LCR) is defined as the ratio of the local solar flux (ql) and DNI as shown in Eq. [START_REF] Li | Prospective fully-coupled multi-level analytical methodology for concentrated solar power plants: General modelling[END_REF]. l ray,e e / q e n S  [START_REF] Du | Optical and radiative properties analysis and optimization study of the gradually-varied volumetric solar receiver[END_REF] l / LCR q DNI  (11

)
where Se is the area of the local element.

Optimization model based on a multi-objective GA

Genetic algorithm (GA) primarily presented by Prof. John Henry Holland in 1975 is a kind of evolutionary algorithm which imitates the processes of biology evolution in nature [START_REF] Holland | Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence[END_REF], and it is viewed as an effective method for optimization problems [START_REF] Xi | Economical evaluation and optimization of organic Rankine cycle with mixture working fluids using R245fa as flame retardant[END_REF][START_REF] Xi | Parametric optimization of regenerative organic Rankine cycle (ORC) for low grade waste heat recovery using genetic algorithm[END_REF][START_REF] Xi | A graphical criterion for working fluid selection and thermodynamic system comparison in waste heat recovery[END_REF][START_REF] Wang | Multi-objective optimization of the aiming strategy for the solar power tower with a cavity receiver by using the non-dominated sorting genetic algorithm[END_REF]. In current model, a revised version called Non-dominated Sorting Genetic Algorithm II (NSGA-II) [START_REF] Deb | A fast and elitist multiobjective genetic algorithm: NSGA-II[END_REF] for the multi-objective optimization is adopted to solve the present problem.

At the beginning of the optimization, a population with N individuals is randomly initialized, and each individual (g) is stored in a vector of (nm+1) genes (gi) as expressed in Eq.( 12). The first nm genes in an individual are integer numbers, and each gi is the ordinal number of the aiming line for corresponding ith mirror (Mi). The last gene m 1 n g  is a real number which represents the width of the aiming plane (Waim). In current optimization, N=50 is used, and the lower bound ( , , , ,

nn g g g g g   (12) LU aim m LU m =1 , , 1 =50mm , 300mm, 1 ii ii g g n i n g g i n            (13)
After that, the procedures including non-dominated sorting, crowding distance sorting, and tournament selection, crossover, mutation, recombination and selection are applied to produce the new population from the old one. This process will be repeated for plenty of generations. In current optimization, the maximum generation number of 500 is used. At the end of the optimization, a Pareto optimal front which provides N optimal individuals can be obtained.

Solution selection from the Pareto front

The Pareto front shows a trade-off between the flux distribution non-uniformity and the optical loss. In practice, a compromise has to be found between the non-uniformity and the optical loss.

Therefore, a method called Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) [START_REF] Tzeng | Multiple attribute decision making: methods and applications[END_REF] and a limitation called efficiency limitation are used to select the optimal solution from the Pareto front.

The procedure of TOPSIS is as follows. First, the non-uniformity index (fST or fMT) and the optical loss (ηloss) for each optimal individual are normalized using Eq.( 14). Then, two distances

i i i i i i i i N N f f f               (14)        
i i i i i i i i i i d f f d f f                                  (15)   * i i i i r d d d     ( 16
)
Where i=1, 2, …, N; f represents fST or fMT;

  For avoiding significant decrease in optical efficiency, an efficiency limitation is also used to choose the solution in the following way. First, a group of optimal individuals, which meet the limitation in Eq.( 17), is selected from the Pareto front. λ in Eq.( 17) is a coefficient that is adopted to limit the optical loss. λ of 0.95 is used in current optimization. It should be noted that the value of λ can be adjusted based on the requirement of the optimization. Second, the individual which has the minimum efficiency in the group is selected as the suggested solution.

    opt opt S2 S1     (17)
where

  opt S2  and   opt S1
 are the efficiencies when S2 and S1 are used, respectively.

After the above two procedures, the efficiency of the solution suggested by the TOPSIS is compared with the suggested solution of the efficiency limitation. If the former is larger than the latter, the solution suggested by the TOPSIS will be chosen as the final optimal solution. Otherwise, the suggested solution of the efficiency limitation will be selected.

Model validation

The MCRT model is validated as follows. First, the LCR profile on the focal plane of an LFR with 30 mirrors is calculated under normal incidence and compared with the experimental results provided by Chemisana et al. [START_REF] Chemisana | Optical performance of solar reflective concentrators: A simple method for optical assessment[END_REF] as shown in Fig. 6, where the mirror slope error is 12.5 mrad. It is seen that two curves correspond with each other, and the relative deviations are less than 1.6%.

Furthermore, the LCR profiles on the tubes of the MTCR in current LFR are computed at a typical condition with the solar altitude (αs) of 45° and the solar azimuth (Αs) of 135°. The S1 is used, and the diffuse reflection in the MTCR is treated as the specular reflection in this case, which is the same as that of SolTrace. The results are compared with those computed by the SolTrace [START_REF] Wendelin | SolTrace: a new optical modeling tool for concentrating solar optics[END_REF] in Fig.

Optimal solutions and flux distributions

Fig. 8 shows the Pareto fronts of the optimizations for the two systems. It can be seen that the optical loss (ηloss) decreases with the ascending non-uniformity index for each system. The solutions selected by the TOPSIS and the efficiency limitation are also marked, and the solutions obtained by the efficiency limitation are selected for the two cases adopting the selection approach introduced in Section 3.3. Details of the optimal solutions are given in Table 2. Fig. 9 illustrates the LCR distributions in the MTCR for the one-line strategy (S1) and GA optimization strategy (S2). It can be found in Fig. 9(a) that the flux distribution non-uniformity is significant for S1 with fMT=1.01, where most rays shine on tube 4 and tube 5, while tube 1 and tube 8 are barely utilized. As a result, the LCRmax of 88.5 appears on tube 5. However, it is seen in Fig. Fig. 10 shows the LCR distributions on the absorber tube in the STRSC for the two aiming strategies (S1, S2). It is presented that the flux is significantly non-uniform for S1 with fST=0.67 and LCRmax=95. [START_REF] Li | Review of methodologies and polices for evaluation of energy efficiency in high energy-consuming industry[END_REF], where only 22% of the total power is on the top half of the tube. However, it is seen that the flux of S2 is quite uniform with fST=0.32 and LCRmax=62.1, where fST and LCRmax are reduced by 52% and 35% compared with those of S1, respectively. It is worth noting that 38% of the total power shines on the top half of the absorber of S2, which could help to reduce the bending of the tube and protect the glass envelope from vacuum failure. Moreover, it can be found that the drop in efficiency of S2 is 3.3 percentage points compared with that of S1. This is because more rays are lost through the gap between the top of the absorber and the CPC when S2 is used. The above results indicate that the optimization strategy can reach a reasonable compromise between the flux distribution non-uniformity and the optical loss under the typical αs,T.

Influence of the number of aiming lines

In the optimization, the number of the aiming lines (naim) is the only parameter of the aiming strategy which is not optimized. So, its effect is further discussed in this section under the above typical condition. Fig. 9(b) and Fig. 11 show the optimal flux distributions in the MTCR at different aiming line numbers (naim). It can be found that the optimal result of Waim varies with the variation of naim. It is also seen that the differences among the flux distributions for all cases with naim=5~19 are not obvious, where fMT and LCRmax are in the ranges of 0.07~0.03 and 34.2~36.3, respectively. Moreover, it is found that ηopt is in the range of 66.0%~66.2% for all cases. These results indicate that naim has little impact on neither flux non-uniformity nor optical efficiency of the MTCR system.

In the following discussion, naim=11 is used for the MTCR. Fig. 12 shows the LCR distributions in the STRSC at different aiming line numbers (naim). It is seen that the optimal result of Waim varies with naim. It is also seen that the influence of naim on the optical efficiency is insignificant when naim=5~19. However, decreasing trends of fst and LCRmax with increasing naim can be observed when naim=5~11. This is because that the rays on the absorber can be dispersed in a better way when more aiming lines are adopted. Furthermore, it is found that fst and LCRmax change little when naim=11~19, and they remain at about 0.34 and 61, respectively. From above results, it can be concluded that naim should be large enough, which is necessary for obtaining a uniform flux distribution in the STRSC. In current STRSC, naim of 11 is suggested. Fig. 12. Optimal flux profiles in STRSC at different aiming line numbers (n aim ) when α s,T =α s =45° and A s =90°.

Employment of an optimal solution in a range of transversal incident angle

It is well known that the mirrors will track the sun during operation, and the time between two tracking actions is usually in the range of several seconds to several minutes, which is very short. Therefore, if the aiming strategy optimization was implemented at each tracking step, which has a corresponding transversal incidence angle (αs,T), there would be too many optimization cases. As a result, the computation time would be huge. For reducing the computation time, a probable method is to use the optimal solution obtained at a specific αs,T in a reasonable large angle range around this specific αs,T.

For the current two LFRs, the optimal solutions obtained at αs,T=45° are tested under other values of αs,T. Fig. 13 and Fig. 14 show the flux distributions in the MTCR at αs,T=42.5° and αs,T=47.5°, respectively. It is presented in Fig. 13(b) and Fig. 14(b) that uniform fluxes in the MTCR can still be obtained by S2, and the fMT of 0.04 is achieved at two angles. The drops in fMT are both 96% compared with those of S1 in Fig. 13(a) and Fig. 14(a). Moreover, the drops in the efficiency of S2 are both 1.9 percentage points compared with those of S1. Fig. 15 shows the flux profiles in the STRSC for two aiming strategies at αs,T=42.5° and αs,T=47.5°. It is seen that the optimal solution at αs,T=45° can still obtain uniform fluxes at αs,T=42.5°

and αs,T=47.5°. It is also seen that the flux profiles for both S1 and S2 change little when αs,T increases from 42.5° to αs,T=47.5°. This is because the incident rays from the mirrors vary little within this small range of 5° for αs,T.

- In addition, another 16 cases given in the first two columns in Table 3 are also examined. The angle interval of αs,T between two adjacent cases is 5°. Results which are similar to those above are found as well. It is found that each optimal solution which is obtained at a specific αs,T for each case can be applied in a range of 5° around this specific αs,T. Therefore, for each LFR, the 17 cases in Table 3 can obtain the uniform fluxes in the whole range of αs,T=10°~90°. As a result, the number of the optimization cases and the computation time are greatly reduced. Detailed discussion of the 17 cases will be conducted in section 5.2.

Application of GA optimization strategy under a real-time condition

It is important for an aiming strategy to obtain sufficiently good results at any time rather than just a specific time point. So, 17 key cases given in the first two columns in Table 3 are used to cover the whole ranges of the time and the transversal incidence angle. The procedures for implementing the GA optimization strategy (S2) are detailed as follows.

First, aiming strategies for the 17 key cases with different values of αs,T,i are optimized using S2 when As=90°, where i is the ordinal number of the cases in Table 3. The optimal solutions for the MTCR and STRSC systems are given in the Appendix. Second, the αs,T(ts) at each possible time point (ts) in a year will be calculated for each LFR. If Eq.( 18) is satisfied, the optimal solution of case i will be selected as the aiming strategy at ts. Third, the tracking angles of the mirrors at ts will be calculated using Eq. [START_REF] Li | Modeling a hybrid methodology for evaluating and forecasting regional energy efficiency in China[END_REF]. Finally, the tracking angles at all possible yearly time points will be calculated in advance and stored in the computer of the tracking system to drive the mirrors.

  o s,T s s,T, 2.5 i t    (18)
In the following paragraphs, the GA strategy (S2) is applied in both the MTCR and STRSC systems on spring equinox to illustrate the application of S2 under a real-time condition, where the LFR is assumed to be located at the latitude of φ=N23.5°. And 17 time points given in the last three columns in Table 3 are studied, where the incident ray at the ith time point has the same αs,T as that of case i. It is seen in Fig. 16 that fMT is greatly reduced from 0.77~1.09 when S1 is used to 0.02~0.06 when S2 is used in the whole range of time. It is also seen in Fig. 17 that hot regions appear on tubes 5 and 6 when S1 is used, while this phenomenon is eliminated by S2 as shown in Fig. 18. In Fig. 17 and Fig. 18, it is visually observed that the southern ends of the tubes are barely shone. Because the rays generally come from the south, and the rays reflected by the southern ends of the mirrors will not hit the southern ends of the MTCR, which leads to this end loss. Moreover, it is observed in Fig. 16 that the maximum flux (ql,max) is reduced from 10.0~92.8 kW•m -2 when S1 is used to 4.6~38.6 kW•m -2 with the adoption of S2. Finally, it can be found that a small drop of 1.0~3.5 percentage points in ηopt can be found when S1 is replaced by S2.

Application in an LFR with an MTCR

Application in an LFR with an STRSC

Fig. 19 shows the optimal result of S2 and the corresponding result of S1 in the STRSC system on the spring equinox. Fig. 20 and Fig. 21 illustrate the typical real-time solar fluxes in the STRSC when S1 and S2 are used, respectively, where the end loss caused by the nonzero incident angle can also be observed. Fig. 22 shows the comparison of the detailed flux profiles on the absorber.

It is seen in Fig. 19 that fST is reduced from 0.59~0.70 when S1 is used to 0.29~0.37 with the adoption of S2. It is found in Fig. 20 and Fig. 22 that hot regions occur at the bottom of the absorber when S1 is used, and the top half of the absorber is barely used. However, the hot spots are efficiently weakened by S2 as shown in Fig. 21 and Fig. 22, and it is observed in Fig. 19 that the maximum flux is reduced from 12.1~101.2 kW•m -2 when S1 is used to 9.9~69.6 kW•m -2 with the adoption of S2. Moreover, it is found that the solar power is distributed quite well around the tube for each case, and about 35~40% of total power is shone on the top half of the tube. Finally, it is seen that a small decrease of 0.2~3.8 percentage points in ηopt is found when S1 is replaced by S2.

Conclusions

This work presents an aiming strategy optimization approach for homogenizing the flux distributions in both the Multi-Tube Cavity Receiver (MTCR) and the Single-Tube Receiver with a Secondary Collector (STRSC) in linear Fresnel reflectors (LFRs). In the approach, the fluxes are computed using an MCRT model, and the optimization is conducted based on a multi-objective GA.

The MCRT model was validated against both experimental and simulation data. The following conclusions are derived.

( Moreover, the optimal solution obtained at a specific value of the transversal incidence angle (αs,T) can be applied in a relatively large range (e.g., 5°) around it.

(2) The number of aiming lines has almost no effect on the optical efficiency and the flux non-uniformity in the MTCR system. For the STRSC system, naim has almost no effect on the efficiency, but it has a visible effect on the flux non-uniformity. It is found that a sufficiently large naim is necessary for obtaining uniform fluxes in the STRSC.

(3) Under a real-time condition, application of the GA approach indicates that the flux distributions in both the MTCR and the STRSC can be homogenized efficaciously in the whole range of the incident angle by the GA approach with a small drop of around 0.2~3.8 percentage points in efficiency compared with those of S1. It is found that the non-uniformity index for the MTCR system is greatly reduced from 0.77~1.09 to 0.02~0.06, and the maximum flux is reduced from 10.0~92.8 kW•m -2 to 4.6~38.6 kW•m -2 when S2 replaces S1. For the STRSC system, the non-uniformity index is decreased from 0.59~0.70 to 0.29~0.37, and the maximum flux is decreased from 12.1~101.2 kW•m -2 to 9.9~69.6 kW•m -2 when S2 replaces S1.

In conclusion, these results indicate that the present GA approach is effective and suitable for optimizing the flux distributions in the receivers of LFRs, which can be beneficial for the further studies and the industrial application.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Sketch of an LFR showing the transfer of the solar ray.

  (a), where all mirrors aim at the center line of the aiming plane. In current work, a new optimization strategy (S2) is presented and illustrated in Fig. 3(b) for distributing the flux in the receiver uniformly. In S2, naim aiming lines are uniformly distributed on the aiming plane, and they are numbered A1 to Anaim from the west to the east. Each mirror would choose an aiming line from the naim lines, and the width of the aiming plane (Waim) in Fig. 3(b) would be optimized as well.

( 2 )Fig. 3 .

 23 Fig. 3. Sketches of two aiming strategies for LFRs.

StartFig. 4 .

 4 Fig. 4. Flowchart for the code of the GA optimization approach.

Fig. 5 .

 5 Fig. 5. Sketches of the meshes in MTCR and STRSC.

  gi are given in Eq.(13).

  are calculated by Eq.[START_REF] Li | A passive anion-exchange membrane direct ethanol fuel cell stack and its applications[END_REF]. Finally, the individual which holds the greatest value of * i r in Eq.(16) will be selected as the suggested solution.

  minimum and maximum values of xi, respectively.

Fig. 6 .Fig. 7 . 5 . Results and discussion 5 . 1

 67551 Fig.[START_REF] Li | Gas-side fouling, erosion and corrosion of heat exchangers for middle/low temperature waste heat utilization: A review on simulation and experiment[END_REF]. Comparison of present result and experimental result in Ref.[START_REF] Chemisana | Optical performance of solar reflective concentrators: A simple method for optical assessment[END_REF].

Fig. 8 .

 8 Fig. 8. Pareto fronts of MTCR and STRCSC systems when α s,T =α s =45° and Α s =90°.

9Fig. 9 .

 9 Fig. 9. Comparison of flux distributions in MTCR between two aiming strategies when α s,T =α s =45° and A s =90°.

Fig. 10 .

 10 Fig.10. Comparison of flux profiles in STRSC between two aiming strategies when α s,T =α s =45° and A s =90°.

Fig. 11 .

 11 Fig. 11. Optimal flux distributions in MTCR at different aiming line numbers (n aim ) when α s,T =α s =45° and A s =90°.

f

  ST LCR max  i,opt

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Comparison of flux distributions in MTCR between two aiming strategies when α s,T =α s =42.5° and A s =90°.

Fig. 16 shows

 16 Fig.16shows the optimal result of S2 and the corresponding result of S1 in the MTCR system

Fig. 16 .

 16 Fig. 16. Optimal result of S2 and the result of S1 in the MTCR system on spring equinox.

  t s = 8.138 h: f MT =0.86, q l,max = 50.9 kW•m -2 , η opt =54.2%.(b) t s = 10.136 h: f MT =1.05, q l,max = 85.8 kW•m -2 , η opt =66.4%.(c) t s =12 h: f MT =1.09, q l,max =91.5 kW•m -2 , η opt =67.0%.

Fig. 17 .

 17 Fig. 17. Real-time flux distributions in MTCR on spring equinox when S1 is used.

  ts= 8.138 h: Waim=242 mm, fMT=0.05, ql,max= 23.2 kW•m -2 , ηopt=51.5%. (b) ts= 10.136 h: Waim=247 mm, fMT=0.04, ql,max= 36.3 kW•m -2 , ηopt=63.7%. (c) ts=12 h: Waim=245 mm, fMT=0.05, ql,max= 36.1 kW•m -2 , ηopt=64.7%.

Fig. 18 .

 18 Fig. 18. Real-time flux distributions in MTCR on spring equinox when S2 is used.

Fig. 19 .Fig. 20 .Fig. 22 .

 192022 Fig. 19. Optimal result of S2 and the result of S1 in the STRSC system on spring equinox.

  ) Comparison of the typical flux distributions between traditional one-line strategy (S1) and the present GA optimization strategy (S2) indicate that a compromise between the flux non-uniformity and the optical efficiency can be reached by S2 in both MTCR and STRSC systems.

Table 1

 1 Geometric and optical parameters of the LFR[START_REF] Sahoo | Steady state hydrothermal analysis of the absorber tubes used in Linear Fresnel Reflector solar thermal system[END_REF][START_REF] Häberle | The Solarmundo Project. Advanced technology for solar thermal power generation[END_REF][START_REF] Abbas | High concentration linear Fresnel reflectors[END_REF][START_REF] Serrano-Aguilera | Thermal 3D model for direct solar steam generation under superheated conditions[END_REF].

	Parameters	Value	Parameters	Value
	Primary mirror number n m	25	Diffuse reflectance of MTCR's wall	0.77
	Width of the field W field	20.4 m	Coating absorptivity in MTCR	0.93
	Width of the mirror W m	0.6 m	Coating diffuse reflectance in MTCR	0.07
	Length of collector L m	100 m	Aperture width of CPC	295 mm
	Distance between two mirrors	0.85 m	Thickness of CPC	5 mm
	Specular reflectance of mirror	0.92	Half acceptance angle of CPC θ a	56°
	slope error of mirror and CPC	1.0 mrad θ omax	3.37 rad
	Height of the tube in LFR H t	8.0 m	r 2	62.5 mm
	Cavity's bottom width of MTCR	655 mm	Tube's outer radius in STRSC r 1	35 mm
	Cavity's top width o of MTCR	300 mm	Envelope's outer radius R g	57.5 mm
	Half acceptance angle of MTCR	56 o	Thickness of envelope	3 mm
	Depth of MTCR	120 mm	Absorptivity/specular reflectance of CPC 0.05/0.95
	Number of tubes in MTCR n t	8	Glass transmittance/absorptivity	0.96/0.02
	Tube outer diameter in MTCR	36 mm	Glass specular reflectance	0.02
	Gap between two tubes in MTCR	3 mm	Glass refraction index	1.47
	Glass cover thickness in MTCR	3 mm	Coating absorptivity in STRSC	0.96
	Absorptivity of MTCR's wall	0.15	Coating diffuse reflectance in STRSC	0.04
	Specular reflectance of MTCR's wall 0.08	Angular radius of sun rays	4.65 mrad

  (b). The equations of the CPC profiles AB and BC in XrYrZr are

	expressed in Eqs.(2) and (3)[62].
	m	()	  16.1 ( 13) 0.10 0.10( 14) ,13   14.3 ( 13) 0.65 0.05( 14) , 20	19 25

Table 2 .

 2 Optimal solutions for MTCR and STRCSC systems when α s,T =α s =45° and Α s =90°.

		i	1	2	3	4	5	6	7	8	9 10 11 12	13
	MTCR,	g i	7	10	2	5	1	6 11	2	7 11	1	1	4
	n aim =11	i	14	15 16 17 18 19 20 21 22 23 24 25	26
		g i	10	3	9	6	4	5	9	9 10	5	7	7 255 mm
		i	1	2	3	4	5	6	7	8	9 10 11 12	13
	STRSC,	g i	4	3	6	1	7	4	1	8	9	1	7 11	2
	n aim =11	i	14	15 16 17 18 19 20 21 22 23 24 25	26
		g i	3	4 11	9	6	9 10	9	6	4	4	4 145mm

5, 130mm, 0.40, 68.8, 64.9% 7, 196mm, 0.41, 66.3, 64.2% 9, 176mm, 0.36, 65.3, 64.2% 11, 145mm, 0.32, 62.1, 64.2% 13, 151mm, 0.34, 60.9, 64.6% 15, 173mm, 0.34, 60.0, 64.1% 19, 198mm, 0.34, 61.4, 64.1% θ / ° LCR θ W E

Table 3 .

 3 Seventeen cases and corresponding time points on spring equinox.

	Case i	α s,T,i /°	t s /h	A s /°	α s /°
	1	10.0	6.716	94.0	10.0
	2	15.0	7.076	96.3	14.9
	3	20.0	7.434	98.6	19.8
	4	25.0	7.788	101.0	24.6
	5	30.0	8.138	103.7	29.3
	6	35.0	8.482	106.5	33.9
	7	40.0	8.822	109.6	38.3
	8	45.0	9.158	113.0	42.6
	9	50.0	9.488	116.9	46.8
	10	55.0	9.814	121.3	50.7
	11	60.0	10.136	126.4	54.4
	12	65.0	10.452	132.4	57.7
	13	70.0	10.766	139.5	60.7
	14	75.0	11.078	147.9	63.3
	15	80.0	11.386	157.5	65.2
	16	85.0	11.694	168.4	66.5
	17	90.0	12.000	180.0	66.9

Table 1 .

 1 Optimal solutions of the MTCR system for seventeen key cases. 1

	Case i	g 1 g 2	g 3	g 4	g 5	g 6	g 7	g 8	g 9	g 10	g 11	g 12	g 13	g 14	g 15 g 16	g 17 g 18	g 19 g 20	g 21 g 22 g 23 g 24 g 25	g 26
	1	4	9	2	6	4	1	3	9	1	6	1	4	6	10	2	8	8	7	3	7	4	2	3	5	2 mm
	2	4 10	2	6	7	2	1	10	1	4	1	3	6	10	6	8	8	9	2	8	4	3	4	7	5 260 mm
	3	4	4	3	8	10	3	8	1	10	9	10	6	4	2	8	9	5	3	4	3	4	5	6	6	9 292 mm
	4	4	3	9	6	2	4	1	11	6	9	1	11	2	4	8	7	9	9	2	8	7	6	6	7	8 260 mm
	5 10	7	9	1	10	6	4	4	1	11	1	1	2	11	8	6	3	9	9	9	10	8	5	3	5 242 mm
	6 10	7	9	1	7	5	5	4	1	11	1	1	2	10	10	6	3	10	9	9	10	9	7	3	5 245 mm
	7 10	7	8	1	7	5	4	1	5	11	1	1	2	11	8	11	4	9	9	9	9	8	6	4	5 246 mm
	8	7 10	2	5	1	6	11	2	7	11	1	1	4	10	3	9	6	4	5	9	9	10	5	7	7 255 mm
	9	8	6	9	2	5	5	3	4	5	11	1	1	1	11	7	8	2	10	10	9	9	9	7	4	5 254 mm
	10	9	7	5	1	7	6	4	4	4	10	1	1	3	11	7	8	1	9	11	11	10	9	6	3	5 256 mm
	11 10	6	5	3	7	6	3	4	4	11	1	1	1	11	8	7	2	11	10	10	9	10	4	4	5 247 mm
	12	9	6	8	2	6	5	4	4	3	10	1	1	2	11	7	7	1	10	11	11	9	9	6	4	5 253 mm
	13	9	8	7	2	7	6	4	4	3	11	1	1	1	11	7	8	3	11	10	10	10	9	5	3	5 251 mm
	14	8	7	8	3	6	6	3	4	1	11	1	1	1	11	11	8	3	10	10	6	10	8	6	4	5 252 mm
	15	8	6	7	2	5	5	2	4	4	11	1	1	1	11	7	8	1	10	10	10	10	9	6	4	5 248 mm
	16	3	7	7	2	6	5	2	4	5	11	1	2	1	11	8	8	1	11	11	9	9	10	6	4	4 247 mm
	17	6	4	6	2	2	8	1	1	8	11	6	8	6	11	11	1	10	4	5	3	10	4	9	3	8 245 mm

Table 2 .

 2 Optimal solutions of the STRSC system for seventeen key cases. 2

	Case i	g 1 g 2	g 3	g 4	g 5	g 6	g 7	g 8	g 9	g 10	g 11	g 12	g 13	g 14	g 15 g 16	g 17 g 18	g 19 g 20	g 21 g 22 g 23 g 24 g 25	g 26
	1	9	8	6	3	4	8	1	7	8	4	10	8	1	2	2	7	4	3	1	3	5	4	5	5	8	236mm
	2	8	5	5	3	7	2	1	1	8	7	2	10	6	6	2	5	3	7	7	7	3	9	4	7	8	189mm
	3	8	4	7	3	8	1	9	1	1	1	7	7	3	8	7	10	6	3	2	7	5	7	10	5	8	169mm
	4	9	6	4	6	1	6	7	1	9	8	2	10	2	2	7	3	5	9	10	6	6	2	6	7	5	141mm
	5 10	7	5	4	5	1	7	8	9	1	2	11	1	10	2	7	7	5	6	9	9	4	4	5	3	135mm
	6	9	3	7	1	4	8	8	11	3	1	3	6	3	2	10	11	10	2	9	10	5	7	4	7	6	151mm
	7	7	4	3	6	2	6	7	1	8	1	2	11	4	9	10	10	10	4	6	8	5	7	11	2	4	151mm
	8	4	3	6	1	7	4	1	8	9	1	7	11	2	3	4	11	9	6	9	10	9	6	4	4	4	145mm
	9	5	3	2	7	9	6	2	7	2	3	7	7	3	10	8	8	5	7	5	10	3	6	8	10	7	198mm
	10 10	5	3	4	6	1	7	1	1	9	9	6	5	1	4	10	5	11	2	11	5	8	9	4	5	147mm
	11 10	5	3	4	6	2	5	1	1	9	8	9	5	2	3	8	4	10	2	11	6	8	9	4	5	154mm
	12	8	8	7	1	3	10	1	1	9	8	1	8	4	9	11	6	11	2	5	5	11	5	9	3	8	155mm
	13 10	4	3	4	7	1	5	3	1	9	8	6	5	1	3	9	4	10	2	11	6	8	9	4	5	165mm
	14 10	4	3	4	8	1	5	1	1	9	9	6	5	1	4	8	3	11	2	11	6	8	9	4	6	159mm
	15 10	4	3	4	6	1	6	1	1	9	9	6	5	2	4	10	4	11	2	11	6	8	9	4	5	159mm
	16 10	3	3	4	6	2	5	1	1	9	8	6	5	2	3	10	4	11	2	11	6	8	9	4	5	161mm
	17 10	8	4	4	2	2	5	7	8	7	2	5	5	4	9	10	4	10	4	5	9	7	7	4	7	201mm

Acknowledgements

The study is supported by the Key Project of National Natural Science Foundation of China (No.51436007) and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51721004).

The authors would also like to thank the Innovative Talents Support Plan of China Postdoctoral Foundation.

Appendix

The optimal solutions of the aiming strategies under seventeen key cases for the MTCR and STRSC systems are given in Table 1 and Table 2, respectively. In the optimization, naim=11.