Optimization of a Heliostat Field by Multiobjective Particle Swarm Optimization (MOPSO) Algorithm Based on Energy, Exergy, and Economic Point of Views

This research is devoted to the energy, exergy, and economic analyses and optimization of a 19 heliostat field. The model of the heliostat solar receiver includes detailed geometric factors related to the 20 optical and thermal losses and efficiencies throughout the year. The main parameters of the thermal 21 performance of this system consist of energy and exergy efficiencies, and economic parameters are 22 investigated. By computing the energy, exergy, and economic analysis tools, they are applied for the analysis 23 of performance, and viability of the system’s operating in Tehran City, including the detailed information of 24 the environmental conditions of that location. For optimization purposes, 7 design variables related to 25 geometric specification of the heliostat field are selected and the related lower and upper bonds are selected. 26 Two target functions considered for the optimization are heliostat field exergy efficiency and payback period. 27 The economic feasibility results of this study reveal that the net present value is 58.84 million US$, the 28 payback period is 6.76 years, and the internal rate of return is 0.16. By considering the MOPSO algorithml, 29 the annual mean exergy efficiency is increased from the 30.9% to 34.3% while the heliostat field payback 30 period in reduced from the 6.76 to 4.3 years. 31


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The concentrated solar power (CSP) called the solar tower, is one promising technology to utilize solar 34 energy. The CSP solar system is a set of mirrors with the tracking system in different lines; sometimes it is 35 called a heliostat field [1]. This device is the core element in solar technology. The system is based on a 36 set of mirrors rotating on two axes, reflecting and concentrating the sunlight at the top of a spot or tower.

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There, the sunlight turns into heat with high-temperature, which can be used to produce steam or a hot showed that by increasing the number of mirrors, the optical efficiency of this type of heliostat was increased 45 too. Since by variation of the number of mirrors from 7 to 9 and 11, the optical performance of this system 46 varied to 20.65%, 27.13%, and 29.13%, respectively. Moreover, the concentration ratio of this device varied 47 from 6.74 to 9.77 due to changing the number of mirrors from 7 to 11. 48 Eddhibi et al. [8] studied a heliostat field, including different losses like shadowing losses, blocking and 49 atmospheric attenuation, and cosine loss in modeling and simulation. This study's results revealed 50 efficiencies associated with the cosine loss, atmospheric attenuation, and shadowing and blocking losses 51 of 82.41%, 95%, and 92%, respectively. The same investigation was conducted by other research groups, 52 with good results' agreement [9,10]. The optimization of a heliostat field with specified geometry was done 53 by Talebizadeh et al. [11] to find the best achievement of the heliostat field for the maximum heat absorption 54 of the solar receiver. Results revealed that increasing the height of the tower by 7.7% and reducing the 55 heliostat field by 19.5% lead to about a 4% increase and a 17% decrease of the heliostat field's total 56 efficiency and total area, respectively. The economic optimization of a heliostat field was conducted by Li 57 et al. [12] to obtain the maximum absorbed solar energy per unit cost of the specific heliostat field (Lhasa).

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In this study, the unit cost of collected energy was evaluated for different parameters including radius of the 59 field, optical parameters of the mirrors, the height of the tower, and the heliostat mirror cost. Results of this 60 optimization showed that increasing the radial distance by 6% leads to an increase of the unit cost of 61 collected energy from 12.49 to 12.98 MJ $ ⁄ . In the optimization of a hybrid combined power cycle and the 62 solar dish made by Saghafifar [13], the layout configuration of the heliostat was analyzed for two 63 arrangements. Also, different economic parameters such as net present value (NPV), payback period (PP), 64 levelised cost of electricity (LCoE), and life cycle saving, Knopf objective function were investigated for 65 these two configurations. Results of heliostat field optimization showed that the weighted efficiency was 66 58.6% for the radial -staggered layout and 58.4% for the spiral layout. This study showed that the LCoE 67 for both layouts is close to 34 $ MW h ⁄ .

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A new method was applied by Zi and Zhifeng [14] for the PS 10 location in China. In this research, the 69 optical effects of the heliostat were investigated considering a solar tracking system in the field layout.

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Heliostat field different parameters including mean optical efficiency, cosine efficiency, blocking and 71 shadowing effects, and atmospheric transmission were evaluated annually. A good agreement was 72 obtained when comparing the results of this study with those of the existing heliostat layout.

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The main innovative aspects of this study can be summarized as:

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 Thermodynamic and economic analyses of a heliostat solar receiver.

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 Evaluation of the exergy lost rate for each component of this typical solar receiver.    139 The distance between adjacent heliostats centers is the characteristic diameter that can be expressed as

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The distance between two adjacent mirrors is a function of the radial length from the tower as the layout 148 is radially staggered. Therefore, the farther from the tower the more space between mirrors. If this space is 149 higher than DM, installing an additional heliostat would be possible. When all these spaces have been filled 150 with additional heliostats, the field is settled. For the i th field, the azimuth angular spacing is obtained as

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[26]: 152 The radial distance between the first row of i th field and the tower (R i ) can be obtained as [26]: and for every line of i th field, the number of heliostats is expressed as [26]: 154 For the i th field, the number of heliostats rows can be obtained [26]: 155 where R is the space between the base of the tower and each heliostat, and σ 1 can be evaluated as [30]: 167 σ 1 = d 3 √(tan (δ s )) 2 + (tan(ε surf )) 2 + (tan(ε track )) 2 where ε track and ε surf are the errors caused by tracking and angular deviation due to surface defects, which 168 are considered to be equal to 1° and 2°, respectively [30]. δ s denotes solar declination, evaluated in 169 Appendix A.

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In Eq. (13) r ap is the effective size of the receiver opening, which can be obtained as [30]: where θ R denotes the angle between the vertical direction and the reflected irradiation, A ap means total 172 aperture area, cos (θ R ) being obtained as [30]: 173 where φ is latitude and α S is the angle of solar altitude given in Appendix A.

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The focus of the present study is not on the energy storage subsystem, but on the heliostat and receiver where h is the specific enthalpy, subscripts in and out are inlet and outlet streams of the control volume, 179 respectively, and Q̇n et is the heat power received by the control volume, which is evaluated as: 180 In the heliostat field, the solar energy input can be evaluated as [31]: where A ap is the total aperture area, G b is the solar direct beam irradiation, and α is the absorption factor of 182 the solar receiver. The thermal power losses of the system are evaluated as [32]:

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Q̇l oss = Q̇l oss,conv + Q̇l oss,rad where subscripts conv and rad mean convection and radiation heat transfer from the solar receiver to the 184 environment. The radiative thermal power loss can be obtained as [32]:

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Q̇l oss,rad = σ. ε re . A re (T re 4 − T amb 4 ) where σ is the Stefan-Boltzman constant, ε is the receiver surface emissivity , A re is the surface area of the 186 solar receiver, and T is its absolute temperature. Subscript amb means ambient.
Uwind and υ m mean wind speed and kinetic viscosity.

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The overall ENE of the proposed system is expressed as: where subscript p refers to the circulation pump, and Q̇n et is the net thermal power received by the receiver 197 and Ẇ is the mechanical power needed for the circulation pump operation, evaluated as where h 2s is the specific enthalpy of the stream leaving the pump if the pumping process is assumed to be 199 isentropic. The pump efficiency is considered 85%.
where T is the absolute temperature, h is the specific enthalpy, and s is the specific entropy. R i is the 205 particular gas constant of chemical species i in the stream, ex chi its specific chemical exergy, x i its mass 206 fraction and y i its molar fraction. g, V, and z are gravitational acceleration, velocity, , and height, 207 respectively. Subscript 0 refers to the environmental (equilibrium) conditions. 3 Heliostats The symbols and subscripts in where Y heating is the capacity of heating production during a year, and k heating denotes the specific cost of 245 that heating.

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The inflation rate effect can be evaluated by[47]: In which n is the number of years. i is the inflation rate which is equal to 3.
where r is the discount factor (considered 3% in this study 253 254

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The MOPSO algorithm is one kind of of the PSO algorithm to solve multi-objective optimization. The

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MOPSO and PSO algorithms use an update of the particle location and velocity in the same manner. As it 258 is known in the PSO algorithm the update for the velocity and location of a particle can be found by 259 calculation of two parameters. These two parameters are the optimal solution that each particle is obtained

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In the above equations, a and d are dimensional search space, ( ) , ( ), ( + 1), ( + 265 1), , , w, c1 and c2, and r1 and r2 are the existing particle velocity, existing particle location, newest 266 particle velocity, newest particle location, the best result that the particle, the best result that the whole 267 population has obtained thus far, inertia weight, acceleration coefficients, and random numbers within the 272 Two selected objective functions for the optimization are as follows:

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Objective I = η exergy (50) The design variables and the range of them considered in this optimization are depicted in Table 4.           presented. In the optimized point, the EXE reaches 38.9% and the PP is reduced to 4.3 years. Table 9 357 shows the optimized decision variables.

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From economic analysis are highlighted a PP is of 6.76 years, slightly higher than the SPP of 6.04 years.

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These are not far from that of many other systems for renewable energy capture and conversion, indicating 473 the economic viability of the system considering its expected lifetime operation for 20 years.    where subscript surf means surface.

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Funding: There is no financial support provided by any specific governmental and institutional organization 501 to complete this manuscript.

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Conflict of Interest: The authors declare that they have no conflict of interest.