Keywords

30.1 Introduction

With appropriate support, Concentrating Solar thermal Power (CSP) could contribute to 9.6% of global electricity from solar power alone by 2050 [1]. CSP has been developed under four technology categories: Power Tower, Parabolic Trough Collectors, Linear Fresnel Reflector (LFR), and Solar Parabolic Dish. Although LFR has shown potential low energy costs due to its simple design, its annual conversation efficiency is still low (8–10%) compared to the other technologies [2]. Mathematical optimization can be used to improve the efficiency of LFR systems and reduce energy costs, thus making this technology more attractive. In line with this, efficiency improvements to an existing porotype were achieved in 2015 using an analytical design approach [3], although the authors only carried out an optical analysis without optimizing the mirror field design variables. In [4], the geometry of a trapezoidal cavity receiver of an LFR plant was designed to minimise heat loss using ray tracing to simulate heat flux patterns on the receiver tube. However, they did not conduct a detailed analysis of the effect of the mirror field on the receiver. Another optical optimization of the arrangement of LFR mirrors was conducted in [5] considering mirror width, spacing, and focal length. The authors assumed a flat horizontal receiver and suggested including a thermal analysis with a more detailed receiver in future work. Barbón et al. [6] studied the lateral variation of solar energy received on an absorber of small-scale LFR plants. Their main focus was end loss and reflected light loss, which is typically disregarded as they are often negligible in large-scale plants. In 2017, Abbas et al. [7] addressed the optical design of an LFR considering different numbers of mirrors, filling factors, and collector widths. However, the authors made use of a set of predesigns for a solar field without incorporating ray tracing models to reduce the required computational resources facilitating the optimization process. The optimal design of LFR systems making use of coupled optical ray-tracing and thermal models and considering both mirror field and receiver design parameters simultaneously has not been sufficiently investigated. The aim of this study is to identify a suitable optimization method which can solve this challenging problem.

30.2 Methodology

A mathematical model is developed by combining energy balance, ray tracing, and thermal modelling. The objective function for the LFR optimal design is given by the total theoretical efficiency (\({\eta }_{total,th}\)) assuming a coupled Carnot cycle [8, 9]. The design variables are the number of mirrors (nm), mirror width (wm), mirror spacing (sm), receiver height (hr), and receiver width (wr).

Seven optimization methods with different characteristics are tested to investigate their efficiency, as well as their different trade-offs between accuracy and required computational effort. Namely, interior-point (IP), sequential quadratic programming (SQP) and active-set (AS) methods are derivative-based; pattern search (PS), Nelder-Mead simplex (NMS) and simulated annealing (SA) methods are derivative-free; whilst genetic algorithm (GA) is both a derivative-free and population-based method. This particular choice of methods is based on the literature and on their availability within Matlab’s optimization toolbox.

The optimization problem is initially reduced to a single design variable, namely the receiver height. The problem is then solved using the PS, NMS, SA and GA methods. The worst-performing algorithm is eliminated (NMS, referred to as fminsearch in Matlab), and the other three are tested on the original five-dimensional optimization problem discussed before. It is fair to note that the number of mirrors (nm) is a discrete variable. In order to avoid a mixed-discrete problem, a four-dimensional continuous problem is solved instead for each of a few selected values of nm.

30.3 Models

Based on the conservation of energy principle, the energy output of the LFR system \(\left({Q}_{o}\right)\) can be found from the difference between the heat input \(\left({Q}_{in}\right)\) to the receiver and the heat loss to ambient air \(\left({Q}_{loss}\right)\) [10].

$${Q}_{o}={Q}_{in}-{Q}_{loss}$$
(30.1)

After reflection on the mirrors, the solar energy incident over the receiver area, \({A}_{r}\), is absorbed by the working fluid as heat, increasing the receiver temperature (\({T}_{r}\)). The absorbed solar energy becomes the heat input to the system (Qin). A portion of absorbed energy is lost to the surrounding, which is at temperature \({T}_{a}\). The heat output of an LFR system can therefore be described by [11, 12]

$${Q}_{o}={I}_{r,abs}{A}_{r}-{A}_{r}{U}_{L}\left({T}_{r}-{T}_{a}\right)$$
(30.2)

where \({I}_{r,abs}\) and \({U}_{L}\) are absorbed solar radiation and heat loss coefficient of the receiver.

There are analytical methods and numerical approaches to calculate \({I}_{r,abs}\) for different LFR systems. SolTrace tool has been used in this study considering its ability to model complex geometries with good accuracy [13]. \({U}_{L}\) is calculated using analytical and empirical correlations considering conduction, convection and radiation losses of the receiver [11, 14].

If solar energy received on the effective aperture of all mirror elements is \({E}_{m}\), the total theoretical efficiency (\({\eta }_{total,th}\)) can be obtained as in Eq. (30.3), assuming coupled Carnot cycle [8, 9].

$${\eta }_{total,th}=\frac{{Q}_{o}\left(1-\frac{{T}_{a}}{{T}_{r}}\right)}{{E}_{m}}$$
(30.3)

The objective function to be optimized is \({\eta }_{total,th}\) and the design variables are the number of mirrors (nm), mirror width (wm), mirror spacing (sm), receiver height (hr) and receiver width (wr), as shown in Fig. 30.1.

Fig. 30.1
A 2 D view and schematic diagram of the L F R system labeled a and b. B has the following labels. Insulation, glass cover, secondary reflector, absorber tube, receiver, primary mirrors, fixed structure, and tracking mechanism.

a Schematic of a 2D view of an LFR system with a trapezoidal cavity receiver, b a schematic diagram of a typical LFR system

Full size image

30.4 Results and Discussion

30.4.1 Candidate Derivative-Free and Population-Based Methods

Two derivative-free methods, pattern search and fminsearch solver in Matlab, were selected and eight simulations were carried out for each method. Figure 30.2a shows that all eight optimization results are gathered around the neighbourhood of the global optimum. These results show a very good agreement with the exhaustive search. Two global optimization methods, Simulated Annealing and GA, were also used to check their behaviour for the objective function. Five simulations were run for Simulated Annealing and three simulations were run for GA. As seen in Fig. 30.2b, all these results lay between 22 and 23 and were very close to the global optimum, hr = 23. All simulations were run using default settings except the function tolerance, which was reduced to 10–3 to minimise the effect of numerical disturbance of the model. It is seen that simulated annealing and GA have shown the best results in consecutive runs showing close solutions to each other.

Fig. 30.2
Two line and scatter plots of total theoretical efficiency versus receiver height. In both, the exhaustive line increases. Pattern and f min searches are plotted between the receiver heights of 21 and 27 and efficiency at 0.25 in a. Simulated annealing and G A are plotted at (23, 0.25) in b.

a Comparison of exhaustive search results against two derivative-free methods b against a population-based method (GA) and simulated annealing

Full size image

30.4.2 A Candidate Optimization Method for a Multidimensional Problem

Optimization was expanded to four continuous variables (wm, sm, hr, wr) while keeping the fifth one (nm) at a few selected discrete values. Thus a four-dimensional continuous optimization problem is solved for each value of nm. Only the PS, SA and GA algorithms are tested here, which are the ones which performed best in the one-dimensional problem. In order to assess the accuracy of the optimizers, an exhaustive search was performed discretizing all four continuous variables (see Table 30.1).

Table 30.1 Design variables and their discretization for exhaustive search
Full size table

The ten best simulation results from the exhaustive search for nm = 10 are shown in Table 30.2. It can be observed that the differences between the efficiency of the global optimum and that of neighbouring designs are very small. In fact, the first two solutions may be deemed global maxima if rounding efficiency to three decimals. Then, the global maxima would be found at \({h}_{r}=9\) m, \({w}_{r}=0.9\) m, \({w}_{m}=0.9\) m, whilst \({s}_{m}\) could take either 0.09 m or 0.13 m. However, the design variables are discretised. For the continuous case, the actual global optimum may lie nearby. Therefore, it is possible that a global search algorithm is able to find even better solutions than this so-called exhaustive search.

Table 30.2 Results of the exhaustive search for nm = 10
Full size table

The optimal designs returned by PS, SA and GA are shown in Table 30.3. It is seen that all solvers achieve a maximum \({\eta }_{total,th}\) between 0.170 and 0.172 (i.e. 17–17.2%). Figure 30.3 shows a comparison of these results in terms of the maximised solution.

Table 30.3 Optimization results of three optimization algorithms for nm = 10
Full size table
Fig. 30.3
A triple bar graph of total theoretical efficiency versus optimization algorithms. In the genetic algorithm, the second and third runs are maximum, while the first run is maximum in simulated annealing.

Comparison of maximised ηtotal,th using three optimization algorithms (nm = 10)

Full size image

The same process of comparing the exhaustive search and optimal solutions was repeated for nm = 30 and nm = 54. Results show a similar behaviour to nm = 10, where an optimal solution was found within the same region of the search-space as the exhaustive search. Although the three optimization algorithms found better solutions, it should be noted that the discretization of continuous search space has greatly affected these results. A comparison of the results of these two different simulations is shown in Fig. 30.4. The average simulation times for the three algorithms are shown in Fig. 30.5.

Fig. 30.4
2 triple bar graphs of total theoretical efficiency versus optimization algorithms. A. The second and third runs are higher in the genetic algorithm. B, the first run is high in the genetic and simulated algorithms and the second run is high in the pattern search.

A comparison of three optimization algorithms: a nm = 30; b nm = 54

Full size image
Fig. 30.5
A triple bar graph of simulation time versus optimization algorithms. At n m = 10, 30, and 54, the pattern search is low, the genetic algorithm is moderate, and the simulated annealing is high.

Average simulation time for all three optimization algorithms

Full size image

30.5 Conclusion

After considering all the simulation results, it is seen that pattern search, simulated annealing and genetic algorithm provided acceptable solutions. The differences between the best and the worst solutions using pattern search, simulated annealing and GA were 0.0012–0.0020, 0.0008–0.0018 and 0.0003–0.0020 respectively. The largest difference found was 0.002, equal to 0.2% of the total theoretical efficiency, which is more than enough for many practical applications. Since all algorithms provided acceptable results, the selection of an algorithm for further work can be decided based on the simulation time and the ease of handling the discrete design variable, number of mirrors, within the optimization codes.

It is clearly seen that the pattern search is 8–10 times faster than the other two algorithms in terms of computational time. Increasing the simulation time for larger mirror numbers is also significantly lower in comparison. The genetic algorithm seems to be slightly better than simulated annealing taking 10–20 min less simulation time. However, it is important to note that for population-based methods like GA, the initial setup is usually time consuming and adding extra variables is relatively easy.