Parametricism allows people to use a computer to generate a lot of designs for evaluation, comparison, and selection. If the choice of the parameters is driven by a genetic algorithm, then the computer will also be able to spontaneously evolve the designs towards one or more objectives. In this article, I use the design of the heliostat field of a concentrated solar power tower as an example to illustrate how this type of generative design may be used to search for optimal designs in engineering practice. As always, I recorded a screencast video that used the daily total output of such a power plant on June 22 as the objective function to speed up the calculation. The evaluation and ranking of different solutions in the real world must use the annual output or profit as the objective function. For the purpose of demonstration, the simulations that I have run for writing this article were all based on a rather coarse grid (only four points per heliostat) and a pretty large time step (only once per hour for solar radiation calculation). In real-world applications, a much more fine-grained grid and a much smaller time step should be used to increase the accuracy of the calculation of the objective function.
Video: The animation of a generative design process of a heliostat field on an area of 75m×75m for a hypothetical solar power tower in Phoenix, AZ.
|Figure 1: A parametric model of the sunflower.|
Heliostat fields can take many forms (the radial stagger layout with different heliostat packing density in multiple zones seems to be the dominant one). One of my earlier (and naïve) attempts was to treat the coordinates of every heliostat as parameters and use genetic algorithms to find optimal coordinates. In principle, there is nothing wrong with this approach. In reality, however, the algorithm tends to generate a lot of heliostat layouts that appear to be random distributions (later on, I realized that the problem is as challenging as protein folding if you know what it is — when there are a lot of heliostats, there are just too many local optima that can easily trap a genetic algorithm to the extent that it would probably never find the global optimum within the computational time frame that we can imagine). While a «messy» layout might in fact generate more electricity than a «neat» one, it is highly unlikely that a serious engineer would recommend such a solution and a serious manager would approve it, especially for large projects that cost hundreds of million of dollars to construct. For one thing, a seemingly stochastic distribution would not present the beauty of the Ivanpah Solar Power Facility through the lens of the famed photographers like Jamey Stillings.
In this article, I chose a biomimetic pattern proposed by Noone, Torrilhon, and Mitsos in 2012 based on Fermat’s spiral as the template. The Fermat spiral can be expressed as a simple parametric equation, which in its discrete form has two parameters: a divergence parameter β that specifies the angle the next point should rotate and a radial parameter b that specifies how far the point should be away from the origin, as shown in Figure 1.
|Figure 2: Possible heliostat field patterns based on Fermat’s spiral.|
When β = 137.508° (the so-called golden angle), we arrive at Vogel’s model that shows the pattern of florets like the ones we see in sunflowers and daisies (Figure 1). Before using a genetic algorithm, I first explored the design possibilities manually by using the spiral layout manager I wrote for Energy3D. Figure 2 shows some of the interesting patterns I came up with that appear to be sufficiently distinct. These patterns may give us some ideas about the solution space.
|Figure 3: Standard genetic algorithm result.|
|Figure 4: Micro genetic algorithm result.|
Then I used the standard genetic algorithm to find a viable solution. In this study, I allowed only four parameters to change: the divergence parameter β, the width and height of the heliostats (which affect the radial parameter b), and the radial expansion ratio (the degree to which the radial distance of the next heliostat should be relative to that of the current one in order to evaluate how much the packing density of the heliostats should decrease with respect to the distance from the tower). Figure 3 shows the result after evaluating 200 different patterns, which seems to have converged to the sunflower pattern. The corresponding divergence parameter β was found to be 139.215°, the size of the heliostats to be 4.63m×3.16m, and the radial expansion ratio to be 0.0003. Note that the difference between β and the golden angle cannot be used alone as the criterion to judge the resemblance of the pattern to the sunflower pattern as the distribution also depends on the size of the heliostat, which affects the parameter b.
I also tried the micro genetic algorithm. Figure 4 shows the best result after evaluating 200 patterns, which looks quite similar to Figure 3 but performs slightly less. The corresponding divergence parameter β was found to be 132.600°, the size of the heliostats to be 4.56m×3.17m, and the radial expansion ratio to be 0.00033.
In conclusion, genetic algorithms seem to be able to generate Fermat spiral patterns that resemble the sunflower pattern, judged from the looks of the final patterns.