B.- Automatic generation of staggered fields as initial solutions
First, it is important to highlight that this procedure is not relevant for the proposed optimizer and its parallel execution. In fact, any other approach could be selected as long as the guidelines given about it were respected to generate good initial solutions. However, as the developed strategy can be of general interest in the application context, it is explained in detail next.
The heuristic designed for automatic generation of radial-staggered fields is inspired by the method proposed in [2]. A scheme of separating heliostats into groups is used. Every group consists of, at least, a `primary' ring and several `staggered' ones (if any). Primary rings define the unitary angular separation between heliostats in the same group (see Fig. 6). As clearly commented in [2, 3], while adding more rows to a same group, distances between heliostats are progressively increased. Hence, it is advisable to reset these parameters periodically, which leads to get different groups of heliostats. However, this aspect will be also exploited to generate multiple different but staggered fields.
Fig. 6.- Concepts of the proposed method for layout generation
The layout generator adapts its output only depending on the properties of heliostats and randomness. All the process is expressed in terms of the characteristic radius of heliostats, r=d/2, the minimum placement radius Rmin and the total number of heliostats to place, H (see Sec. 2). The only extra value is the maximum number of rows that could be included in the same group, let it be G. It is needed to randomly define how many rows every group will have. In any case, to avoid uncertainty, this variable has been fixed to G=6. This decision is based on the size of target plants and considering that in [1] the number of rows per group is in that range. A polar coordinate system is also superimposed to the initial one for convenience as done in [2]. North is at 0° and East is at 90° while radii are still computed in relation to the receiver base.
Fig. 7.- Computation of the angular unit of a group
Fig. 8.- Sample staggered field generated with the proposed method
The method starts by setting the radius of the first row, R0, which is always a primary one, Rmin+d/2. This is the minimum radius that respects the constraints linked to Rmin. Then, the angular space taken by every heliostat in the primary row, let it be α, is calculated. This is the unitary angular separation previously mentioned and it is obtained with Eq. (6), which is derived from Fig. 7. As can be seen, the angle computed is slightly smaller that the real one that would be necessary. This approximation, which is possible because no heliostat will be strictly next to a previous one, aims to save some potential room. Finally, the number of additional rows in the same group is computed as a random integer between 0 and G. These steps are required when a new group is started. After them, the first row is defined by adding more heliostats until the field limit is reached. All of them in a row share the same radius and an azimuth iα, where i is the index in that row. Staggering is simply achieved by doing i = 0, 2, ... for rows 0, 2, ... while i = 1, 3, ... for rows 1, 3, ... of that group. When a certain row Rz is complete, the next one in the same group has a radius equal to Rz+1 = Rz + d. However, if a new group is started, the radius of its first row is set to Rz+1 = Rz + 2d. These ideas are depicted in Fig. 6.
| α = 4 asin( r / (2R) ) | Eq. (6) |
Resulting fields are forced to feature east-west symmetry. Hence, when a heliostat is added in the east, if the total has not been reached yet, its equivalent can be also generated. Specifically, for a heliostat at position (Rz, nα), its symmetric one in the west would be at (Rz, 360° - nα). Besides, it must be noted that Rmax is not taken into account. In fact, for a high value of H and/or a small valid region, this constraint could be occasionally violated by some heliostats. Those fields could be then evaluated as infeasible solutions. However, based on their structure, that aspect could be temporarily ignored to maximize their contribution to pure random solutions. This is simply an implementation decision, though.
The procedure described is repeated until H heliostats have been placed. As required, i) it does not need optimizing any parameter and ii) a wide variety of different fields can be obtained due to its random component. They are not expected to be optimal but solid proposals to be studied. Otherwise, it would be extremely difficult for any pattern-free optimizer to autonomously achieve this kind of fields. Finally, in Fig. 8, a sample field of 400 heliostats placed as described is shown. Inputs are d/2 = 4.672565 m and Rmin = 20.0 m. As can be seen, the field has been limited to 90° (east) as for the test problem used at experimentation. Hence, it is important to mention that, if the optimizer is forced to keep the field in a certain angular range, this condition should be also inherited by this procedure. In any case, its consideration is also trivial: when a new heliostat has an azimuth angle higher than the limit, its deployment is canceled and the current row is finished.
References
[1] Ramos, A., Ramos, F.: Strategies in tower solar power plant optimization. Solar Energy 86(9), 2536–2548 (2012)[2] Siala, F.M.F., Elayeb, M.E.: Mathematical formulation of a graphical method for a no-blocking heliostat field layout. Renewable energy 23(1), 77–92 (2001)
[3] Stine, W.B., Geyer, M.: Power from the Sun. Public website: http://www.powerfromthesun.net/book.html (2001)