Solving Optimization Problem Of Space Factor Of Multiple__CPV Trackers Using “Butterfly Approach”
Solving Optimization Problem Of Space Factor Of Multiple CPV Trackers Using “Butterfly Approach”
Kenji Araki
Daido Steel Co., Ltd. 9 Takiharu-cho, Minami-ku, Nagoya 457-8712 Japan Abstract. Optimization of land use to multi-tracker CPV system is discussed by mathematical approach. Optimization problem using butterfly plot (Contour plot on the shading to adjacent tracker) is discussed to seek optimum allocation pattern. With initial solutions given by this optimum allocation pattern, numerical optimization calculation is done to obtain the optimum allocation including, skew angle to the North-South axis, aspect ratio between X and Y pitch and optimum panel aspect ratio. It is suggested that there are two candidate of optimum allocation pattern.
Keywords: CPV, Tracker, Space Factor, Optimization, Shading Loss
PACS: 88.40.fc Solar Energy Modeling and Analysis; 88.40.jp Multi-junction Solar Cells
INTRODUCTION
Generating as much energy as possible from a given land is one of the common interests to CPV technologies. Many papers on calculation of shading loss were published [1] [2] for increasing space factor (ratio of total panel area per given land). Most of them were simple geometrical shading calculation without consideration of PV-specific physics and even without experimental proof. However, some of them including our old researches consider spectrum effects, mismatching loss, seasonal sunshine duration fluctuation and possibility of skewed array configuration. The shading loss model considering CPV-related constrains was proved to meet to the field performance [3]. At the same time, design methods for practical use of the land based on that precise calculation were proposed and were actually used for CPV system design [3].
Beside such practical approach, it is also important to develop mathematical solution of “Optimization problem of land use” or specifically, “maximization problem of space factor with given accepted shading loss”. With this approach, we will be able to know what crucial factors to shading loss calculations are and thus improve the algorithm by enhancing accuracy or simplifying. It is also useful to set the “initial value of iterative or interactive simulations” by giving mathematical solution from given conditions without consideration of practical constrains. IMPLICATIONS FROM “BUTTERFLY”
Butterfly contour plot is very useful tool for understanding geometrical constrains on allocation of trackers. The original concept was invented by Dobón [4] [5]. The butterfly plot is a contour plot that represents the shading influence to adjacent arrays moving parallel to the one located in the plot origin. The butterfly plot gives a rough idea on the possible and preferred position (distance along X or Y direction, for example) of the adjacent trackers and thus gives a hint to allocation optimization.
The original butterfly plot only considers the ratio of shadow on the panel from an adjacent tracker. It is apparent the energy harvest from the array is different from the value directly calculated from the shaded ratio. Several improvement including weighting and integration with CPV performance model were done and that improved method actually used for the design of 30 kW CPV power plant [6] [7] [8].
The improved butterfly is shown in Figure 1, considering seasonal fluctuation of DNI, spectrum effect by different height of sun, nonlinear response of CPV output, partial shading of strings, overlay of shadows from different trackers, and influence from
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Note that the shape of contour is completely different from the original butterfly from Dobón [4] [5]. The shape of contour has a great influence to simulation result and advanced butterfly was chosen in this study.
One of the apparent implications from butterfly shape of the contour plot (Figure 1) is a hint to give the best direction of array allocation.
from butterfly plot.
For example, the distance in given directions in the butterfly contour roughly corresponds to the pitch of the tracker allocation. Precisely speaking, a single tracker receives influence from multiple surrounding trackers and a butterfly only says about the influence to a single adjacent tracker, but it gives information from the most significant tracker and gives good implication. Figure 2 shows examples of the pitch of trackers in N-S direction, E-W direction and 45 degree directions from Figure 1. Apparently, the pitch in N-S direction is smaller than that of E-W direction. This means that the allocation pitch in E-W direction tends to be larger than that of N-S direction because E-W direction is more sensitive to shading. In case that the trackers are allocated in 10 degree off from N-S and E-W axis, the pitch in both X and Y directions in the butterfly contour are larger than that of exactly N-S and E-W directions. Therefore, unless there is constrains from site boundary for example, allocation along N-S and E-W directions is a better choice. Targeting the solution of the maximum number of trackers in a given land area, namely the best and maximum space factor, the product of X-pitch and Y-pitch will be minimum. The minimum product will be often given, if X = Y, that is another candidate of the optimum allocation, corresponding 45 degrees off from both X and Y directions, that is often called Chessboard allocation.
APPROACH FOR OPTIMIZATION
The real optimum solution may vary according to conditions. Although butterfly gives direct geometrical hint to allocation directions and reasonable initial values of X and Y pitch, the real optimization should be done by numerical calculation.
FIGURE 3. Explanation of optimum calculation on the butterfly; the target is to minimize the area of rectangular (hatched area), cross product of two pointing vectors on the contour from given shading loss.
One of the practical initial value set of the optimum direction and distance is given from the minimum value of cross product of a pair of orthogonal vectors along the contour from given shading loss (Figure 3). That product corresponds to the unit of the area that one tracker occupies in the infinite land as well as “mesh” in one of the interactive method of tracker allocation [3].
One of the example of the optimum allocation pattern results is shown in Figure 4.
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In this calculation, all the variables and parameters were treated as non-dimensional. For example, the panel dimensions (width and height) and allocation dimensions (X pitch and Y pitch) were converted to aspect ratios of the panel and the pitch from the given space factor. Figure 4 indicates the shading loss that is not directly calculated by the cross product of a pair of orthogonal vectors along the contour from given space factor. Since one of the purposes of this study is to obtain general solution of typical conditions, the following crucial conditions were omitted. (1) Mismatching loss by partial shading of string is ignored for avoiding complexity. (2) Edge effect of the array, for example, the tracker in the north-west corner does not receive shading loss from the one in the east. In other words, the simulation is done by infinite number of trackers.
Figure 4 suggests two candidates of optimization. One is the 0 degree skew, namely trackers are aligned exactly along N-S and E-W directions. In this allocation the aspect ratio is below 1 (approx. 0.8 in Figure 4), namely the E-W pitch is greater than N-S pitch. Another candidate is 45 degree skew, namely tracker are allocated like a chessboard. In this allocation, tracker pitches in two directions is the same, namely aspect ratio is 1.
This trend is insistent to different sets of parameters. The optimum aspect ratio may be different, but the fact that there are two candidates of optimum allocation patterns (0 degree skew and 45 degree skew) seems to be consistent.
RESULTS AND DISCUSSIONS
Supposing that there are consistently two
candidates (0 degree rectangular and 45 degree chessboard pattern), shading loss calculations were done. The algorithm of calculation of shading loss considering nonlinear correction of CPV output influenced by seasonal factor is found in the literatures [2] [5] and [7].
The target of this optimization calculation is to find the optimum panel shape (height/width ratio of the panel), the optimum aspect ratio of N-S and E-W pitches and corresponding optimized shading loss by both chessboard allocation and N-S/E-W rectangular allocation. The common parameters in the calculation are found in Table 1.
TABLE 1. Common parameter table to optimization
Parameter Value
Space Factor 0.2 Amplitude of seasonal DNI modulation 0.5
The calculation results are shown in Figure 5 to 8 with different levels of latitudes. In each figures, the X
axis corresponds to aspect ratio of the mesh, namely the ratio of X pitch and Y pitch. Note that 0.27 of aspect ratio is minimum because of interference of rotation circle of each tracker with given space factor. loss: 0 degree of latitude. loss: 20 degree of latitude. loss: 40 degree of latitude.
loss: 60 degree of latitude.
The following implications were obtained from these calculation results;
(1) The optimum aspect ratio of the X-Y allocation pitch is strong function of the panel shape (aspect ratio of panel), especially in low latitude area. It is not wise to use high aspect (more than 1).
(2) Most of the cases, chessboard allocation gives less shading loss than that of optimized N-S / E-W rectangular allocation.
Generally speaking, not presented in the Figure 5 to 8, the optimum allocation pattern receives great influence from the aspect ratio of tracker panel and seasonal fluctuation of sunshine duration (seasonal fluctuation of DNI). The area where sunshine duration is good in summer has a great advantage of packing many trackers in a given land by sacrificing energy production in winter.
This approach was used for the design of 50 kW CPV demonstration plant by NGCPV project [9]. The system performance has been monitored by NGCPV European partners and the designed shading loss was 2 % [10].
CONCLUSIONS
Optimization problem using butterfly plot (Contour plot on the shading to adjacent trackers) and numerical calculation were discussed and suggested that there are two candidates of optimum allocations that may be used to initial value or decreasing degree of freedom to optimization calculation.
Optimization of tracker allocation is done by the model confirmed by a field evaluation with the help of above approach. The shading loss is influenced by many factors and optimum conditions, for example aspect ration of the panel varies accordingly.
ACKNOWLEDGMENTS
This work has been partially supported by New Energy and Industrial Technology Development Organization (NEDO) in Japan as a part of the “Research and Development on Innovative Solar Cells (International Research Center for Innovative Solar Cell Program)” and by the European Commission through the funding of the project NGCPV EUROPE JAPAN (EU Ref. European Union’s Seventh Programme for research, technological development and demonstration under grant agreement No 283798).
REFERENCES
1. P.J.Pérez, G.Almonacid, J.Aguilera, P.G.Vidal, J. de la
Casa, I.Luque-Heredia, M. Doménech. “Multi-trackers
Systems Calculation of Losses due to Self-shadowing”,
Proc. 4th ICSC, 2008, pp. 201.
2. H. Capdevila, A. Marola, Andrea and M. Herrerías, High
resolution shading modeling and performance simulation
of sun-tracking photovoltaic systems, AIP Conference
Proceedings, 2013, 1556, 201-204
3. K. Araki, “Two interactive and practical methods for
optimization of tracker allocation in a given land,” CPV-
8. AIP Conference Proceedings, Volume 1477, 2012, pp.
244-247
4. J. Monedero and F. Dobón, “Two-Axis Tracking System
Tetra-Track: System Description and Study for Optimized Space Factor”, CD-ROM of Proc. SCC2003,
2003
5. J. Monedero, F. Dobón, A. Lugo, P.Valera, R.Osuna, L.
Acosta, G.N. Marichal, “Minimizing Energy Shadow
Losses for Large PV Plants”, Proc. 3rd WCPEC, 2003
6. K. Araki, I. Kumagai and H. Nagai, “Theory and
Experimental Proof of Shading Loss of Multi-Trackers
CPV System”, Proc. 5th World Conference on Photovoltaic Energy Conversion, 2010, 114-117
7. K. Araki, T. Yano, and Y. Kuroda, “30 kW Concentrator
Photovoltaic System Using Dome-shaped Fresnel Lenses”, Optics Express, Vol. 18, Issue S1, 2010, pp.
A53-A63
8. Y. Kemmoku et al., “Examination of the Output
Characteristic Curve of a 500X Concentrator PV System
Considering Seasonal Variation of the System Characteristic”, Proc. 22nd EU-PVSEC, 2007, pp. 152
9. K. Araki, C. Domínguez, H. Nagai, I. Luque-Heredia, K.
Nishioka, K. Hobo, I. Anton, M. Futo, G. Sala, K.
Tamura and I. Kumagai, “Improvement of CPV System
Performance by NGCPV Collaboration”, 38th IEEE PVSC, 2012
10. M. Castro, C. Domínguez, R. Nu?ez, I. Antón, G. Sala,
and K. Araki, “Detailed effects of wind on the field
performance of a 50 KW CPV demonstration plant,” 9th
International Conference on Concentrating Photovoltaic
Systems, Miyazaki, Japan, 2013. AIP Conference Proceedings, 1556, 256-260, (2013).
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