Application of Minimum Energy Effect to Numerical Reconstruction of Insolation Curves
Abstract
:1. Introduction
1.1. Generally on Solar Insolation
- Costly construction;
- Relatively low efficiency;
- Limited service life;
- Environmental issues associated with subsequent disposal;
- Usability depending on geographical location (especially latitude);
- Unstable output power in connection with the pollution of panels, but especially with weather conditions (clouds and precipitation, air pollution of natural origin, and due to human activity).
1.2. Processing of Solar Data
1.3. An Overview of Numerical Methods for Solving Solar “Field” Distributions
1.4. Motivation
1.5. Structure of the Article
2. Minimum Energy Effect
Feynmann’s Approach to the Problem
- A: time t0, height h0
- B: time t2, height h2
- C: time t1
- for t belonging to <t0, t1>
- for t belonging to <t1, t2>
3. Analytical Calculation of Solar Radiation
4. Application of the MEE to Solar Insolation Analysis
4.1. Partial Quadratization
- from sunrise to 9:00 a.m.;
- from 9:00 a.m. to 3:00 p.m.;
- from 15:00 p.m. to sunset.
4.2. Application to Insolation Curve
- (A)
- A universal solution covering the period from sunrise to sunset:
- (B)
- The solution by dividing the period of sunshine into three intervals.
Principle of the Method Application in a Single Time Interval
4.3. Possible Approaches to Solution
5. Method Verification
Point A: | 9:00, | 0.92595772 kW/m2 | (9; 0.92595772) |
Point B: | 15:00, | 0.925957 kW/m2 | (15; 0.925957) |
Point A: | 0:00, | 0 kW/m2 | (A(x) = 0; A(y) = 0) |
Point B: | 6:00, | −0.00000072 kW/m2 | (B(x) = 6, B(y) = −0.00000072) |
The Effect of the Number of Elements on the Calculation Accuracy
6. Discussion on Results and Challenges
6.1. Influence of Adaptive Meshing
6.2. Higher-Order Elements
6.3. Subsequent Operations Such as Integral, Derivatives, etc.
6.4. Nonlinear Weighting Functions
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Day | R2 Whole Day | R2 for Three Daily Intervals | ||
---|---|---|---|---|
Sunrise–Sunset | Sunrise–9:00 | 9:00–15:00 | 15:00–Sunset | |
Day35 | 0.96698216 | 0.99987956 | 0.99438629 | 0.99987956 |
Day80 | 0.94087229 | 0.99629058 | 0.99789852 | 0.99629058 |
Day126 | 0.93315426 | 0.99188302 | 0.99910400 | 0.99188302 |
Day172 | 0.93689507 | 0.99162372 | 0.99927887 | 0.99162372 |
Day219 | 0.93501843 | 0.99202784 | 0.99908987 | 0.99202784 |
Day266 | 0.94193598 | 0.99646385 | 0.99783519 | 0.99646385 |
Day310 | 0.96693848 | 0.99989190 | 0.99433739 | 0.99989190 |
Day355 | 0.98107886 | 0.99993787 | 0.99086938 | 0.99993787 |
C1(y) [kW/m2] | S [Wh/m2] |
---|---|
0.03 | −0.0010728 |
0.04 | −0.0012970 |
0.05 | −0.0014546 |
0.06 | −0.0015456 |
0.0686395 * | −0.0015704 |
0.07 | −0.0015698 |
0.08 | −0.0015274 |
0.09 | −0.0014184 |
Number of Elements [-] | S [Wh/m2] | S [%] |
---|---|---|
2 | −0.001570476 | 75.00 |
4 | −0.001963062 | 93.75 |
6 | −0.002093936 | 97.22 |
12 | −0.002079394 | 99.31 |
24 | −0.002090300 | 99.83 |
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Maga, D.; Hrad, J.; Hajek, J.; Othman, A. Application of Minimum Energy Effect to Numerical Reconstruction of Insolation Curves. Energies 2021, 14, 5313. https://doi.org/10.3390/en14175313
Maga D, Hrad J, Hajek J, Othman A. Application of Minimum Energy Effect to Numerical Reconstruction of Insolation Curves. Energies. 2021; 14(17):5313. https://doi.org/10.3390/en14175313
Chicago/Turabian StyleMaga, Dusan, Jaromir Hrad, Jiri Hajek, and Akeel Othman. 2021. "Application of Minimum Energy Effect to Numerical Reconstruction of Insolation Curves" Energies 14, no. 17: 5313. https://doi.org/10.3390/en14175313