[2023 National Mathematical Modeling Competition] Ideas for solving problem A

Ideas for solving problem A in the 2023 National Mathematical Modeling Competition

Question 1:

To calculate the annual average optical efficiency and annual average output thermal power of the heliostat field, as well as the annual average annual output thermal power per unit mirror area, we can follow the following steps:

1. Traverse the positions of all heliostats and calculate the optical efficiency of each heliostat based on the given heliostat size and installation height.

2. Calculate the direct normal irradiance (DNI) of each heliostat using the formula for solar altitude and solar azimuth.

3. Calculate the output thermal power of each heliostat, based on the optical efficiency and DNI values.

4. The output thermal power of all heliostats is added to obtain the annual average thermal output output of the entire heliostat field.

5. Based on the area of ​​the heliostat, calculate the annual average thermal power output per unit mirror area.

The following are the calculation steps for question 1:

Step 1: Calculate the optical efficiency of each heliostat.

The optical efficiency of each heliostat can be calculated for a given heliostat size and mounting height.

Step 2: Calculate the DNI for each heliostat.

Calculate the DNI value for each heliostat position using the formulas for solar altitude angle and solar azimuth angle.

Step 3: Calculate the thermal power output of each heliostat.

According to the optical efficiency of the heliostat and the value of DNI, the output thermal power of each heliostat is calculated.

Step 4: Calculate the annual average thermal power output of the entire heliostat field.

The output thermal power of all heliostats is added to obtain the annual average thermal output output of the entire heliostat field.

Step 5: Calculate the annual average output thermal power per unit mirror area.

Divide the annual average thermal output power of the entire heliostat field by the total area to obtain the annual average thermal output power per unit mirror area.

Note that these calculations require numerical calculations based on actual data and formulas, so calculation software or programming languages ​​are required for actual calculations. The calculation results can be filled in the table according to the format of Table 1 and Table 2.

Question 2:

Question 2 requires that the parameters of the heliostat field be designed so that the heliostat field reaches the rated annual average output thermal power and the annual average output thermal power per unit mirror area is as far as possible under the condition that the given heliostat size and installation height are the same. big. This is an optimization problem that can be solved using mathematical optimization methods.

First, we can model problem 2 as a mathematical programming problem, defining decision variables, objective function and constraints:

Decision variables:

• Heliostat position coordinates (x, y): The position coordinates of each heliostat.
• Heliostat Size: The size of the heliostat, determined by length and width.
• Installation height: The installation height of the heliostat.
• Number of heliostats: Total number of heliostats.

Objective function:

• The objective function is the annual average output thermal power per unit mirror area, that is, to maximize the annual average output thermal power per unit mirror area.

Restrictions:

• The position coordinates of all heliostats must be within the circular area.
• Heliostats cannot block each other.
• The area and installation height of the heliostat must comply with the specified range.
• The annual average thermal power output of the heliostat field must reach the rated power.

This problem can then be solved using mathematical optimization methods such as linear programming or nonlinear programming to find optimal heliostat field design parameters. The results obtained include the position coordinates of the absorption tower, heliostat size, installation height, position coordinates and other information. Fill in the table according to the format of Table 1, Table 2 and Table 3, and add the position coordinates of the absorption tower, each fixed The sunglass size, installation height, and position coordinates are saved in the result2.xlsx file.

Question 3:

Question 3 requires that the parameters of the heliostat field be redesigned under the condition that the size of the heliostat can be different and the installation height can also be different, so that when the heliostat field reaches the rated power, the average annual thermal power output per unit mirror area can be achieved. As big as possible. This is also an optimization problem and can be solved using mathematical optimization methods.

Similar to problem 2, first, we can model problem 3 as a mathematical programming problem, defining decision variables, objective functions and constraints. Decision variables include heliostat position coordinates, heliostat size, installation height, number of heliostats, etc. The objective function is to maximize the annual average thermal output per unit mirror area. Constraints include restrictions on the position of the heliostat, the shading relationship between the heliostats, limitations on the size and installation height of the heliostat, and the conditions for the annual average thermal output to reach the rated power.

This problem is then solved using mathematical optimization methods to find optimal heliostat field design parameters. The results obtained from the solution include the position coordinates of the absorption tower, the size of each heliostat, the installation height, the position coordinates and other information,