The optimized design of the heliostat field for Question A of the 2023 National College Student Mathematical Modeling Contest. The fourth edition of the ideas and detailed model formulas for Question A of the National Competition has been written. The catalog is as follows:
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1. Restatement of the problem... 1
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2. Problem analysis... 1
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3. Model assumptions... 6
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4. Establishment and solution of problem 1 model... 6
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4.1 Establishment of heliostat field coordinate system... 6
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4.2 Calculation of heliostat related parameters and vertex coordinates... 7
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4.2.1 Calculation of normal vector and pitch angle of heliostat mirror... 7
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4.2.2 Calculation of heliostat vertex coordinates... 8
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4.3 Establishment of efficiency model... 10
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4.3.1 Cosine efficiency calculation model... 11
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4.3.2 Shadow occlusion efficiency model... 13
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4.3.3 Collector cut-off efficiency model... 16
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4.4 Solution to Problem 1 Model... 19
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5. Establishment and solution of problem 2 model... 20
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5.1 Simplification of model solution variable dimensions... 20
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5.2 The geometric layout of the mirror field... 21
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5.2.1 Radial grid method... 21
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5.2.2 Intensive Campo rule arrangement method... 22
For detailed model and subsequent update code downloads, see the address at the end of the article. Will continue to update
Question A of the 2023 National Competition: Building a new power system with new energy as the main body is an important measure for my country to achieve the goals of "carbon peaking" and "carbon neutrality". Tower solar thermal power generation is a new low-carbon, environmentally friendly, clean energy technology.
The heliostat is the basic component of the tower solar thermal power station (hereinafter referred to as the tower power station) to collect solar energy. Its base is composed of a longitudinal shaft and a horizontal shaft, and the plane reflector is installed on the horizontal shaft. The axis of the longitudinal rotation axis is perpendicular to the ground and can control the azimuth angle of the reflector. The axis of the horizontal rotating shaft is parallel to the ground, and the pitch angle of the reflector can be controlled. The schematic diagram of the heliostat and base is shown in Figure 1. The height of the intersection point of the two rotating axes (also the center of the heliostat) from the ground is called the installation height of the heliostat. Tower power stations use a large number of heliostats to form an array, called a heliostat field. The heliostat reflects the sunlight and gathers it to the heat collector installed on the top of the absorption tower in the mirror field, heats the heat transfer medium in it, stores the solar energy in the form of heat energy, and then realizes the conversion from heat energy to electric energy through heat exchange. Sunlight is not a parallel ray, but a cone-shaped ray with a certain cone angle, so the incident ray from the sun reflected by any point of the heliostat is also a cone-shaped ray[2]. When the heliostat is working, the control system controls the normal direction of the heliostat in real time according to the position of the sun, so that the light emitted from the center of the sun is reflected by the center of the heliostat and then points to the center of the collector. The height of the collector center above the ground is called the absorption tower height.
It is now planned to build a circular heliostat field in a circular area with the center located at 98.5∘ east longitude, 39.4∘ north latitude, 3000m altitude, and 350m radius (Figure 2). With the center of the circular area as the origin, the due east direction as the positive axis, the due north direction as the positive axis, and the upward direction perpendicular to the ground as the positive z-axis, a coordinate system is established, which is called the mirror field coordinate system.
The planned absorption tower height is 80m, and the collector is a cylindrical surface-receiving collector with a height of 8m and a diameter of 7m. No heliostats will be installed within 100m around the absorption tower, leaving open space to build workshops for installing power generation, energy storage, control and other equipment. The shape of a heliostat is a plane rectangle, and its upper and lower sides are always parallel to the ground. The distance between these two sides is called the mirror height, and the distance between the left and right sides of the mirror is called the mirror width. Usually, the mirror width is different. Less than the mirror height (constraint 1). The side length of the mirror is between 2m and 8m, and the installation height is between 2m and 6m (the model independent variable optimization solution interval limit of problem 2 and problem 3), the installation height must ensure that the mirror will not touch the ground when it rotates around the horizontal axis (constraint Condition 2). Due to the need for maintenance and cleaning of vehicles, the distance between the centers of adjacent heliostat bases is required to be at least 5m longer than the mirror width. (Constraint 3)
To simplify calculations, the calculation time points for all "annual average" indicators in this question are 9:00, 10:30, 12:00, 13:30, and 15:00 on the 21st of each month, local time. (Only 12*5 time points need to be calculated)
Topic A of the 2023 National College Student Mathematical Modeling National Competition:
Please build a model to solve the following problems:
problem analysis
National Competition Question A Question 1: If the absorption tower is built at the center of the circular heliostat field, the size of the heliostats is 6m×6m, and the installation height is 4m, and the positions (coordinates) of the centers of all heliostats are given It is known that some intermediate quantities can be calculated according to the relevant formulas in the appendix) (hereinafter referred to as the heliostat position, see the attachment for relevant data). Please calculate the annual average optical efficiency, annual average output thermal power, and unit mirror surface of the heliostat field. Annual average thermal power output by area (see the appendix for the definitions of optical efficiency and thermal output power). Please fill in the results in the format of Table 1 and Table 2 respectively.
Key points: (The relevant formulas have been given in the appendix to establish a formula for calculating the annual average optical efficiency, annual average output thermal power, and annual average output thermal power per unit mirror area of the heliostat field under the conditions given in Question 1. The mathematical model can be solved directly. The accuracy of this question will affect the following questions, so accurate modeling is required)
National Competition A Question 2: According to the design requirements, the rated annual average thermal output power of the heliostat field (hereinafter referred to as the rated power) is 60MW. If all heliostats have the same size and installation height, please design the following parameters of the heliostat field (you need to establish a model based on question 1. The model in question 1 is an internal model, and an optimized model is set externally): Absorption tower Position coordinates (2 variables), heliostat size (1 variable), installation height (1 variable), number of heliostats (1 variable, N), heliostat position (2N variables) (optimization variable), so that the annual average thermal power output per unit mirror area is as large as possible (optimization objective function) under the condition that the heliostat field reaches the rated power (constraint condition 4). Please fill in the results into the tables according to the formats of Tables 1, 2, and 3 respectively, and save the position coordinates of the absorption tower, heliostat size, installation height, and position coordinates to the result2.xlsx file in the format specified by the template.
Question 3 of National Competition A: If the size of the heliostat can be different, the installation height can also be different, and the rated power setting is the same as that of question 2. Please redesign the parameters of the heliostat field so that the heliostat field can reach the rated power condition Under this condition, the annual average thermal output per unit mirror area should be as large as possible. Please fill in the results in the form of Table 1, Table 2 and Table 3 respectively, and the position coordinates of the absorption tower (2 variables), the size of each heliostat (N variables), and the installation height (N variables) , the position coordinates (2N variables) are saved to the result3.xlsx file in the format specified by the template. (This is the same as question 2, except that there are more variables to solve and the complexity of the problem is further increased)
It can be seen that Problems 2 and 3 are optimized based on the geometric model of Problem 1, so they mainly focus on the establishment of the geometric model of Problem 1. Later optimization problems can be solved using various optimization algorithms, but at the same time there are many optimization variables here, so Certain considerations need to be made in the simplification of the optimization problem and the selection of the algorithm, which is also a focus of the final award.
Another important point is the establishment of the geometric model for question 1. The modeling accuracy of this question has a great impact on the results. The judges can directly distinguish the awards based on the correctness of the answers. Therefore, students who want to compete for awards need to pay attention to this question.
Appendix related calculation formulas
Understanding of the formula: The solar altitude angle is related to three variables, local latitude, solar hour angle, and solar declination angle. Among them, the latitude is known, the solar hour angle is related to the time (5 times in a day), and the solar declination angle is related to the number of days (a total of 12 months, the 21st of each month, so there are 12 days). So for each time (a total of 5*12 times), the solar altitude angle is determined.
Formula understanding: The altitude is known, and the solar altitude angle has been calculated previously, so for each time (a total of 5*12 times), the DNI is determined.
Formula understanding: DNI has been calculated, so the output thermal power is only related to area and optical efficiency. The lighting area should be a fixed value, that is, length times width
The calculation steps are as follows: given the coordinate position of the mirror center, dHR can be calculated directly, and then the atmospheric transmittance is calculated. The specular reflectance is a constant value, so there are still 3 efficiencies that need to be modeled and calculated (this is the main factor we need to establish model)
For detailed model and code downloads, see the address at the end of the article.
Model assumptions
(1) The heliostat field is circular. The heliostat field adopts a radial staggered layout plan and is evenly arranged circumferentially with the central tower as the center of the circle. In order to highlight the impact of the sun's position on the shadow shielding efficiency, the heliostats in the mirror field are based on the principle of not causing mechanical collision, and a dense simulated mirror field is established.
(2) The mirror field plane is an ideal horizontal plane, and the heights of the columns of all heliostats are consistent. All heliostats have the same specifications and are rectangular. The heliostats adopt the altitude-azimuth tracking method, and it is assumed that the heliostat mirror is an ideal plane.
Establishment and solution of problem 1 model
Establishment of heliostat field coordinate system
The heliostat fields of tower-type solar thermal power stations are mostly arranged circumferentially with the heat collecting tower as the center. Whether it is a circular radiating mirror field or a square wheat field-shaped mirror field, the space right angle with the heat collecting tower as the coordinate origin can be used. The coordinate system represents the orientation of each heliostat in the mirror field. In order to describe the position of the sun and the heliostat at the same time, this paper will use the horizontal coordinate system with the intersection point of the central axis of the heat absorbing tower and the heliostat field plane as the coordinate origin as the heliostat field coordinate system. That is, the geometric center of the base of the heat absorbing tower is taken as the origin O of the coordinate system, the due east direction is designated as the positive semi-axis of the coordinate system X, the true north direction is designated as the positive semi-axis of the coordinate system Y, and the zenith direction is designated as the positive semi-axis of the coordinate system Z. Semi-axis, the established three-dimensional space rectangular coordinate system of the mirror field is shown in the figure.
rotation matrix
When performing coordinate conversion, it is necessary to determine the conversion relationship between the mirror coordinate system and the ground coordinate system, that is, a rotation matrix needs to be used.
The rotation matrix for rotation around the x\y\z axis is as follows: Rotation around the x-axis (clockwise rotation on the yz plane)
Rotate around the y-axis (clockwise in the zx plane)
Rotate around the z-axis (clockwise in the xy plane)
As can be seen from the title, when the heliostat is working, the control system controls the normal direction of the heliostat in real time according to the position of the sun, so that the light emitted from the center point of the sun is reflected by the center of the heliostat and then directed to the center of the collector. According to this condition, the direction of the mirror can be determined, which is the transformation relationship between the mirror coordinate system and the ground coordinate system, that is, the rotation matrix.
Calculation of heliostat related parameters and vertex coordinates
Because the overall efficiency of the heliostat field is not a simple superposition or multiplication of the efficiencies of individual heliostats, it is often necessary to determine the target location by calculating the coordinates of each vertex of the heliostat in space in the simulation study of the heliostat. Interrelationship between heliostats and adjacent heliostats. Especially in the calculation process of shadow occlusion efficiency, it is necessary to project the target heliostat to the plane where other heliostats are located to determine whether shadow or occlusion occurs. Therefore, the first step in establishing the heliostat field efficiency model is to calculate the heliostat Mirror related parameters and vertex coordinates.
For complete and detailed model and code downloads, please see: 2023 Mathematical Modeling National Competition Question A Idea Model