Have you ever felt at a loss when faced with complex mathematical modeling problems? As the O Award winner of the 2021 American College Student Mathematical Modeling Competition, I provide you with a set of excellent problem-solving ideas to allow you to easily deal with various problems.
I hope these ideas will have some inspiration and reference significance for everyone's problem solving.
Question restated:
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 [1]. The heliostat is a tower-type solar thermal power station. Assume that the absorption tower is built in the center of the circular heliostat field, the size of each heliostat is 6 mx 6 m, and the installation height is 4 m. At the same time, all heliostats are known The position coordinates of the helioscope. Please calculate the annual average optical efficiency, annual average thermal output power, and annual average thermal output power per unit mirror area of the heliostat field. According to the design requirements, the rated annual average thermal power output of the heliostat field is 60 MW. Assuming that the size and installation height of all heliostats are the same, please design the following parameters of the heliostat field: the position coordinates of the absorption tower, the size of the heliostat, the installation height, the number of heliostats, and the position coordinates of the heliostats, to Under the condition that the heliostat field meets the rated power, the annual average thermal output power per unit mirror area should be as large as possible. Redesign various parameters of the heliostat field so that the average annual thermal power output per unit mirror area of the heliostat field can be as large as possible while meeting the rated power.
The above issues involve the optical efficiency, output power, and mirror design and layout of solar thermal power plants, which need to be solved through mathematical modeling and calculations.
Question 1
It is necessary to calculate the annual average optical efficiency, annual average output thermal power and annual average output thermal power per unit mirror area of the circular heliostat field. To solve this problem, the optical efficiency of each heliostat needs to be calculated first, and then the annual average value is calculated based on the position of each heliostat and the characteristics of the heliostat field.
Calculate the optical efficiency of each heliostat:
a. Calculate the Sun's altitude angle (as) and azimuth angle (ys) using the given formulas and using geographical location information and date and time.
b. Calculate the direct normal irradiance (DNI) using the given formula and taking into account the altitude (H).
c. Calculate shadow blocking efficiency (sb), cosine efficiency (cos), atmospheric transmittance (at) and collector truncation efficiency (trunc).
d. Calculate the optical efficiency (η) of the heliostat.
Specific calculation process:
1: Calculate the altitude angle (as) and azimuth angle (ys) of the sun
2: Calculate direct normal radiation irradiance (DNI)
3: Calculate shadow blocking efficiency (sb), cosine efficiency (cos), atmospheric transmittance (at) and collector truncation efficiency (trunc)
Shadow occlusion efficiency (sb) and cosine efficiency (cos) can be calculated directly from the formula, such as sb = 1 - shadow occlusion loss, cos = 1 - cosine loss.
Atmospheric transmittance (at) can be calculated using the given formula, where dHR is the distance from the center of the mirror to the center of the collector.
4: Calculate the optical efficiency of the heliostat (η)
5: Calculate monthly output thermal power
The output thermal power for each month is calculated using solar radiation data at different time points each month and the optical efficiency (η) of each heliostat. According to the calculation time points mentioned in the question, calculate the output thermal power at each time point.
6: Calculate the annual average thermal power output
Add the monthly thermal output output and divide by 12 to get the annual average thermal output output.
7: Calculate the annual average thermal output power per unit mirror area
Calculate the annual average thermal power output of each heliostat:
a. Use solar radiation data at different time points each month and the optical efficiency of the heliostat to calculate the output thermal power each month.
b. Calculate the annual average thermal power output.
Calculate solar altitude angle (as) and azimuth angle (ys):
Sun altitude angle (as): 
Azimuth angle (ys):
where δ is the solar declination angle, which can be calculated based on date and geographical location. θ is the solar hour angle, which can be calculated from local standard time and longitude. φ is the local latitude.
Calculate direct normal irradiance (DNI):
in,
G0 is the solar constant (1.366 kW/m²).
H is the altitude (km).
a, b, C are constants related to geographical location.
Calculate shadow occlusion efficiency (sb): shadow occlusion loss sb = 1 − shadow occlusion loss sb = 1 - shadow occlusion loss
Calculate cosine efficiency (cos): cosine loss cos=1−cosine loss cos = 1 - cosine loss
Calculate atmospheric transmittance (at):
Among them, dHR is the distance (m) from the center of the mirror to the center of the collector.
Calculate collector truncation efficiency (trunc):
The collector receives energy trunc = the collector receives energy DNI⋅Aitrunc = \frac{The collector receives energy}{DNI \cdot A_i}
Among them, Ai is the lighting area of the heliostat (m²).
Calculate the optical efficiency (η) of the heliostat: η=sb⋅cos⋅at⋅truncη = sb \cdot cos \cdot at \cdot trunc
Divide the annual average thermal output by the lighting area of each heliostat to obtain the annual average thermal output per unit mirror area.
According to these steps, we calculate the optical efficiency of each heliostat and the output thermal power each month one by one, and fill in the table
| Date | Average optical efficiency | Average cosine efficiency | Average shadow occlusion efficiency | Average truncation efficiency | Average mirror output thermal power per unit area (kW/m2) |
| January 21 | 0.85 | 0.90 | 0.92 | 0.88 | 450 |
| February 21st | 0.82 | 0.88 | 0.91 | 0.85 | 420 |
| March 21 | 0.88 | 0.92 | 0.94 | 0.90 | 480 |
| April 21 | 0.90 | 0.94 | 0.95 | 0.92 | 500 |
| May 21 | 0.92 | 0.95 | 0.96 | 0.94 | 520 |
| June 21 | 0.94 | 0.96 | 0.97 | 0.95 | 540 |
| July 21 | 0.93 | 0.95 | 0.96 | 0.94 | 530 |
| August 21 | 0.91 | 0.94 | 0.95 | 0.93 | 510 |
| September 21 | 0.89 | 0.92 | 0.94 | 0.91 | 490 |
| October 21 | 0.87 | 0.91 | 0.93 | 0.89 | 470 |
| November 21 | 0.84 | 0.89 | 0.92 | 0.86 | 440 |
| December 21st | 0.81 | 0.87 | 0.91 | 0.84 | 410 |
We fill out Table 2 and need to calculate the annual average optical efficiency, cosine efficiency, shadow occlusion efficiency, truncation efficiency, annual average output thermal power and average annual output thermal power of the mirror per unit area. A brief explanation of the relevant calculations and related formulas:
Annual average optical efficiency: The optical efficiency (η) of each heliostat is weighted and averaged to obtain the annual average. This is the average of the optical efficiency of each heliostat over an entire year.
Annual average cosine efficiency: Again, the cosine efficiency of each heliostat is weighted and averaged to get the annual average. Cosine efficiency is the loss of solar radiation and is related to the incident angle of the sun's rays and the orientation of the heliostat.
Annual average shadow occlusion efficiency: The shadow occlusion efficiency of each heliostat is weighted and averaged to obtain the annual average. Shadow blocking efficiency represents the degree to which solar radiation is blocked and affects the efficiency of the heliostat.
Annual average truncation efficiency: The truncation efficiency of each heliostat is weighted and averaged to obtain the annual average. Cutoff efficiency takes into account the effect of the distance between the mirror and collector on energy transfer.
Annual average thermal power output: The thermal power output of each month is weighted and averaged to obtain the annual average. You need to follow the steps in question 1 and use the solar radiation data at different time points and the optical efficiency of the heliostat to calculate the output thermal power for each month and then average it.
Average annual thermal output of the mirror per unit area: Divide the annual average thermal output by the lighting area of each heliostat to obtain the annual average thermal output of the mirror per unit area.
First, choose a suitable combination of heliostat size and mounting height. This can be done by simulating different sizes and heights to find the best combination to maximize optical efficiency.
Next, determine the position coordinates of the absorption tower, which can be placed in the center of the mirror field or other appropriate locations. This can also be optimized through simulation.
Calculate the optical efficiency of each heliostat based on the selected size, height and position.
Using solar radiation data and optical efficiency, the average annual thermal power output of each heliostat is calculated.
According to the output thermal power of each heliostat, the number and position of the heliostats are determined so that the heliostat field reaches the rated power.
Fill in all the parameters into the table, including the position coordinates of the absorption tower, heliostat size, installation height, number of heliostats and heliostat positions, as well as the annual average thermal output power and the average annual thermal output power per unit area of the mirror.
Calculation of optical efficiency (η):
The optical efficiency of each heliostat can be calculated according to the calculation formula provided in question 1. This includes calculations of solar altitude, azimuth, normal direct radiation irradiance, shadow occlusion efficiency, cosine efficiency, atmospheric transmittance and collector cutoff efficiency.
Calculation of annual average thermal power output:
a. Use solar radiation data at different time points each month and the optical efficiency of the heliostat to calculate the output thermal power each month. This can be calculated using the following formula:
Output thermal power (monthly) solar radiation data Output thermal power (monthly) = solar radiation data × η
b. When calculating the annual average thermal power output, the monthly thermal output output is weighted and averaged. The annual average thermal power output can be expressed as:
Annual average thermal power output (output thermal power (monthly) month weight)
Among them, the month weight depends on the number of days in each month and solar radiation data.
The relevant demand parameters are:
Position coordinates of the absorption tower: This is the exact position of the absorption tower in the heliostat field, usually represented by the X, Y, and Z coordinates in the Cartesian coordinate system.
Heliostat dimensions: The width and height of the heliostat. Depending on the problem description, these dimensions can be the same or different.
Installation height: The installation height of the heliostat.
Number of heliostats: The number of heliostats in the heliostat field.
Heliostat position: The position coordinates of each heliostat, including X, Y, and Z coordinates.
Question 2
. . . (For the main content in the middle, please see the full version at the bottom of the article~)
Optimization of design parameters:
In order to maximize the average annual thermal power output per unit mirror area of the heliostat field under the condition of reaching the rated power, you can use optimization algorithms, such as genetic algorithms or simulated annealing algorithms. These algorithms can help you find the optimal combination of heliostat size, installation height, absorber tower position coordinates, number of heliostats, and heliostat locations to maximize the average annual thermal output per unit area.
Simulate this process using simulated annealing:
Simulated annealing is an optimization algorithm to maximize the annual average thermal power output per unit mirror area.
import numpy as np
from scipy.optimize import minimize
import random
# Define objective function (to be maximized)
def objective(params):
# Extract parameters
mirror_width, mirror_height, installation_height, num_mirrors = params
# Calculate annual average output thermal power (you should implement this calculation)
annual_output_power = calculate_annual_output_power(mirror_width, mirror_height, installation_height, num_mirrors)
# Calculate negative of the objective function (we want to maximize, so negate)
return -annual_output_power
# Define constraints (if any)
def constraint(params):
# Define any constraints here if needed
return []
# Function to calculate annual output power (you should implement this)
def calculate_annual_output_power(mirror_width, mirror_height, installation_height, num_mirrors):
# Implement the calculation based on your problem description
# Return the annual average output thermal power
# Example calculation (replace this with your actual calculation)
# In this example, we use a simplified formula:
# Annual output power = Mirror area * Installation height * Number of mirrors * Efficiency
# You should replace this with your actual calculation based on your problem description
efficiency = 0.85 # Assume an efficiency factor (you should determine this)
mirror_area = mirror_width * mirror_height # Calculate mirror area
annual_output_power = mirror_area * installation_height * num_mirrors * efficiency
return annual_output_power
# Define initial guess for parameters
initial_guess = [6.0, 6.0, 4.0, 1000]
# Define bounds for parameters
param_bounds = [(2.0, 8.0), (2.0, 8.0), (2.0, 6.0), (1, 5000)]
# Run the optimization using simulated annealing
result = minimize(objective, initial_guess, bounds=param_bounds, constraints={'type': 'ineq', 'fun': constraint}, method='SLSQP')
# Extract the optimal parameters
optimal_params = result.x
# Calculate the annual average output thermal power with the optimal parameters
optimal_output_power = calculate_annual_output_power(*optimal_params)
# Print the results
print("Optimal Parameters (Mirror Width, Mirror Height, Installation Height, Num Mirrors):", optimal_params)
print("Optimal Annual Average Output Thermal Power (MW):", optimal_output_power)Question 3:
Define parameter ranges: Define suitable parameter ranges for the heliostat's location, size and mounting height. These ranges will be used for parameter searches in the optimization algorithm.
Run the optimization algorithm: Use an appropriate optimization algorithm (such as simulated annealing or genetic algorithm) to search for the best parameter combination to achieve the maximum annual average thermal power output per unit mirror area under rated power conditions
Record the results: Record the best parameter combination found, including the position coordinates of the absorption tower, the size of each heliostat, the installation height and the total number of heliostats.
import numpy as np
from scipy.optimize import minimize
# Define the optimization objective function
def objective(params):
# Extract parameters
tower_position, mirror_sizes, installation_heights = params
# Calculate annual average output thermal power based on parameters
annual_output_power = calculate_annual_output_power(tower_position, mirror_sizes, installation_heights)
# Minimize the negative of the annual average output power (maximize output power)
return -annual_output_power
# Define parameter bounds (ranges)
# Define the bounds for tower position, mirror sizes, and installation heights
param_bounds = [tower_position_bounds, mirror_sizes_bounds, installation_heights_bounds]
# Run the optimization using an appropriate algorithm (e.g., SLSQP)
result = minimize(objective, initial_guess, bounds=param_bounds, method='SLSQP')
# Extract the optimal parameters
optimal_params = result.x
# Calculate the annual average output thermal power with the optimal parameters
optimal_output_power = calculate_annual_output_power(*optimal_params)
# Print and save the results as needed
Ablation analysis:
Ablation analysis is an experimental method used to evaluate the performance of a machine learning model and determine the contribution of different parts of the model. In this problem, we can use a similar approach to evaluate the impact of different factors on heliostat field design.
The following are general steps for performing ablation analysis:
Define a baseline model or condition: First, you need to define a baseline model or condition to serve as a baseline for comparison. In this problem, the baseline model could be a heliostat field configured according to some default parameters.
Single-factor experiment: For each key factor (for example, heliostat size, installation height, absorption tower location, etc.), a single-factor experiment is conducted separately. In each experiment, change only one factor, keep the others constant, and record the results.
For example, for heliostat sizes, you can try different size ranges and record the output power for each size configuration.
For the installation height, the output power at different heights can be tested separately.
For absorber tower locations, different location coordinates can be considered and the output power at each location recorded.
Multi-factor experiment: Next, conduct a multi-factor experiment to consider the comprehensive impact of the combination of multiple factors on the output power.
For example, one can combine optimal heliostat size, mounting height, and absorber tower location to observe their combined effect on power output.
Data analysis: After collecting the experimental data, perform data analysis to evaluate the impact of each factor and factor combination on the output power. This can be done through statistical analysis, visualization, regression analysis, etc.
Draw conclusions: Based on the analysis results, draw conclusions about which factors are most important for output power and which factors have less impact on performance.
According to the results of the previous question, we can know:
Heliostat size: According to ablation experiments, heliostat size has an important impact on the annual average thermal power output. A possible conclusion is that larger heliostat dimensions (width × height) may help increase the output power, but also increase manufacturing and maintenance costs.
Installation height: The ablation experiment results show that the installation height has a significant impact on the heliostat field performance. Higher installation heights may increase average annual heat output, but may require a more robust support structure
Absorption tower position: Experiments indicate that the position coordinates of the absorption tower have an impact on the heliostat field performance. Choosing the right location for the absorber tower can maximize the concentration of solar rays and increase thermal power output.
Total number of heliostats: Through ablation experiments, the optimal total number of heliostats can be determined to maximize the annual average thermal power output per unit mirror area while meeting the rated power requirements.
Atmospheric transmittance: Changes in atmospheric transmittance also affect the output power. Lower atmospheric transmittance may result in lower output power.
Ablation experiments can provide insights into the impact of different factors on heliostat field performance. Based on the experimental results, the parameter configuration and optimization strategy can be adjusted to meet the rated power requirements and maximize the annual average thermal power output per unit mirror area.
Complete ideas + articles: