Full analysis of code papers for the 2023 Tianfu Cup Room Temperature Superconductivity + Renewable Energy Application Mathematical Modeling Competition

Question restatement

Question requirements: Assume that a country has multiple renewable energy power generation sites distributed in different geographical locations. Each power generation site varies in the amount and time period of energy it produces. At the same time, it is assumed that room temperature superconducting materials LK-6, LK-66, and LK-666 have been developed, which can achieve zero-resistance transmission of current at room temperature. Competitors are asked to conduct mathematical modeling and optimization based on the following scenarios:

1. Given the locations, energy production curves and transmission distances of multiple renewable energy power generation sites.

2. Given the transmission characteristics of room temperature superconducting materials, including transmission efficiency, maximum transmission power, etc.

3. Consider the cost and efficiency of power grid transmission, such as transmission loss, equipment cost and other factors.

4. Goal: Optimize the transmission and storage scheme of renewable energy to minimize overall energy loss and maximize transmission efficiency. Ask the contestants to design a mathematical model to answer the following questions:

Q1) How to reasonably divide the transmission paths of room temperature superconducting materials to minimize the total energy loss?

Question analysis

To reasonably divide the transmission paths of room temperature superconducting materials to minimize the total energy loss, we can use dynamic programming methods to solve it. This method will consider each site as a starting point and gradually select the next site to achieve the goal of minimizing the total energy loss.

The following are the specific steps and analysis:

Step 1: Define state and objective functions

Define the state: We can define a state dp[i], which represents the minimum total energy loss from the starting point to site i.
Define the objective function: Our goal is to minimize the total energy loss, that is, dp[n], where n is the end site.

Step 2: State transition equation

Consider a transmission from site j to site i. We can use the following state transition equation:
$dp[i] = \min_{j} \left( dp[j] + m_{j \to i} *e_{i} \right)$
Among them, m(j→i) represents the transmission loss cost from site j to site i, and e_i represents the energy generation of site i in this time period.

Step 3: Initialization and Iteration

Initialization: Set dp[starting point] = 0, and other dp[i] are initialized to a larger value (representing infinity).
Iteration: Calculate dp[i] step by step according to the state transition equation.

Step 4: Path reconstruction

Starting from the end point n, by comparing different dp[i] values, the path that minimizes the total energy loss is found.
You can backtrack step by step from the end point to the starting point to determine the previous site in the transmission path for each site.

Through dynamic programming methods, we can find the optimal transmission path from the starting point to the end point to minimize the total energy loss. In actual application, the best transmission path can be obtained by substituting data into the formula for calculation and iteration.

Code:

The code calculates the minimum total energy loss from the starting point to the end point in a step-by-step iterative manner, taking into account different sites, room temperature superconducting materials, time periods and transmission distances. Finally, the result of minimum total energy loss is output.

Q2) On the premise of ensuring stable operation of the power grid, how to adjust the transmission plan of renewable energy to maximize transmission efficiency?

Step 1: Define variables and objective function

Define variables: For each pair of power generation sites $i$ to $j$, introduce a variable $x_{ij}$ to represent the transmission volume from site $i$ to site $j$.
Define the objective function: Our goal is to maximize the weighted sum of transmission efficiency, that is, the sum of transmission volumes times the transmission efficiency. Therefore, the objective function can be expressed as: maximize $\sum_{i,j} x_{ij}*{\eta }_{ij}$
where eta is the transmission efficiency

Step 2: Constraints

Transmission restrictions: For each pair of sites $i$ to $j$, the transmission volume $x_{ij}$ should meet the following restrictions: 0≤xij≤maximum $0 \leq x_{ij} \leq{}P _{ij}$ transmission power ij Where, the maximum transmission power P_{ij} is from The maximum transmission power of room temperature superconducting material available from site $i$ to site $j$.
Stability constraints: Considering the stability of the power grid, ensure that the energy supply of each site does not exceed the energy it generates: $\sum_j x_{ij} \leq \text{E}_i$ , where E is the amount of energy generated.
Transmission efficiency constraint: Since transmission efficiency is the proportion of energy loss during transmission, the transmission efficiency should be calculated based on the transmission loss cost: $e_{ij} = \frac{1}{1 + l_{ij} \times \sum_t x_{ijt}}$ where e {ij} is the transmission loss cost from site i to site j, \sum_t x {ijt} is Total transfer volume.

Step 3: Solve the optimization problem. Combine the objective function and constraints to obtain a complete linear programming problem. Using a linear programming solver, such as in Python scipy.optimize.linprog, the optimal transmission scheme that maximizes transmission efficiency can be solved.

Code:

Through this code, we can find renewable energy transmission solutions that maximize transmission efficiency while maintaining grid stability.

Q3) Considering the production cost of room temperature superconducting materials and the cost of renewable energy power generation, how to weigh the economic benefits and environmental benefits and formulate a comprehensive optimization strategy?

Considering the production cost of room temperature superconducting materials and the cost of renewable energy power generation, how to make a trade-off between economic benefits and environmental benefits and formulate a comprehensive optimization strategy involves the trade-off of multiple factors, including transmission efficiency, cost, Environmental protection, etc. We can solve this problem by building a multi-objective optimization model and use multi-objective optimization algorithms to find comprehensive optimization strategies.

Step 1: Define the multi-objective optimization model

Define the objective function: Our goal is to find a balance between economic costs and environmental benefits, so we can define two objective functions: minimizing the total cost and maximizing the environmental benefits.
- Minimize total cost: $f_1 = \sum_{i,j,t}x_{j}\cdot(C_{ij} + T_{j})$
- Maximize environmental benefits: $f_2 = \sum_{i,j,t} x_{jt} \cdot E_{jt}$
- In these formulas, $x_{ijt}$ represents the transmission volume, $C_{ij}$ represents the sum of the production cost and the transmission cost, $T_{ijt}$ represents the transmission cost, and $E_{ijt}$ represents the environmental benefit.

Step 2: Constraints

Transmission limits: Transmission volume limits and stability constraints, similar to previous modeling.

Step 3: Multi-objective optimization algorithm Use multi-objective optimization algorithms, such as weighted sum method, ε constraint method, Pareto optimization and other methods, to find the balance point between economic benefits and environmental benefits, that is, a set of Pareto optimal solutions.

Step 4: Analyze the trade-off results. Analyze the Pareto optimal solution set and select the most appropriate comprehensive optimization strategy based on actual needs and preferences. This may involve weighing and considering factors such as cost, environmental benefits, and economic benefits.

Among them, the method of analyzing the Pareto optimal solution set is as follows:

1. Visualize the Pareto optimal solution set: Use scatter plots or line graphs to visualize the solutions in the Pareto optimal solution set in the target space. The horizontal axis and vertical axis can represent different objective functions respectively. This will help you visualize the relative position and distribution of solutions.

2. Choose the right solution: Consider the actual context of the problem and your preferences and choose the solution that best meets your needs. Sometimes, not all Pareto solutions are equivalent, so you may be biased towards certain goals. For example, if you're more economical, you can choose a lower-cost solution.

3. Establish objective weights: Objective weights can be introduced to transform multi-objective problems into single-objective problems. According to the relative importance of the goals, a weight is assigned to each goal, and the objective function is weighted and summed to merge multiple goals into one goal. In this way, the solution in the Pareto optimal solution set can be transformed into the solution of the single-objective problem.

4. Sensitivity analysis: Analyze the sensitivity of the objective function to parameters, that is, the changes in the objective function when the parameter values are slightly changed. This can help you understand how the set of Pareto optimal solutions is affected by parameter changes and determine whether your decision is still valid.

5. Understand the trade-offs: The solutions in the Pareto optimal solution set provide the trade-offs between different objectives. You can see how the trade-offs between different objectives depend on different factors by looking at the distribution of solutions. This helps you understand the complexity of the problem and potential decision-making strategies.

6. Formulate decision-making strategies: Based on the analysis results, formulate comprehensive decision-making strategies. You can balance economic and environmental benefits based on different scenarios and preferences. This may involve developing a series of strategies to deal with different situations based on the solutions in the Pareto optimal solution set.

7. Experiment and verification: After selecting one or more solutions as a decision-making solution, conduct experiments and verification. Apply the chosen solution to a real situation, observe its effect and evaluate it. This will help you validate the soundness of your analysis and the feasibility of your decisions.

code

Solving multi-objective optimization problems often involves the use of multi-objective optimization algorithms, such as the Pareto front method.

Here, we use deapa library in Python to implement the NSGA-II algorithm to solve synthetic optimization problems. We will assume two objective functions: minimizing total costs and maximizing environmental benefits.

This code demonstrates how to use the NSGA-II algorithm to solve multi-objective optimization problems, helping you find the right balance between economic and environmental benefits.

For a more detailed analysis and a complete version of the code, see here:

2023 Tianfu Cup Room Temperature Superconductivity + Renewable Energy Application Mathematical Modeling Competition Code Paper Full Analysis-csdn