【原】2021年高中数模美赛A题中文及解法思路

目录在此点击1 2022-12-08 发布于湖北

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2021年高中数模美赛A题中文及解法思路

解法思路：

（1）计算家中1600平方英尺的非网格住宅需要的电量，计算最长无阳光天数，求出这些天总需要有电量。根据总电量选相应电池，根据价格和电池存储比值，求出“最佳”电池存储系统。

（2）根据个人需求和偏好，调整模型参数，为任何家庭选择最佳的电池存储。

（3）通过重力势能和电能电能的相互转化,实现像电池一样的储能功能,计算水泥高度与电量关系，求出需要水泥量和高度。讨论使用水泥电池储存太阳能的一些优点和缺点。

2021 HiMCM

问题A：存储太阳

你的团队正在帮助计划使用太阳能为一个1600平方英尺的偏远地区的住宅供电。你需要计划足够的能量，以支持家庭在夜间和阴天的能量需求。你做了一些研究，发现当你的太阳能电池板不能产生足够的电能时，你可以从电网（即电力公司）获取电能，或者使用储能系统。由于房子位于偏远地区，连接电网的成本非常昂贵，因此你决定脱离电网，投资于储能。

出身背景能量存储系统允许你捕获电能，将其作为另一种形式的能量（电池、热能、机械能）存储，然后在需要时使用。这些存储单元的目的是存储阳光充足的白天所产生的能量，以便在太阳能电池板不能产生足够的能量来满足需求（夜间或云层覆盖）时使用，或者用于存储和转移多余的能量。在使用储能系统的太阳能家庭中，大多数使用某种电池。一些房主只有一个大电池，而其他人可能使用“电池组”（连接两个或更多电池）。能量储存可能很昂贵，所以房主应该选择适合他们情况的系统。图1显示了离网能源的一般概念。

图1：离网能源系统

在选择储能系统时，需要考虑许多标准。以下是一些最常见的决策标准，以及哪种电池规格对每种标准最重要。为了一次为更多的家庭供电，您需要一个具有高连续额定功率的电池。为了给能量密集型设备供电（在短时间内需要更多的电力），您的电池应具有较高的瞬时额定功率。要让您的家运行更长时间，请寻找可用容量更高的电池。如果你想最大限度地利用每千瓦时的电量，那就找一个往返效率更高的电池。

 铅酸（浸水或密封）和锂离子电池是不同使用水平的全时离网供电的理想选择。

 铅酸电池已经存在了很长一段时间，以其低廉的价格和可靠性而闻名。

 锂离子电池更贵，但不需要维护。其他选择包括镍镉电池和液流电池。

 寻找磷酸铁锂（LFP）电池（一种锂离子电池），以获得循环次数最多的最长寿命。

 对于可能的绝对最高安全等级（尽管它们都是安全的），请查看LFP电池。

要求

1.考虑您正在规划的1600平方英尺的非网格住宅。

a、首先列出一系列问题，以确定您的能源需求，从而分析您的太阳能存储需求。讨论问题的可能答案范围。为了让您开始，这里有几个问题：

 有多少人会在这个家里使用能源？

 家中哪些物品需要能量，需要多少能量？

 家里的人什么时候用能源？

b、使用第1.a部分中的分析，以及

问题背景和您认为重要的任何其他因素，以开发数学模型或算法，为您的离网家庭选择“最佳”电池存储系统。

c、考虑可用的电池存储选项，并使用您的模型为您的离网家庭选择最佳选项。问题陈述末尾的图表提供了几个但不是所有的电池选项。讨论您的选择。

2.根据要求1调整和概括您的模型，使其能够适应个人需求和偏好，从而为任何家庭选择最佳的电池存储。讨论您对模型所做的更改。评估您的模型。

3.最近，瑞典的研究人员发现水泥可以用来储存能量[1]。混凝土是用水泥制成的，用于建筑、人行道、桥梁和无数其他结构。

a、确定使用水泥电池储存太阳能的一些优点和缺点。请描述您如何将水泥用作离网家庭或任何家庭的电池？

b、确定并讨论您需要的其他信息，以便对水泥电池的使用情况与当前可用的太阳能电池进行建模和比较。注意：不需要创建模型。

4.写一篇一页的非技术新闻文章，描述您的太阳能电池存储决策模型。在您的文章中包含关于水泥电池未来可能性的任何建议。总页数不超过25页的PDF解决方案应包括：

 一页汇总表。

 目录。

 您的完整解决方案。

 一页文章。

 参考列表。

注：HiMCM比赛现在有25页的限制。您提交的所有方面都达到25页的限制（摘要表、目录、参考列表和任何

附录）。

术语汇编

连续额定功率：电池可连续提供的功率（直到电池电量耗尽）的千瓦（kW）或安培（A）。

储能系统：一种系统，它捕获电力，以另一种形式存储，然后取回以供以后使用。

瞬时额定功率：电池可以在短时间内（但不能长时间维持）提供的功率的千瓦（kW）或安培（A）。

电池寿命：电池继续工作的时间长度，即通过预期运行年限、预期吞吐量（使用kWh）和预期循环次数（充电和放电）进行测量。

脱离电网：不使用或依赖公共电力供应（或其他公用设施）。

往返效率：每投入一个单位，你将从电池中获得的电量单位的百分比。

可用容量：电池能够储存并供应给您家的电量，单位为千瓦时（kWh）或安培小时（Ah）。

参考

[1] 张、艾玛·Q和唐露萍。2021，“可充电混凝土电池”11号楼，3号楼：103。https:///10.3390/buildings11030103

[1] 这只是太阳能存储系统可用电池的一小部分。

[2] 估计包括10%的电池缺陷和转换器损耗。

[3] 铅酸蓄电池的损耗为15-20%kWh-千瓦时

表中的缩写

2021 HiMCM

Problem A: Storing the Sun

Your team is helping to plan the use of solar power to provide electricity to a 1600 square-foot home being built in a remote area. You need to plan for enough energy to support the energy requirements of the home at night and on a cloudy day. You have done some research and found that you can either pull energy from the grid (i.e. a power company) when your solar panels aren’t producing enough, or use an energy storage system. As the house is in a remote area, the cost of connecting to the grid is very expensive, so you decide to go off-the-grid and invest in energy storage.

Background.An energy storage system allows you to capture electricity, store it as another form of energy (battery, thermal, mechanical) and then have it available to use when needed. The purpose of these storage units is to store energy produced during sunny daylight hours for use when the solar panels do not produce enough energy for the demand (night or cloud covered), or for storage and transfer of excess energy. Of the solar-powered homes that use an energy storage system, most use some sort of battery. Some homeowners have only one large battery, while others may use a “bank of batteries” (two or more batteries connected). Energy storage can be expensive and so homeowners should choose a system that is appropriate for their situation. Figure 1 shows the general concept of off-the-grid energy.

Figure 1: Off-the-Grid Energy System

In choosing an energy storage system there are many criteria to consider. Here are a few of the most common decision criteria, as well as which battery specifications matter most for each criterion.

· To power more of your home at once, you will want a battery with a high continuous power rating.

· To power a more energy-intensive appliance (requiring more power in short bursts), your battery should have a high instantaneous power rating.

· To run your home for a longer amount of time, look for a battery with a higher usable capacity.

· If you want to get the most out of every kilowatt-hour of electricity you put into your battery, look for a battery with a higher round-trip efficiency.

· Lead-acid (flooded or sealed) and lithium-ion batteries are ideal for a full-time, off-grid supply of different levels of use.

· Lead-acid batteries have been around for a long time and are known for their low prices and reliability.

· Lithium-ion batteries are more expensive, but require no maintenance. Other options are nickel cadmium and flow batteries.

· Look for lithium iron phosphate (LFP) batteries (a type of lithium-ion battery) to get the longest lifetime that you can cycle the most times.

· For the absolute highest safety rating possible (although they are all safe), look to LFP batteries.

Requirements

1. Consider the 1600 square-foot off-the-grid home you are planning.

a. Analyze your solar-power storage requirements by first making a list of questions to determine your energy needs. Discuss the range of possible answers to your questions. To get you started, here are a few questions:

· How many people will be using energy in this home?

· What items in the home will need energy and how much energy will they need?

· When will people in the home use energy?

b. Use your analysis from part 1.a., along with the criteria and considerations from the

problem background and any other factors you consider important, to develop a mathematical model or algorithm for choosing the “best” battery storage system for your off the-grid home.

c. Consider available battery storage options and use your model to choose the best option for your off-the-grid home. The chart at the end of the problem statement provides several, but not all, battery options. Discuss your choice.

2. Adjust and generalize your model from Requirement 1 so that it is adaptable to individual needs and preferences to choose the best battery storage for any home. Discuss the changes you make to your model. Evaluate your model.

3. Recently, researchers in Sweden discovered that cement could be used to store energy[1].

Concrete, which is made with cement, is used to build buildings, sidewalks, bridges, and countless other structures.

a. Identify some of the advantages and disadvantages of using cement batteries to store solar power. Describe how might you incorporate cement as a battery for your off-the-grid home or for any home?

b. Determine and discuss the additional information you would need in order for you to model and compare the use of cement batteries to currently available batteries for solar power storage. Note: You are NOT required to create the model.

| ©2021 by COMAP, Inc. | www.comap.com | www.mathmodels.org | @comapmath | info@comap.com |4. Write a one-page non-technical news article describing your solar power battery storage decision model. Include in your article any recommendations for the future possibilities of a cement battery.

Your PDF solution of no more than 25 total pages should include:

· One-page Summary Sheet.

· Table of Contents.

· Your complete solution.

· One-page Article.

· References list.

Note:The HiMCM Contest now has a 25-page limit. All aspects of your submission count toward the 25-page limit (Summary Sheet, Table of Contents, Reference List and any

Appendices).

Glossary

Continuous Power Rating: the kilowatts (kW) or Amps (A) of power that the battery can provide continuously (until the battery runs out of power).

Energy Storage System: a system that captures electricity, stores it in another form, and then retrieves it for use at a later time.

Instantaneous Power Rating: the kilowatts (kW) or Amps (A) of power that the battery can provide in short bursts (but not sustain over a long period).

Lifetime (of battery): the length of time your battery will continue working which is

measured by expected years of operation, expected throughput (kWh of use), and expected number of cycles (charging and discharging).

Off-The-Grid:not using or depending upon public supply of electricity (or other utilities).

Round-Trip Efficiency: the number of units of electricity you’ll get out of a battery for every unit you put into it measured in a percentage.

Usable Capacity: the amount of electricity, in kilowatt-hours (kWh) or Amp-hours (Ah) that a battery is able to store and supply to your home.

Reference

[1] Zhang, Emma Q., and Luping Tang. 2021. "Rechargeable Concrete

Battery"Buildings 11, no. 3: 103. https:///10.3390/buildings11030103

[1] This is just a small sample of available batteries for solar storage systems.

[2] Estimate to include 10% battery imperfection and converter loss.

[3] Losses from lead-acid batteries are 15-20%

kWh - Kilowatt Hours