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### 知识点生成 #### 一、美国大学生数学建模竞赛(MCM/ICM) - **背景介绍**：美国大学生数学建模竞赛（MCM/ICM）是由美国数学及其应用联合会主办的一项国际性赛事，是全球最具影响力的数学建模竞赛之一。每年吸引来自世界各地的数千支队伍参赛，旨在通过解决现实世界中的问题来培养学生的数学建模能力、团队合作精神和创新能力。 - **比赛分类**：MCM/ICM分为两个主要部分：MCM(Mathematical Contest in Modeling)侧重于数学方法的应用，而ICM(Interdisciplinary Contest in Modeling)则更强调跨学科的方法。 - **奖项设置**：竞赛设有不同级别的奖项，其中最高奖项为Outstanding Winner（特等奖/O奖），获奖论文不仅在学术上有极高的价值，还能够提供创新的解决方案。 #### 二、2024年美赛特等奖论文概述 - **论文题目**：“动态水坝模型：一种多层次、以人为本的大湖水量调节方法” - **研究背景**：大湖作为北美的心脏地带，其水资源对于周边环境和社会至关重要。为了更好地保护大湖地区，研究人员提出了对现有Plan 2014进行扩展和完善的想法。 - **主要研究内容**： - **问题定义**：优化大湖区域的水位调节策略，以满足人类需求同时保护生态系统。 - **模型构建**：将大湖视为动态流网络，并开发了两种控制算法来确定最优的水坝调度方案。 - **算法设计**：第一种控制算法用于确定每个湖泊在一定时间范围内可通过Sault St. Marie河的Compensating Works水坝和Moses-Saunders水坝调节达到的最优水位。第二种控制算法则在日尺度上规划水坝的操作计划，以实现既满足各方利益又避免极端事件发生的水位目标。 - **参数设定**：模型采用机械式方法模拟实际水流，需要准确设定反映自然与人工过程的参数。考虑到大湖系统的复杂性和不确定性，研究者采用了综合方法来应对这些挑战。 - **创新点**：本研究的独特之处在于它提出了一种多层次的人本主义方法，通过结合长期和短期控制算法来平衡各种需求，并确保系统的稳定运行。 #### 三、动态水坝模型的核心要素 - **动态流网络**：该模型将大湖系统视为一个动态变化的流网络，其中水流受到多种因素的影响，包括天气、气候变化等。 - **控制算法**： - **多月尺度控制算法**：此算法的目标是在较长的时间尺度上找到最优化的水位方案，以适应季节性的变化和其他宏观条件。 - **日尺度控制算法**：该算法着重于短期内的具体操作，如每日的水坝开关控制，确保实际操作能够达到预定的水位目标。 - **参数优化**：为了提高模型的准确性和实用性，研究者需要仔细校准反映自然流程和人为干预的参数。这涉及到对历史数据的分析以及对未来情景的预测。 - **风险管理**：模型的设计考虑到了极端事件的风险管理，如洪水或低水位情况，以确保整个系统的安全性和可持续性。 #### 四、模型的应用前景 - **实践意义**：这种基于数学建模的动态水坝调节方案不仅可以应用于大湖区域，还可以推广到其他面临类似水资源管理挑战的地区。 - **跨学科合作**：项目本身展示了数学、计算机科学、环境科学等多个领域的交叉合作，为解决复杂的现实问题提供了新的视角和方法。 - **未来展望**：随着技术的进步和更多数据的积累，此类模型有望得到进一步的优化和完善，为实现更加精细化的水资源管理提供有力支持。这篇2024年美赛特等奖论文不仅展示了高水平的数学建模能力和技术创新，还为解决实际问题提供了有价值的思路和解决方案。通过深入分析和应用这类模型，我们可以更好地应对未来的水资源挑战，实现可持续发展。

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Problem Chosen

2024

MCM/ICM

Summary Sheet

Team Control Number

2429211

Dynamic Dams Model: A Multigranular, Human-Centered Approach For

Modeling Water In The Great Lakes

Summary

The Great Lakes serve as the heart of North America with their ebbs and ﬂows providing the

lifeblood of both the surrounding environment and societies. Because of the lake’s incredible

importance, it is in the best interest of the world to protect them. Plan 2014, proposed by the

International Joint Committee (ĲC), manages the water in the Great Lakes by building up water

behind both the Compensating Works dam at the Sault St. Marie River and Moses-Saunders dam

and taking action as needed. To better serve the Great Lakes region, we seek to expand Plan 2014

to regulate water levels optimizing for human needs and the preservation of ecosystems.

To determine optimal dam scheduling we modeled the Great Lakes as a dynamic ﬂow network

and created two control algorithms that utilize linear programming to solve for optimal dam

schedules and water levels. The ﬁrst control algorithm determines the optimal water level in each

lake that is achievable through use of the Compensating Works and Moses-Saunders dams over the

course of a multiple-month time horizon. After the optimal water levels have been determined, we

employ another control algorithm to plan how to schedule the dams over a daily time horizon to

achieve the water levels that equitably beneﬁt stakeholders, but still adhere to the larger schedule. By

utilizing these two control algorithms, it is possible to generate dam schedules that meet stakeholder

needs while avoiding catastrophic events like ﬂooding or dangerously low water levels.

An important aspect of our model is that it is mechanistic. Since we model real ﬂows, it is

important to determine accurate parameters to encode natural and artiﬁcial processes. However,

the Great Lakes are a highly complex system, and they are inﬂuenced by several stochastic and

highly volatile processes. To account for this complexity, we utilized a data-based approach to best

coincide with the data that the ĲC will have access to. By utilizing time lagged cross-correlation,

we were able to determine the relationship of ﬂow rates between Great Lakes. We also utilized

linear regression to model the relationship between the height of each Great Lake and the rate

of ﬂow in its distributaries. This allowed us to include necessary complexities while maintaining

accurate and faithful parameters.

Our model provides a number of key insights into eﬀective management of the Great Lakes. We

observe that using adaptive control algorithms, it is possible to determine schedules that avoid

ﬂooding in most cases and allow the ĲC to be prepared when ﬂooding is inevitable and to inﬂuence

the location and time of the ﬂood. Moreover, we observe that despite the large amount of time it

takes for Lake Ontario to experience the eﬀects of changes in Lake Superior, it is crucial to manage

Lake Superior with the downstream eﬀects on Lake Ontario in mind. When Lake Ontario faces a

challenging climate event, eﬀective scheduling at Lake Superior can mitigate the damage and allow

Lake Ontario to maintain optimal water levels.

Keywords: Linear Program; Dynamic Flow Network; Human-Centered; Multigranular Approach.

Team # 2429211 Page 3 of 24

1 Introduction

1.1 Background

Figure 1: A cloudless view of all of the Great

Lakes taken by a NASA satellite in August 2010

[13].

The Great Lakes are one of the most promi-

nent features of the North American continent;

they not only make up 20 percent of the world’s

freshwater, but they also have a tremendous im-

pact on the economy, climate, and ecosystems

of the United States and Canada. The Great

Lakes region is one of the largest economies

in the world, supporting 1.5 million people and

forming the backbone for other industries which

are valued at over 6 trillion dollars [8]. Addi-

tionally, they also are the source of life for over

3500 species of plants and animals across a

wide range of biomes making them indispens-

able to the North American ecosystems [18].

Due to the invaluable beneﬁt that the Great

Lakes provide us, it is of the utmost importance

to society that they are ﬂourishing in the best

way possible.

To ensure that the Great Lakes community is well-serviced, it is necessary to analyze the impacts

on the stakeholders dependent on the lakes. The stakeholders identiﬁed in the report by the Inter-

national Joint Committee (ĲC) are: domestic water supply, commercial navigation, hydropower,

environment, recreational boaters, docking companies, and shoreline (riparian) property owners

[10].

The current methodology of managing the water ﬂow in Lake Ontario is based on Plan 2014.

Plan 2014 prioritizes natural ﬂow out of Lake Ontario into the St. Lawrence River while continually

checking if a number of critical points have been reached. When a critical point has been reached,

the ĲC takes action to control the level of water in the dam [2]. After the initiation of Plan 2014

in 2017, record precipitation hit the area and Lake Ontario experienced severe ﬂooding [7]. This

caused millions of dollars of damage and increased the risk for invasive species (like zebra mussels),

harmful algae bloom, and sewage blockage [15] [19] [10]. This event brought the eﬀectiveness of

the plan under scrutiny. Flooding has proven to be a continual threat to the lives and livelihoods

of the stakeholders. Thus, ICM has set out to formulate water level control mechanisms that could

prevent such dire events.

1.2 Restatement Of The Question

To address the concerns of the ĲC, we propose a number of human-centered algorithms that

control the water levels of the Great Lakes to best address the residents’ concerns around Lake

Ontario and the St. Lawrence River. Our models and algorithms are ﬁt to address the needs of

various stakeholders, while ensuring that the water levels within the lakes remain safe. Motivated

by the ﬂooding damage from 2017, our model focuses on achieving water levels and currents that

Team # 2429211 Page 4 of 24

beneﬁt the stakeholders subject to hard constraints on the risk of ﬂooding and low-water levels. Our

outlook diﬀers signiﬁcantly from that of Plan 2014 to be more resilient in a highly volatile climate.

We propose using a linear program to model the needs of the stakeholders with respect to the water

level and modulating it with dynamic control of dams that account for the transportation times of

the water. Inspired by the successes of Plan 2014 as a method to solve a complex constrained

scheduling problem and equipped with the knowledge of where Plan 2014 falls short, we propose

a model that utilizes the Compensating Works dam at the Sault. St. Marie River as well as the

Moses-Saunders dam to do the following:

1. Center the human impacts by modeling the needs of the stakeholders

2. Robustness to severe climate events

3. Dynamic control of the system

1.3 Overview of Linear Programs

Our control algorithms are implemented as linear programs. Linear programming is a widely-

used optimization technique in the ﬁeld of operations research and network science. It has various

applications from policymaking to power plant deployment to production logistics to watershed

management [11, 20]. Linear programs rapidly found applications across the natural sciences,

industry, government, and many other ﬁelds, and are a natural way to model many scheduling

problems where it is important to understand how to plan a series of actions to maximize an

objective (typically proﬁt or some other measure of overall utility) subject to some constraints (e.g.

there might only be so much time available to us) [21]. Within the realm of environmental decisions,

linear programming has been applied to solve complex questions like green-energy planning [5].

Inspired by the success of linear programs in related ﬁelds, our control algorithms employ linear

programming to determine how to utilize the dams within the Great Lakes. The advantages to be

expected from utilizing linear programming are as follows:

• Algorithms for solving linear programs are widely available and extremely fast. Due to the

eﬀectiveness of the famous Simplex algorithm and its variants, linear programs can be solved

with incredible speed which is advantageous for testing out a variety of weather scenarios.

• Linear programs are interpretable. Practitioners need access to a white-box control algo-

rithm where it is possible to understand their decision.

• Linear programs are not deﬁned with speciﬁc parameter values in mind. If practitioners need

to revise their predictions for environmental factors, they can easily rerun the model with the

adjusted parameters.

However, it’s also important to acknowledge the downsides that accompany linear programs:

• The objective and constraints must be linear. This limits the ways we can model our system

considerably and, as such, it is important to recognize parts of the model that are remnants

of the required linearity.

Team # 2429211 Page 5 of 24

• When a linear program is impossible to solve, it is diﬃcult to visualize what is going wrong.

If our control algorithms determine that there is no valid schedule that meets all constraints,

it will be diﬃcult for the practitioner to ﬁnd a way to determine a valid schedule.

By keeping these disadvantages in mind, we designed our control algorithms to be eﬀective even

with the drawbacks of linear programming.

1.4 Assumptions And Justiﬁcations

As mentioned in the problem statement, the dynamic network ﬂow problem is wicked. As such,

we make use of several assumptions to manage the complexity of our models.

↓

Figure 2: Since water levels do not vary

much relative to total depth, we can approxi-

mate the lake surface area as being constant.

We can then convert ﬂow to marginal height

by dividing by surface area.

Hydrology: The lake’s sizes remain con-

stant and our system can be simpliﬁed

to ﬁve nodes by not factoring in the

auxiliaries like lake St. Claire.

Justiﬁcation: Keeping the lake sizes con-

stant provides us with a consistent con-

version metric, as seen in ﬁgure 2 be-

tween ﬂow and height for each lake.

Based on the correlation data for the

ﬂow rates, the other nodes are negligi-

ble as compared to the main outﬂows

between the Great Lakes.

Systemic: Dams can be fully opened or closed within an hour and they have zero downtime.

Additionally, the lag times calculated from correlation data are accurate (Figure 2).

Justiﬁcation: We want to encapsulate granular dam schedules to minimize ﬂooding and drought of

the Great Lakes community. Also, we need the lag times to be accurate to simulate the water

ﬂowing between the lakes.

Scaling: The regressions we found for the ﬂow rates from the lakes are linear (Figure 3) and the

time horizon of the data is on the order of a few months.

Justiﬁcation: The regressions are done on the order of the change in height of the lakes, so the

ﬂow rates for our purposes should be accurate. Since the time scale of our time lags is on the

order of a few months, our time horizon encapsulates the lag time and our model should be

used dynamically based on current data.

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