Problem Chosen
C

2024 MCM/ICM Summary Sheet

Team Control Number
2419588

Optimizing Great Lakes Water Management: An Adaptive
Hydrological Network Approach
 Summary
The Great Lakes, holding 21% of the world's fresh surface water, face complex water level management challenges due to natural and human factors. This study presents an Adaptive Hydrological Network Simulation and an Equilibrium Stake-
 holder Satisfaction (ESS) Model to address these challenges. The network model
represents the lakes as interconnected nodes and edges, simplifying the system's
 complexity and facilitating analysis. The ESS Model integrates stakeholder prefer-
ences, prioritizing their diverse interests in water level management. The study introduces a modified dam control algorithm based on Model Pre-
dictive Control (MPC), enhancing the responsiveness of water level regulation to
 environmental changes. The algorithm's performance is assessed using 2017 data,
revealing its sensitivity to high water levels and the need for improved adaptation
 ls to extreme weather events. The ESS Model's stakeholder satisfaction functions are
tailored to reflect the preferences of shipping companies, dock managers, environ-
e mentalists, property owners, recreational users, and hydro-power generation com d panies, with weights assigned to balance their influence. o The research concludes that the proposed models and algorithms show prom-
ise in managing the Great Lakes' water levels effectively, considering ecological, so-
 cial, and economic factors. However, there are limitations, such as the model's simm plification of natural factors and the need for empirical validation of stakeholder  H satisfaction functions. Future work should focus on refining the model to better cap-
ture the system's complexity and improve predictive accuracy.
 AT Keywords: Adaptive Hydrological Network; Equilibrium Stakeholder Satisfaction;
Model Predictive Control; Great Lakes Water Management; Environmental Sensitiv-
 M ity Analysis 
 

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Content
1 Introduction...................................................................................................1
1.1 Problem Background ....................................................................................................1 1.2 Restatement of the Problem.........................................................................................1 1.3 Our Work .......................................................................................................................1
 2 Assumptions and Justifications ..................................................................2  3 Notations..........................................................................................................3
4 Adaptive Hydrological Network Simulation...........................................3
4.1 Introduction to the Network Model...........................................................................3
 4.2 Model Framework: Representation of Nodes and Edges .......................................3
4.3 Mathematical Formulation of the Network Model..................................................4 4.4 River Flow between Lakes...........................................................................................4
 4.5 Incorporating Uncontrollable Factors ........................................................................8
5 Equilibrium Stakeholder Satisfaction (ESS) Model.............................11
 ls 5.1 Overview of the ESS Model.......................................................................................11
5.2 Stakeholder Analysis ..................................................................................................11
e 5.3 Stakeholder Satisfaction and Weighting..................................................................13  d 5.4 Calculation of Optimal Water Level ........................................................................15 o 6 Dam Control Algorithm Implementation...............................................16  6.1 Control Objectives and Performance Metrics .........................................................16 m 6.2 MPC Model..................................................................................................................16  H 7 Dam Control Algorithm .............................................................................17
7.1 Analysis of Plan2014...................................................................................................17
 T 7.2 The modification of the Rule Curve .........................................................................18 A 8 Assessing Control Algorithm Sensitivity in 2017 .................................18
8.1 Data Collection and Model Simulation....................................................................18
 M 8.2 Simulation Results and Analysis ..............................................................................19
8.3 Sensitivity Assessment of Control Algorithms.......................................................19
 8.4 Stakeholder Impact Evaluation.................................................................................19  8.5 Algorithm Sensitivity to Environmental Changes .................................................20
9 Model Evaluation and Further Discussion .............................................20
9.1 Strengths.......................................................................................................................20 9.2 Weaknesses ..................................................................................................................20
 9.3 Further Discussion ......................................................................................................20  10 Conclusion...................................................................................................21
References ......................................................................................................... 22

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1 Introduction

1.1 Problem Background

Managing the Great Lakes' water levels, crucial for regional ecosystems, econo-

mies, and communities in the US and Canada, is complex. These lakes, with 21% of

the world's fresh surface water, face seasonal changes due to natural phenomena like precipitation and evaporation, and human impacts from industri-



alization and water regulation structures, such as the Soo



Locks and Moses-Saunders

Dam. Overseen by the Interna-

 tional Joint Commission (IJC),

efforts to balance stakeholder

interests are challenged by cli-

 ls mate change-induced fluctua-

Figure 1 Top view of the Great Lakes

e tions. This urgency underscores

 the need for effective water

d level management strategies, highlighting the importance of addressing this trans-

o boundary environmental issue.

 1.2 Restatement of the Problem

m The core problem involves devising a method to regulate the water levels effec-

 H tively, ensuring that the diverse needs of all stakeholders are met. This encompasses:

 Problem 1: Determining the optimal water levels for the five Great Lakes

 T throughout the year, considering the varying desires and the associated costs

and benefits for each stakeholder group.

A  Problem 2: Developing algorithms that can dynamically maintain these opti-

mal water levels based on real-time inflow and outflow data.

M  Problem 3: Assessing the algorithms' sensitivity, particularly for dam outflows,

 using 2017 data to forecast stakeholder satisfaction.

 

Problem 4: Assessing the algorithms' responsiveness to environmental changes, such as variations in precipitation, winter snowpack, and ice jams, which could
 significantly impact water levels.
Problem 5: Specifically focusing on Lake Ontario, given recent concerns over
 its water level management, to analyze the factors and stakeholders uniquely

 influencing it.
1.3 Our Work
To avoid complicated description , intuitively reflect our work process, the flow

chart is show as the following Figure 1

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    els Figure 1. Our Work  d 2 Assumptions and Justifications o To quantify the demands of the various stakeholders, we make the following as sumptions. m Assumption 1: The Great Lakes are considered as a complex network system.  H Justification: This assumption simplifies the complexity of the
model by representing the relationships between lakes and rivers
 T as a network, facilitating the analysis and understanding of interac-
tions among the lakes.
A Assumption 2: The surface area of the lakes remains relatively stable over long periods.  M Justification: This assumption is based on historical data and geographical studies, suggesting that the surface area of the lakes does  not change significantly, thus serving as a basis for calculating water level changes. Assumption 3: There is a stable relationship between lake water
 Justification: This assumption is based on the relative stability of
lake geomorphological features and the typical relationship between water levels and storage volume, known as the stage-storage
curve.
Assumption 4: River flow is primarily related to the water level of upstream lakes.
Justification: This assumption is grounded in the principles of fluid dynamics, where water flows from higher to lower water levels, and the flow rate is influenced by the water level difference and river characteristics.

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Assumption 5: Precipitation is the main factor affecting lake storage volumes, and

evaporation is the primary pathway for water loss in the Great

Lakes system.

Justification: This assumption is based on an understanding of the

Great Lakes' water balance studies, where precipitation and evapo-

 ration are key natural processes affecting the water balance.
Assumption 6: Ice jams have a significant impact on the hydrodynamics of the Great Lakes region.

 Justification: This assumption takes into account the potential ef-
fects of ice jams on river flow during winter, which is crucial for simulating and predicting water level changes during the winter

months.
3 Notations



In our paper, Lake Superior is referred to as 1, Lake Michigan and Lake Huron are referred to as 2, Lake Erie is referred to as 3, and Lake Ontario is referred to as
 4.
The key mathematical notations used in this paper are listed in Table 1.

 els Symbol  d  o 
12

Table 1: Notations used in this paper
Description Water level of  Flow of the river connecting , Compensating Works of the Soo Locks

Unit
 3-1

 m 4 Adaptive Hydrological Network Simulation

 H 4.1 Introduction to the Network Model Network models, graphing systems as interconnected nodes and edges, are key
 T for complex system analysis. The Great Lakes' intricate water dynamics require adA vanced hydrological modeling. This research uses a network model to understand the
macro-level water interactions, focusing on system-wide insights. The model provides
 M a theoretical basis for Great Lakes water management, improving control over com-
plex system interactions.
 4.2 Model Framework: Representation of Nodes and Edges

We view the Great Lakes Basin as a complex network system, with each lake considered as a node within the system, denoted by , where i represents the lake's se-
 quence number. The water level  of each node is its key attribute. Notably, since
Lake Michigan and Lake Huron are geographically adjacent without a clear river sep-
 aration and are often considered a single body of water based on multiple studies and

 provided data, we analyze these two lakes as one node in our network model. This
approach simplifies the model's complexity while maintaining an accurate description of the lake system's overall dynamics.

Every river connecting two lakes is represented as an edge  connecting nodes

 and , with the flow rate from lake i to lake j denoted by . For rivers controlled

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by dams, a control mechanism  is added to these edges.
4.3 Mathematical Formulation of the Network Model

Water level serves as an indicator of the lake surface height, while flow rate re-

flects the dynamic changes in the water volume of a lake. In studying the hydrological

characteristics of lakes, our goal is to reveal the intrinsic connection between water

level and flow rate. Given that the geomorphological characteristics of each lake are relatively stable, we can anticipate a stable relationship between water level and water
 volume, i.e., the water stage-storage curve. However, the complexity of lake geomor-
phology means this relationship can be influenced by various factors. According to
 the Great Lakes Commission, the surface area of the lakes remains relatively constant

 over long periods. Statistical analysis of historical data has yielded the following val-
ues for average water levels and standard deviations:

Table 2: Average Water Levels and Standard Deviations for the Great Lakes

Lake

Average Water Level (m) Standard Deviation (m)

 Superior

183.35

0.25

Michigan and Huron

176.33

0.45

 ls St. Clair

175.10

0.37

e Erie

174.28

0.32

 d Ontario

74.83

0.29

 o Given the relative stability of lake surface areas and the minor overall impact of m water level changes on lakes, it is reasonable to use the lake's surface area as a basis
for calculating water level changes.
 H In the Great Lakes system, for any given lake , the change in water level  over  T a given time interval  can be represented by the following formula:
A (1)  Here, Inflowtotal, represents the total inflow into lake , Outflowtotal, represents M the total outflow from lake , is the precipitation over lake ,  is the evap-
oration from lake , and  is the surface area of lake .
 The flow rate  from lake  to lake  can be described by a function , which re-
flects the relationship between the water level difference ( and ) and the charac-
 teristics of the river connecting them. The function f may include the effects of dam

control () and can be linear or nonlinear; the specific model will be explored in detail in subsequent research.
 4.4 River Flow between Lakes In this section, we delve into the river flow rate  from lake  to lake , defined
 as:

(2)

with the aim of identifying the unknown function .

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4.4.1 Exploratory Data Analysis (EDA)

In this section, we perform Exploratory Data Analysis (EDA) on the Great Lakes

system's water level and flow rate data to identify basic features, trends, and patterns.

EDA is vital for understanding data structure, spotting outliers, and preparing for

deeper analysis and modeling. Initially, we cleaned the data thoroughly to ensure

 analysis accuracy. We hypothesized that river flow rates depend on water level dif-
ferences between upstream and downstream lakes. To verify this, we analyzed water

 level differences between such lakes and their correlation with connecting rivers' flow
rates using Pearson, Spearman, and Kendall Correlation methods. These methods evaluate linear, non-linear, and rank-order variable relationships. Example analysis

 with Lake Superior, Lake Michigan, Lake Huron, and the St. Mary's River provided
specific correlation coefficients:

1. Pearson Correlation: -0.29 2. Spearman Correlation: -0.31
 3. Kendall Correlation: -0.20
And visualized the data sets accordingly:

THmodels Figure 3 Correlation between Water
Level Differences and River Flow
A Rates in the St. Mary's River

Figure 4. Histogram of Cross-Correlation Function (CCF) for Water Level
Differences and River Flow Rates

 M The analysis revealed a very low correlation between water level differences  and river flow rates, exhibiting a negative correlation, which contradicts the ex-
pected physical phenomena. Given this, we inferred that the effect of water level dif-
 ferences on river flow might have a time lag. To further explore this possibility, we

 employed the Cross-Correlation Function (CCF) to analyze correlations at different
time lags and generated corresponding histograms.

 The histogram results showed that, within a time lag range of 0 to 12 months,
the correlation between water level differences and river flow rates remained weak. This finding prompted us to reassess our initial hypothesis that water level differ-

ences directly influence river flow rates.

4.4.2 Determining the Function 

Upon further analysis of the physical scenario involving river flow rates and lake

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water levels in the Great Lakes system, we drew a preliminary conclusion based on

hydrodynamic principles: the flow rate of rivers is primarily related to the water level

of upstream lakes. This conclusion is grounded in the fundamental principles of fluid

dynamics, namely, that water flows from higher to lower water levels, and the rate of

flow is influenced by the water level difference and channel characteristics (such as

the slope and width of the channel).







 els Figure 5. Physical Model of River Flow Rates and Lake Water Levels in the Great  Lakes System
d In this physical model, the water level of downstream lakes does not directly afo fect the supply of water from upstream lakes to rivers. Instead, the water level of up stream lakes determines the amount of water that can pour into the rivers. When the m water level of an upstream lake is higher than the elevation at the river's mouth, water  H naturally flows downstream. This process is driven by gravity, with water flowing
along the channel until a new equilibrium state is reached. Therefore, the key factors
 T determining river flow rate are the water level of the upstream lake and the height A difference at the river's mouth.
To further validate the direct correlation between the water level of upstream
 lakes and river flow rates, we conducted a detailed correlation analysis between the M water level of Lake Superior and the flow rate of the St. Mary's River. The analysis  results showed a Pearson Correlation of 0.91, a Spearman Correlation of 0.89, and a

 Kendall Correlation of 0.72, all indicating a significant positive correlation without
evidence of time lag effects.





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Figure 6. Correlation Analysis of Upstream Lake Water Levels and River

 Figure 7. Heatmap of Correlation
Coefficients for Upstream Lake Water

Flow Rates

Levels and River Flow Rates

To rule out coincidence, our study was extended to include all upstream lakes

 and their downstream rivers, employing matrix analysis to evaluate correlation coef-

ficients, visualized via heatmap. The findings revealed a significant linear positive

correlation across upstream lakes' water levels and river flow rates, confirming the

 ls hypothesized linear relationship applicable throughout the Great Lakes system. This

e evidence underpins the use of linear modeling for forecasting and examining the dy-

 d namic interplay between upstream lake water levels and river flow rates.

Upon examining the correlations within the Great Lakes system, we noted a rel-

o atively weaker correlation between Lake Ontario and the St. Lawrence River (correla-

 tion coefficient of 0.62), while other lakes showed strong linear positive correlations

m with river flow rates (correlation coefficients over 0.9). These observations validate

 H employing a linear model to represent the relationship between upstream lake water

levels and river flow rates, formulated as:

 T (3)

A where  and  are coefficients to be determined.

The weaker correlation for Lake Ontario and the St. Lawrence River was initially

 M speculated to be due to the confluence of the Ottawa River with the St. Lawrence River.

However, upon verification of the monitoring station data (station number 04264331),

 it was found that this station does not receive inflow from the Ottawa River. Further

literature research revealed the root cause: the construction of dams as part of the "Lake Ontario  St. Lawrence Plan 2014" to control water levels, a human activity that
 disrupted the natural correlation between water levels and flow rates. The official doc-
ument of the plan provided the pre-project release relationship formula for Lake On-
 tario's outflow and its water level:



(4)

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 els Figure 8. Heatmap of Correlation Coefficients for Upstream Lake Water Levels d and River Flow Rates
o After conducting a linear fitting analysis of the relationship between water levels  and flow rates for all lakes, we obtained the following results table. This table details m the linear fitting coefficients for each lake, providing a scientific basis for subsequent  H water resource management and decision-making.
Table 3: Linear Fitting Coefficients for Lake and River Pairs

 T Pair

Relationship

A Lake Superior - St. Mary's River

12 = 12(1713.681 - 312180.54)

 Lake Michigan and Lake Huron - St. M Clair River

23 = 1504.202 - 259900.25

 Lake St. Clair - Detroit River

34 = 1952.983 - 336406.51

 Lake Erie - Niagara River
Lake Ontario - St. Lawrence River

45 = 2088.92  4 - 357993.82 56 = 555.82356(5 - 0.035 - 69.474)1.5

4.5 Incorporating Uncontrollable Factors
In addressing the water balance issues within the Great Lakes basin, we inevita-
 bly encounter numerous complex factors that affect lake water levels and flow rates.
However, in this study, we will focus primarily on precipitation and evaporation, as
 these two components play a pivotal role in the lake water balance. According to re-

search by Neff and Nicholas (2004), precipitation and evaporation are the main factors

in the lake water balance, significantly impacting the long-term variations in lake wa-

ter levels. We will now model and analyze precipitation and evaporation separately.

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4.5.1 Precipitation

Precipitation often acts as the largest influencing factor on lake storage volumes,

but its impact varies across different regions due to distinct geological and geomor-

phological characteristics. Intuitively, we know that precipitation  (in mm) positively

correlates with water storage brought to lakes, represented by  + , where we model

each lake individually in an exploratory manner.
(5)
 We collected data on the daily change in water level , controlled inflows
-1,, controlled outflows ,+1, and precipitation  for lake  to fit and determine the
 precipitation function  for each lake. This approach allows for a nuanced understanding of how precipitation impacts
 each lake's water balance, accounting for regional differences and contributing to

more accurate water management and decision-making strategies. Through fitting these models, we can better predict the effects of precipitation on lake water levels,
 adjusting for controlled inflows and outflows, and thereby enhancing our ability to
manage the Great Lakes' water resources effectively. 4.5.2 Evaporating
 ls Evaporation is a primary pathway for water loss in the Great Lakes system, playe ing a crucial role in the hydrological cycle and water resource management of the en tire basin. We employs the Lumped Modeling approach proposed by Croley (1989), d which integrates the Monin-Obukhov similarity theory with a heat storage model to o provide a comprehensive computational framework for simulating evaporative fluxes  over the Great Lakes. m The process begins with the calculation of saturated specific humidity () using  H the Arden-Buck equation, which describes the specific humidity at saturation under
given temperature and atmospheric pressure conditions. The formula for saturated
 T water vapor pressure () is as follows: A (6)
 Where  is the saturated water vapor pressure in millibars (mb), which can be M calculated based on temperature  and atmospheric pressure (Pa). The actual spe cific humidity (qa) reflects the actual water vapor content in the air, and its calculation

is given by:



(7)

 Where  is the water vapor pressure (mb), which can be derived from relative
humidity (RH) and atmospheric pressure (Pa).

 Subsequently, the bulk evaporation coefficient () is determined, a key parame-
ter describing the efficiency of water vapor transfer from the water surface to the atmosphere. The calculation of  involves atmospheric stability analysis, typically

based

on

the

Monin-Obukhov

length

(  )

and

stability

parameter

(

 

),

requiring

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meteorological data such as wind speed (), air temperature (), and relative humidity (RH).
The daily evaporation rate () is calculated using the following formula:

(8)

In this equation,  represents the evaporation rate,  is the bulk evaporation coefficient,  and  are the specific humidities at the water surface and in the air, re-
 spectively,  is the wind speed,  is the density of water, and  is the density of air. Compute  based on the algorithm shown in the accompanying figure and the
 equation provided in the text. The equation and algorithm are given in the paper.

 The meanings and units of the physical quantities in the formula are shown in the
following table:

Symbol

Description

Unit



reference height





roughness length



 

mean wind speed at reference height Z above the surface   -1



friction velocity

  -1

 ls 

Von Karman's constant

1, 2
e 

stability-dependent parameters

Moninabukhov length



 d 

absolute temperature of near-surface air





acceleration due to gravity

  -2

o 

daily evaporation rate



 The wind speed , humidity , and temperature  are variables that need to be

m obtained by collecting meteorological and hydrological data. The remaining variables

 H can be derived from existing work or reasonably set. Through these steps, our model

enables a dynamic and accurate simulation of the evaporation process for the Great

 T Lakes basin, which is essential for understanding the hydrodynamics of the lakes and

 MA informing water resource management strategies.







Figure 9. Calculation of 

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4.5.3 Ice jams In the development of a water balance model for the Great Lakes, the phenome-
non of ice jams plays a crucial role in affecting the hydrodynamics of the region. Ice jams occur during winter when the lake surface freezes and the ice can obstruct the flow of rivers, leading to potential flooding and altered water levels. To accurately represent this phenomenon in our model, we introduce the ice jam factor (\ ), which quantifies the severity of the ice jam and its impact on river flow.
 The mathematical relationship between the actual river flow () and the ice jam
factor () is expressed as follows:
 (9)  Here,  represents the river flow prior to the occurrence of an ice jam. This
equation allows us to model the reduction in river flow due to ice jams, where  ranges from 0 (no ice jam, full flow) to 1 (complete ice jam, no flow). By incorporating this relationship into our water balance model, we can simulate the effects of ice jams on
 the Great Lakes' hydrology, providing a more nuanced understanding of the system's
behavior during the winter months and informing strategies for water resource management and flood mitigation.
 ls 5 Equilibrium Stakeholder Satisfaction (ESS) Model  e 5.1 Overview of the ESS Model d The Equilibrium Stakeholder Satisfaction (ESS) Model is a novel approach deo signed to optimize water level management in the Great Lakes, with a focus on Lake  Ontario and the St. Lawrence River. It addresses the need to balance the diverse and m sometimes conflicting interests of stakeholders in shipping, power generation, conser H vation, recreation, and property values.
The ESS Model's objective is to find an optimal water level that maximizes stake-
 T holder satisfaction throughout the year, considering the Great Lakes' complex hydro-
logical dynamics, including precipitation, evaporation, runoff, and human activities.
A It uses mathematical and hydrological analyses to evaluate the effects of water level  changes on stakeholders and prioritizes these effects based on their importance. M Developed in response to the International Joint Commission's (IJC) call for ad vanced management strategies, the ESS Model integrates stakeholder satisfaction into
water level management decisions, considering technical, hydrological, social, economic, and environmental factors.
 5.2 Stakeholder Analysis In the development of the Equilibrium Stakeholder Satisfaction (ESS) Model for
 Lake Ontario and the St. Lawrence River, a thorough stakeholder analysis is essential  to ensure the model accurately reflects the diverse interests and influences of those
affected by water level management. This analysis categorizes stakeholders into six primary groups, each with distinct preferences and stakes in water level outcomes. Herein, we delve into the specific interests of these groups and assess their potential impact on water level management through an influence-interest matrix.

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5.2.1 Identification and Preferences of Stakeholders

Shipping Companies: They require high and stable water levels in the St. Law-

rence River for safe navigation and operational efficiency, as fluctuations can disrupt

vessel passage and supply chains.

Dock Managers and Montreal Residents: They advocate for consistent and low

 water levels to prevent flooding and protect waterfront properties and infrastructure,
impacting dock operations and community well-being.

 Environmentalists: They call for water levels that follow natural seasonal varia-
tions to support ecosystem health, with higher spring levels for species breeding and waterway cleansing.

 Property Owners along Lake Ontario: These stakeholders prefer stable water
levels to prevent shoreline erosion and protect their investments from water damage

and land loss. Recreational Boaters and Fishermen: This group favors stable, accessible water
 levels to ensure marina and launch ramp usability for boating and fishing activities. Hydro-Power Generation Companies: They seek to manage water levels effectively for hydroelectric power generation, utilizing high levels as a natural reservoir
 ls to meet energy demands efficiently. e 5.2.2 Influence-Interest Matrix  The matrix below evaluates each stakeholder group based on their level of interd est in water level management and their capacity to influence decision-making proo cesses:
 Table 4: Influence-Interest Matrix for Stakeholder Groups

m Stakeholder Group

Level of Interest Potential to Influence

 H Shipping Companies

High

High

 T Dock Managers and Montreal Residents

High

Medium

A Environmentalists

High

Medium

 M Property Owners

High

Low

Recreational Boaters and Fishermen

Medium

Low

 Hydro-Power Generation Companies

High

High

High Interest & High Influence: Shipping companies and hydro-power genera-
 tion companies possess significant resources to advocate for their interests, making
them key players in water level management discussions.
 High Interest & Medium Influence: Environmentalists and dock managers/res-

 idents are highly vested in water level outcomes but have moderate influence, largely
due to public support and localized impact. High Interest & Low Influence: Property owners, despite their direct stake, often

lack the organized representation needed to exert substantial influence.

Medium Interest & Low Influence: Recreational users, while affected by water

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levels, typically do not have the collective capacity or economic impact to significantly

sway management decisions.

5.3 Stakeholder Satisfaction and Weighting

The ESS Model addresses Lake Ontario and St. Lawrence River water level man-

agement by incorporating stakeholder satisfaction into its analytical framework. It

uses mathematical functions to quantify how stakeholder satisfaction changes with water levels, providing a basis for optimizing water levels. Weights are assigned to
 these functions to represent the stakeholders' relative influence, ensuring a balanced
consideration of diverse interests and an equitable outcome.
 5.3.1 Satisfaction Functions

 The Equilibrium Stakeholder Satisfaction (ESS) Model quantifies how water level
variations impact stakeholder contentment within Lake Ontario and the St. Lawrence River, employing distinct satisfaction functions for each group. Generalized Satisfaction Function:
 The general formula for calculating stakeholder satisfaction () with respect to water
levels () is:
 ls () = 1 - |-|  e  : Satisfaction score for stakeholder . d  : Current water level. o  : Ideal water level for stakeholder .   : Acceptable range of water levels for stakeholder . m Tailored Satisfaction Functions:  H 1. Shipping Companies (1):  T a) Objective: Ensure navigable conditions for medium-sized vessels in the
St. Lawrence River.
A b) Parameters: Vessels' design specifications require a minimum navigation depth of 8.3 meters to maintain a satisfaction level of 1. This translates to a  M Lake Ontario water level of at least 74.4 meters. Below 68 meters, navigation becomes impossible, dropping satisfaction to 0.
 c) Function: The satisfaction function for shipping companies, considering  water depth as the primary factor, is modeled as:



(10)

2.

 Dock Managers and Montreal Residents (2):
a) Objective: Maintain water levels that prevent flooding while ensuring

port operation efficiency.

b) Parameters: Analysis of the Port of Montreal data and the 2017 flood dis-

aster reveals an optimal water level range between 74.4 meters and 76.4

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meters for Lake Ontario to mitigate flood risks and maintain port opera-

tions. Satisfaction is maximized (1) within this range and decreases outside

of it due to increased flood risk or operational inefficiency.

c) Function: Satisfaction for this group inversely correlates with water levels

beyond the identified safe operational range:

3. Environmentalists (3):

(11)

a)

 Objective: Support ecological health through pronounced seasonal water
level variations.

4.

b) Parameters: Environmentalists advocate for a water level range that promotes biodiversity and natural shoreline processes. The satisfaction level
 peaks when Lake Ontario's water levels facilitate natural habitat preserva-
tion and seasonal variations.
 ls c) Function: Given the preference for natural fluctuation, the satisfaction function for environmentalists might resemble a sinusoidal pattern over e the course of a year, reflecting higher satisfaction with greater seasonal  d amplitude. However, for simplicity and in line with available data, a modo ified linear approach can be applied, focusing on the range of water levels  that are deemed ecologically beneficial: m (12)
 H Property Owners, Recreational Boaters, and Fishermen (4 & 5):  AT a) Objective: Maintain water levels conducive to property integrity and rec-
reational activities.
 b) Parameters: These stakeholders prefer stable water levels to minimize eroM sion and facilitate recreational activities. Ideal conditions are identified be tween 74 meters and 76 meters for Lake Ontario, where satisfaction is at its

c)

peak. Function: The satisfaction function for property owners and recreational
 users emphasizes stability within the preferred range, decreasing as levels  move outside this range due to increased risks or reduced accessibility:

 (13)
Here, the range centers around 75 meters, with satisfaction declining linearly as

water levels diverge from this midpoint.

5. Hydro-Power Generation Companies (6):

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a) Objective: Maximize energy generation efficiency through optimal water

level management.

b) Parameters: Hydroelectric power generation is optimized at higher water

levels, with operational thresholds based on the design and capacity of fa-

cilities like the Moses-Saunders Dam. An ideal water level for maximizing

 power generation efficiency and safety is established at or above 75.5 me-
ters.

c)

 Function: The satisfaction function reflects a preference for higher water
levels, plateauing once operational maximums are reached to indicate no further gains in satisfaction:

 (14)

5.3.2 Weight Assignment

 To enable the ESS Model to equitably manage the varied stakeholder interests in

the water levels of Lake Ontario and the St. Lawrence River, structured weight alloca-

 ls tion is crucial. These weights measure each stakeholder group's relative impact, steer-

e ing the optimization towards solutions that meet comprehensive economic, ecological,

 and social goals. A succinct overview of the weight distribution follows:

d Table 5: Stakeholder Satisfaction Functions and Assigned Weights

o Stakeholder Group

Assigned Weight

 Shipping Companies

0.20

m Dock Managers and Montreal Residents

0.15

Environmentalists

0.15

 H Property Owners

0.10

 T Recreational Boaters and Fishermen

0.10

Hydro-Power Generation Companies

0.20

A The weights assigned in the ESS Model represent the stakeholders' relative influ-

ence in decision-making, considering their economic, ecological, and social impacts,

 M as well as policy influence. These weights, ranging from 0 to 1, are based on the Influ-

ence-Interest Matrix. Shipping and hydropower stakeholders have the highest

 weights due to their economic significance. Environmentalists and dock manag-

ers/residents receive moderate weights for their influence on public opinion and local

 economies. Property owners and recreational users are assigned the lowest weights,

reflecting their localized concerns.
5.4 Calculation of Optimal Water Level
 5.4.1 Aggregation of Stakeholder Satisfaction To determine the optimal water level, , that maximizes stakeholder satisfac-
 tion for Lake Ontario and the St. Lawrence River, we apply a weighted satisfaction

model. This involves assigning weights to each stakeholder's satisfaction function

based on their importance, and then aggregating these to calculate a collective satis-

faction score.

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Each stakeholder's weighted satisfaction score is defined as: (15)

1. (): Weighted satisfaction score for stakeholder  at water level .

2. : Assigned weight reflecting stakeholder 's influence. 3. (): Satisfaction function for stakeholder , indicating satisfaction at
water level .
 The overall satisfaction, (), across all stakeholders is the sum of individual  weighted scores:



(16)

The model seeks  where:
(17)
 This method identifies the water level that, considering all stakeholder weights
and preferences, achieves the highest aggregate satisfaction, representing an equilib-
 ls rium point of collective benefit. e 6 Dam Control Algorithm Implementation  d 6.1 Control Objectives and Performance Metrics o The control strategy for the Great Lakes water system aims to achieve optimal  water level regulation across interconnected lakes and rivers. Based on previously esm tablished hydrological and ideal water level models, this strategy utilizes Model Pre H dictive Control (MPC) to formulate an optimization problem at each control interval.
The objective is to minimize the deviation between actual water levels and their ideal
 T states while considering system constraints and operational costs.
6.2 MPC Model
A The optimization problem is defined as follows:  First, we define the objective function. The objective function aims to minimize M the sum of the weighted squared deviations of water levels and inter-lake exchange  flows from their ideal values over a prediction horizon of 0 weeks. This is repre-

 sented by the performance metric J:

(18)

 The constraints include maintaining water levels within safe operational limits
and ensuring that the exchange flows between lakes do not exceed their capacity:

 (19) (20)
The control input at each time step is the release volume from each lake to the

downstream lake, which is subject to the constraints and the objective function:

(21)

where () is the control action for  at  week.

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To solve this optimization problem, a genetic algorithm (GA) is employed. By leveraging the power of genetic algorithms, the control strategy is capable of efficiently navigating the complex search space of the optimization problem, ensuring the Great Lakes water system operates within the desired parameters while minimizing environmental and economic impacts.
7 Dam Control Algorithm
 7.1 Analysis of Plan2014
7.1.1 Bv7 Algorithm
 The Plan 2014 Algorithm, as detailed in the "Lake Ontario - St. Lawrence River  Plan 2014," is a sophisticated water management tool designed to regulate the water
levels of Lake Ontario and the St. Lawrence River. The algorithm is based on a combination of mechanistic release rules, known as "Bv7," and discretionary decisions made by the International Lake Ontario - St. Lawrence River Board (ILOSLRS) to address
 deviations from the specified flows under certain conditions. The algorithm uses a sliding rule curve, which is a function of the pre-project stage-discharge relationship adjusted for recent long-term supply conditions ,+1.
 ls For periods of above-normal supply, the lake release is determined by: e (22)
 d Conversely, for periods of below-normal supply, the lake release is calculated as:  o (23)
m The algorithm also incorporates flow limits to prevent extreme fluctuations in  H river flows, ensuring stable conditions for navigation, ice formation, and environmen-
tal health. These limits include the J Limit, M Limit, I Limit, and L Limit, which re-
 T spectively control the maximum change in flow, balance low levels for navigation,
ensure ice stability, and maintain adequate levels for navigation and overall flow.
A 7.1.2 The Strength & Weakness of Bv7 Algorithm  The Bv7 algorithm, as implemented in the Plan 2014, demonstrates several M strengths in its water management capabilities. It incorporates a robust predictive  framework that accounts for a variety of hydrological conditions, ensuring a stable
water exchange between Lake Ontario and the St. Lawrence River. This algorithm has been effective in balancing the diverse interests of navigation, power generation, and
 ecological health, providing a comprehensive approach to water regulation. However, the algorithm's performance in 2017 highlighted a critical weakness: its
 sensitivity to high water levels. The model's response to the significant precipitation  event that summer was not as agile as required, leading to elevated water levels and
potential flood risks. This indicates a need for enhanced sensitivity in the algorithm, particularly in its ability to anticipate and react to extreme weather events and rapid changes in water supply.

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(24)

7.2 The modification of the Rule Curve

Our proposed control strategy is grounded in Model Predictive Control (MPC), a

method that anticipates the system's state over a future horizon and optimizes the

control inputs at each control interval. The essence of MPC is to transform the control problem into an optimization problem over a finite time horizon. Given the limitations
 of our meteorological forecasting capabilities and the need to prevent drastic changes
in dam outflow that could disrupt the ecological environment, MPC offers a suitable
 control approach.

 We acknowledge the value of the control techniques employed in Plan 2014 and
aim to enhance the model's sensitivity to water levels by refining its release rules. To

this end, we have introduced a modified equation for the lake outflow when the supply is above normal levels:
 (25)

 ls In

this

equation,

the

term

(()

-

 )

captures

the

deviation

of

the

water

level

e from the ideal state, with  serving a role analogous to 1 in the original Plan 2014  model. The inclusion of this term allows the model to respond more dynamically to d deviations in water levels, ensuring a more sensitive and adaptive control strategy. o The parameter  represents a scaling factor that adjusts the sensitivity of the  model to water level deviations. By tuning  and , we can fine-tune the model's rem sponsiveness to both short-term fluctuations and long-term trends in water levels,  H thus achieving a balance between ecological protection and efficient water manage-
ment.
 T Using J as the control objective, we optimized and fitted the model parameters
using annual data.

A 8 Assessing Control Algorithm Sensitivity in 2017

 M In 2017, the Great Lakes experienced significant fluctuations in water levels,  prompting a reassessment of the control algorithms for dam outflows. This chapter

investigates whether the existing algorithms adequately adjusted to these fluctuations, with a focus on aligning water management practices more closely with the varying
 needs of stakeholders. The analysis aims to determine if modifications to the algo-
rithms could result in more satisfactory water levels for all parties involved.
 8.1 Data Collection and Model Simulation

 Utilizing detailed hydrological and environmental data for 2017, sourced from
the U.S. Army Corps of Engineers' Great Lakes Information website, we incorporated key indicators such as precipitation and evaporation rates into our network model.

This data foundation enabled the simulation of the Great Lakes' hydrological dynam-

ics throughout the calendar year.

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8.2 Simulation Results and Analysis
The following charts compare the actual monthly water levels of the lakes in 2017 (provided in the appendix) with the simulated data from the network model:
From the charts, it is evident that, with the exception of a few days where the simulated data deviate significantly from the optimal water levels, the majority of the data closely approximate the ideal water levels.
8.3 Sensitivity Assessment of Control Algorithms
 we dissect the moments when the simulated water levels diverged from actual
observations, pinpointing the algorithms' responsiveness to rapid environmental
 shifts. Specific instances of deviation--particularly during periods of abrupt precipi-
tation changes or evaporation spikes--serve as case studies. Through regression analysis, we quantify the algorithms' delay in response and the accuracy of their adjust-
 ments, identifying patterns that suggest areas for algorithmic refinement. This scru-
tiny reveals how the algorithms' current configurations might be fine-tuned to enhance their predictive accuracy and response speed, thereby minimizing discrepan-
 cies between simulated and actual water levels. MATHmodels Figure 10. 2017 Simulation
 8.4 Stakeholder Impact Evaluation The evaluation of stakeholder impacts centers on correlating the simulated water level adjustments with predefined satisfaction indices for each stakeholder group. By
 overlaying the periods of significant algorithmic deviation with stakeholder satisfac-
tion scores, we identify which stakeholder groups were most adversely affected by
 suboptimal water management. For instance, a detailed analysis is conducted on how
deviations impacted shipping schedules, hydroelectric power generation, and ecological conservation efforts. This examination not only highlights the direct consequences of algorithmic performance on each stakeholder group but also proposes targeted improvements to ensure that future algorithm adjustments more effectively balance the competing needs of all parties involved in the Great Lakes ecosystem.

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8.5 Algorithm Sensitivity to Environmental Changes
The sensitivity of our control algorithms to environmental conditions--specifically, precipitation, winter snowpack, and ice jams--is critically assessed in this section. By integrating environmental variability into our network model, we observed the algorithms' performance under a range of scenarios reflective of the diverse climatic challenges the Great Lakes face annually.
Precipitation: The algorithms demonstrated a calibrated response to increased
 precipitation, adjusting outflows to mitigate potential flooding. However, the reaction
time to sudden, heavy rainfall events highlighted a need for more rapid adaptation
 capabilities.  Winter Snowpack: The gradual melting of winter snowpack presents a predicta-
ble rise in water levels, to which the algorithms generally adapted well. The model simulations for spring thaws showed alignment with expected water level increases, suggesting adequate sensitivity to snowpack melting rates.
 Ice Jams: Ice jams, which can cause abrupt and localized water level rises, tested
the limits of the algorithms' responsiveness. Instances of ice jams within the simulation period revealed occasional delays in outflow adjustments, underscoring an area
 ls for algorithmic improvement in handling such unpredictable events. e 9 Model Evaluation and Further Discussion  d 9.1 Strengths o The model excels in offering a macro-level view of the Great Lakes system  through a comprehensive network of lakes and rivers, balancing system complexity m with overall dynamics accuracy. It incorporates stakeholder preferences into water  H level management via the Equilibrium Stakeholder Satisfaction (ESS) Model, ensuring
equitable decision-making. Enhanced predictive accuracy is achieved by integrating
 T meteorological and hydrological data, with Model Predictive Control (MPC) and ge-
netic algorithms optimizing dam control strategies to minimize environmental and
A economic impacts.  9.2 Weaknesses M The model's representation of the Great Lakes as network nodes overlooks time  lags caused by the lakes' vast areas, affecting water level responses. It excludes  groundwater, tributaries, and surface runoff impacts on the hydrological balance.
Simplifications in modeling natural factors like precipitation and evaporation may not fully mirror real-world conditions, including ice cover effects. Stakeholder satisfaction
 functions, based on linear relationships and assumptions, may not truly reflect stake-
holder preferences, potentially leading to suboptimal management decisions.
 9.3 Further Discussion Enhancements are needed to capture the Great Lakes system's complexity better and improve predictive accuracy. Future research should include more detailed spatial and temporal data, integrate groundwater and surface runoff, and employ advanced meteorological data for natural factor modeling. Stakeholder satisfaction functions require empirical validation to ensure they accurately represent diverse interests.

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By addressing these limitations and refining its components, the model can become a more effective tool for sustainable water resource management in the region.
10 Conclusion
This research has successfully developed and implemented an Adaptive Hydrolog-
 ical Network Simulation and an Equilibrium Stakeholder Satisfaction (ESS) Model to
address the intricate water level management challenges of the Great Lakes. Our ap-
 proach has demonstrated the potential to effectively balance the diverse and often
conflicting interests of various stakeholders, including shipping, power generation, conservation, and recreational activities, within the constraints of the Great Lakes' hy-
 drological dynamics. The modified dam control algorithm, based on Model Predictive Control (MPC), has shown improved responsiveness to environmental changes, particularly in the face of extreme weather events. The ESS Model's stakeholder satisfaction functions have been
 carefully calibrated to reflect the nuanced preferences of each stakeholder group, en-
suring that the water level management decisions are equitable and considerate of all parties involved.
 ls Despite the promising results, our study acknowledges several areas for future ime provement. The model's simplification of natural factors, such as precipitation and  evaporation, may not fully capture the complexities of real-world conditions, includd ing the effects of ice cover. Additionally, the stakeholder satisfaction functions, while o tailored, require empirical validation to ensure they accurately represent the true pref erences of the stakeholders. m Looking ahead, further research should focus on refining the model to incorporate  H more detailed spatial and temporal data, integrate groundwater and surface runoff
dynamics, and utilize advanced meteorological data for more accurate natural factor
 T modeling. Empirical validation of stakeholder satisfaction functions is also crucial to A ensure that the model's predictions align with real-world stakeholder preferences and
outcomes.
 By addressing these limitations and enhancing the model's components, we believe M that our approach can become a more robust and effective tool for sustainable water  resource management in the Great Lakes region, contributing to the long-term ecolog ical health and economic vitality of this vital freshwater system.
 

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References

[1] Wilcox, D.A., Thompson, T.A., Booth, R.K., & Nicholas, J.R. (2007). Lake-level var-

iability and water availability in the Great Lakes. U.S. Geological Survey Circular

1311, 25 p.

 [2] U.S. Army Corps of Engineers. (n.d.). Water level data. Retrieved February 6, 2024,

from

https://www.lre.usace.army.mil/Missions/Great-Lakes-Infor-

mation/Great-Lakes-Information-2/Water-Level-Data/

 [3] Dias, N. L., Hoeltgebaum, L. E. B., & Santos, I. (2023). STAEBLE: A surface-temperature- and available-energy-based lake evaporation model. Water Resources Research, 59, e2022WR033012. https://doi.org/10.1029/2022WR033012

 [4] Neff, B.P., & Nicholas, J.R. (2005). Uncertainty in the Great Lakes Water Balance. U.S. Geological Survey Scientific Investigations Report 2004-5100, 42 p.
[5] Lenters, J.D., Anderton, J.B., Blanken, P., Spence, C., & Suyker, A.E. (2013). As-

sessing the Impacts of Climate Variability and Change on Great Lakes Evapora-
tion. In D. Brown, D. Bidwell, & L. Briley (Eds.), 2011 Project Reports. Great Lakes
 Integrated Sciences and Assessments (GLISA) Center. Retrieved February 6, 2024,
from http://glisaclimate.org/media/GLISA_Lake_Evaporation.pdf
[6] Wikipedia contributors. (2024, February 3). Great Lakes. In *Wikipedia, The Free
 ls Encyclopedia*. Retrieved 00:55, February 6, 2024, from https://en.wikipe-
dia.org/w/index.php?title=Great_Lakes&oldid=1202856604
e [7] Lin, D. H. (2018). Saint Lawrence Seaway and its regulations and handbook: Re d quirements for maximum vessel dimensions etc. *Journal of Navigation*, 10(5),
79-88.
o [8] Cuthbert, M. O., Taylor, R. G., Favreau, G., Todd, M. C., Shamsudduha, M., Vill holth, K. G., MacDonald, A. M., Scanlon, B. R., Kotchoni, D. O. V., Vouillamoz, J.m M., Lawson, F. M. A., Adjomayi, P. A., Kashaigili, J., Seddon, D., Sorensen, J. P. R.,
Ebrahim, G. Y., Owor, M., Nyenje, P. M., Nazoumou, Y., ... Kukuric, N. (2019).
 H Observed controls on resilience of groundwater to climate variability in sub-Sa-
haran Africa. *Nature*, 572, 230-234.
 T [9] Quinn, F. H. (1979). An Improved Aerodynamic Evaporation Technique for Large Lakes With Application to the International Field Year for the Great Lakes. Water A Resources Research, 15(4), 935-940.
[10] Croley, T. E. II (1989). LUMPED MODELING OF LAURENTIAN GREAT
 M LAKES EVAPORATION, HEAT STORAGE, AND ENERGY FLUXES FOR FORE-
CASTING AND SIMULATION. NOAA Technical Memorandum ERL GLERL-70.
 National Oceanic and Atmospheric Research Administration Laboratories.

 [11] International Joint Commission. (2014). Lake Ontario - St. Lawrence River Plan 2014.





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OPTIFLOW :

ENHANCING GREAT

LAKES MANAGEMENT



KEY MODEL FEATURES





 



 Real-Time Adaptive Simulation  ls Adjusts to environmental changes e for responsive water level
 d management.

Stakeholder-Centric Design
Equitably balances diverse group needs, spanning from industry to
ecology.

Advanced Data Utilization
Utilizes historical data to significantly enhance prediction
accuracy and reliability.

 o WHY CHOOSE OUR MODEL?  Hm Precision and Efficiency: Our model introduces flexible dam controls, improving the speed  T and accuracy of water level adjustments for unmatched precision.
Adaptability: It accounts for climate change and seasonal variations, enabling flexible
A responses to extreme weather, ensuring environmental adaptability.  M Resource Optimization: Through precise data analysis, our model rationalizes water  resource allocation, balancing needs across sectors for optimized resource allocation.

 Economic Advantage: By maintaining navigable shipping routes and enhancing power
generation, our model boosts economic benefits.



 Embracing Innovation for a

Sustainable Water Future

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AI Usage Instructions
I. Purpose and Background In the context of the American Collegiate Mathematical Contest in Modeling, our team aimed to enhance problem-solving efficiency by applying AI tools, particularly in the areas of text translation, literature content summarization,
 and code optimization. Our goal was to leverage AI technology to accelerate
the research process and ensure that our solutions remain competitive in the
 international arena.  II. AI Text Translation
We processed a significant number of foreign academic papers and technical documents related to mathematical modeling and data analysis. By comparing the results of manual and AI translations, we found that AI tools excelled in
 handling professional terminology and complex sentence structures. However,
manual proofreading was still necessary in certain cases to ensure precision.
 ls Team members generally agreed that AI translation tools saved considerable
time and improved work efficiency, but suggested that key sections should be
e reviewed manually.  d III. Literature Content Summarization o We employed AI analysis tools to automatically extract key information from  literature. We processed a variety of documents, including mathematical m models, algorithm optimization, and practical application cases. The AI tools  H successfully identified and summarized the core concepts, research methods,
and major findings in the literature, providing theoretical support for our
 T model construction. By quickly obtaining the essence of the literature, we were
able to more effectively integrate existing knowledge and form innovative
A problem-solving approaches.  M IV. Code Optimization
We utilized AI tools to improve code quality. Our project involved multiple
 programming languages, including Python, MATLAB, and C++. The AI tools
helped us identify redundant parts, potential errors, and performance
 bottlenecks in the code, significantly enhancing the code's efficiency and
maintainability. Team members appreciated the level of intelligence of the code optimization tools but also pointed out that the suggestions from AI tools
 needed further validation in handling complex logic.  V. Summary and Recommendations
AI technology has demonstrated significant potential in text translation, literature content summarization, and code optimization, significantly improving work efficiency and quality. Although AI tools have performed well

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in certain areas, there are still limitations in accuracy and context understanding, especially when dealing with complex and highly specialized texts.
MATHmodels 

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