Problem Chosen
D

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 (C), 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
 ls 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
e course of a multiple-month time horizon. After the optimal water levels have been determined, we  d employ another control algorithm to plan how to schedule the dams over a daily time horizon to o 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
 m 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
 H important to determine accurate parameters to encode natural and artificial processes. However,  T 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
A coincide with the data that the C 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
 M 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 C 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.

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Contents

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1 Introduction

3

1.1 1.2

 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Restatement Of The Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3

2

1.3 Overview of Linear Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 1.4 Assumptions And Justifications . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Macroschedule Model

4 5
6

3 4
5 6

2.1 Dynamic Network Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 2.2 Ideal Water Determination Algorithm . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Linear Program Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 ls 2.4 Methods for Stakeholder Management . . . . . . . . . . . . . . . . . . . . . . . . e 2.5 Data-Based Model Of The Hydrosphere . . . . . . . . . . . . . . . . . . . . . . .  d 2.6 Additional Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  o Microschedule Model  Hm Model Insights  T 4.1 Dynamic Control Of The System . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Robustness to Severe Climate Events . . . . . . . . . . . . . . . . . . . . . . . . .
 MA Sensitivity Analysis  Conclusions  6.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 7 7 9 10 12
12
13 14 15
16
17 18

7 8

6.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
 Future Exploration  Memo To The C

18 19 20

A Notation

21

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1 Introduction

1.1 Background

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].
 ls Due to the invaluable benefit that the Great Figure 1: A cloudless view of all of the Great
Lakes provide us, it is of the utmost importance Lakes taken by a NASA satellite in August 2010
e to society that they are flourishing in the best [13].  d way possible. o 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 Interm national Joint Committee (C) are: domestic water supply, commercial navigation, hydropower,
environment, recreational boaters, docking companies, and shoreline (riparian) property owners
 H [10].  T The current methodology of managing the water flow in Lake Ontario is based on Plan 2014. A 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 C takes action to control the level of water in the dam [2]. After the initiation of Plan 2014 M 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 C, 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

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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-
 ls 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
e management [11, 20]. Linear programs rapidly found applications across the natural sciences,  d industry, government, and many other fields, and are a natural way to model many scheduling o 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. m 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].
 H 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
 T expected from utilizing linear programming are as follows: A  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 M 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.

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 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.

 i  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.

 ls Justification: Keeping the lake sizes con-

W e stant provides us with a consistent con-

version metric, as seen in figure 2 be-

 d tween flow and height for each lake.

o Based on the correlation data for the

 Figure 2: Since water levels do not vary

flow rates, the other nodes are negligi-

m much relative to total depth, we can approxi-

ble as compared to the main outflows

mate the lake surface area as being constant.

between the Great Lakes.

 H We can then convert flow to marginal height

by dividing by surface area.

 AT 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).
 M 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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2 Macroschedule Model

Here, we describe our algorithm for determining the optimal achievable water levels in the great lakes over a large time-period.

2.1 Dynamic Network Formulation
We begin with a high-level overview of the model. The Great Lake water level problem can be
 studied as a dynamic network flow problem on the network shown in Figure 3. There are two arcs
that we have some level of control over. The Sault St. Marie Dam allows us to send a constrained
 amount of water from Lake Superior to Lakes Huron and Michigan. Water then flows naturally

 from Lakes Huron and Michigan into Lake Erie and from Lake Erie to Lake Ontario. Through the
Moses-Saunders Dam, we can let a constrained amount of water out of Lake Ontario into the St. Lawrence River.

The reason why our analysis differs from the well-studied static flow problem is because it takes time for water to flow between each node. For instance, it takes roughly one month for flows in
 Lake Superior to be observed in Lakes Huron and Michigan; it takes roughly three months to be
observed in Lake Ontario. Thus, the network flow becomes dynamicwhich significantly increases
 ls the problem's modeling and computational difficulty.

 de Superior  o 1,1  Hm H.M

Erie

Ontario

2

St. Law

 AT Figure 3: Network used in the model.  is the capacity of the arc and  is the transit time of the
arc. All unlabeled arcs correspond to uncontrolled flows of water.
 M More formally, the dynamic flow problem is defined over a network  = (, , , , +, -).  The vertex set is given by   +  - where + represent source nodes, - represent sink nodes,

 and  represent transit nodes. Each edge  has a corresponding capacity  and transit time  [6].
The goal is to find the min-cost flow over a fixed time horizon  that satisfies a particular objective.

To make this problem tractable, we employ Ford and Fulkerson's time-expansion method to convert the problem into static flow [14]. We discretize the model over a fixed timestep, and
 we create copies of each node for each one over the entire time-horizon. We then assign edges
based on the transit times between nodes. With this, the problem is now described by a network
  = (, , , +, -). I.e. an augmented network that includes vertex copies but no transit times.

This formulation is amenable to static-flow analysis.

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2.2 Ideal Water Determination Algorithm
To determine the ideal water levels for each body of water, we developed a utility function for each shareholder that determines their optimal water level based on the historical averages. This optimal water level is determined based on the fluctuations in the water level and what each
 of the stakeholders listed as their ideal scenario from the C report. While navigation, docks,
hydropower, and recreational boaters all prefer a fixed level of water, riparian owners, domestic water supply, and the environment prefer natural oscillations in water level [10]. To account for
 this discrepancy, we simulated sinusoidal oscillations about the mean height of the lake for their
preferred timescales and amplitudes scaled compared to the natural processes:
Environment: The environment prefers natural annual water level oscillations to best align with
 the natural mating seasons of species like wetland birds [12].
Domestic Water Supply: To preserve high water quality, the domestic water supply needs to
 oscillate close to the average to prevent the inflow of sewage and harmful algae blooms
[15][10].
 ls Riparian Owners: The riparian owners prefer lower water levels with smaller amplitude oscillations, with higher frequency to increase the amount of sand re-deposition and dune e replenishment [10].
 d Navigation: Navigation prefers steadily high water levels because lower water levels can increase o costs by 30 percent and double emissions in the Great Lakes [16]. Additionally, they want it  to not flood because they need ships to get through [10].
m Hydropower: The dams prefer their flow rates to be constant or near their ideal operating range of  H around 7, 0003 , so they prefer to be above the average water levels, similar to navigation, to

 T maintain that [9]. A Boaters: Boaters also prefer to be in a similar range of slightly above the average water level to
improve the speed they can travel through the water during the day [10].
 M Docks: For the shipping ports in Montreal, they prefer to have constant low water levels so the amount of the ideal amount of water coming downstream from the Moses-Saunders Dam  should be minimized during working hours.  There are conflicting interests among the stakeholders, and it is impossible to please each stakeholder. To account for this, we introduced a weighting system into our model to allow each
 stakeholder to be considered with different severity. As a result, our ideal water plots look like  figure 4 compared to the historical data
2.3 Linear Program Formulation
Here we describe the Linear Program (LP) we propose to determine long-term dam schedules for the Great Lakes. We consider two sets that our decision variables and parameters belong to: (1) Bodies of Water and (2) Time. Our decision variables are the amount of water we let flow from the

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183.45

Lake Superior ideal water level curve

Ideal

Historic

75.1

Lake Ontario ideal water level curve

Ideal Historic

183.40

75.0

Water level (m) Water level (m)

183.35 183.30

74.9 74.8



183.25 183.20

74.7 74.6



0

50

100 150 200 250 300 350

Day of year

0

50

100

150 200 Day of year

250

300

350

Figure 4: Plot of the ideal water levels (solid orange lines) based on stakeholder needs for Lake Superior and Lake Ontario compared to the historical averages (dotted blue lines) from the dataset.

 Superior to the St. Mary River at each time-step and the amount of water we let flow from Lake  ls Ontario to the St. Lawrence River1. The objective of the LP is to minimize the absolute difference
between the achieved water level and the "ideal" water level. The LP is constrained by limits on the
e water level ensuring no flooding occurs and the water level is never critically low. The variables  d and parameters and described in the first table in Appendix A. The algebraic form of the linear o program is shown below:

 minimize



 |,

-

^ , |

+

  (1 - ) | 

-

^ |

m ,

 

 H subject to 1, = 1,-1 - -1 + 1,-1

 T 2,

=

2,-1

+

1 2



-

-

(2 2, -1+2) 2

+

2,-1

  [1, ]   [1, ]

A 3,

=

3,-1

+

(2 2, - +2) 3

-

(3 3, -1+3) 3

+

3,-1

  [1, ]

 M 4,

=

4,-1

+

(3 3, -1+3) 4

-

-1

+ 4,-1

  [1, ]

,  
 ,      1   2

  ,   [1, ]   ,   [1, ]   [1, ]   [1, ]

 =  -1 + (1 - )

  [1, ]

1In Appendix A, more variables are listed, but these are completely determined by equality constraints and are thus not listed as decision variables here.

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All decision variables are nonnegative. The first four constraints capture how the height of

water in each lake changes over time by tracking how much water enters the lake and how much

leaves. We use the index - to capture the fact that there is a lag between the changes in Lake

Superior and the changes at other lakes. The 5th and 6th constraints allow us to ensure that no

flooding or dangerously-low water levels occur. To capture the fact that dams can only let out so

 much water at a time, we introduce the 7th and 8th constraints to bound the amount of water we
can let out of the dams. Lastly, we track the flow in the St. Lawrence River with the 9th constraint.

 2.4 Methods for Stakeholder Management As C notes, this problem is particularly difficult because of the often-conflicting requests from

 the various stakeholders around the Lake Ontario and St. Lawrence River regions. Our algorithm
provides a method to weight stakeholder interests, however, it relies on the practitioner to assign

these weights manually. This has the potential to create divides between groups of stakeholders. Thus, some sort of heuristic should be used to equitably assign these weights on a per-stakeholder basis.
 We propose the following methodology to determine fair weightings between ideal water level
curves:

 els  We create an initial set of weights 0 and set them as parameters in the model.  d  Then we simulate roughly five months2, and observe the forecasted ideal water levels.  o  We then proceed with the schedule for two months.

m We simulate beyond the necessary time horizon to ensure that our model accounts for future  H impacts of flows in the other lakes. In particular, we designate a three-month buffer since this is
the time taken for an effect in Lake Superior to be observed in Lake Ontario.
 T After adopting the given schedule for the two month period, we calculate the net error between A the real observed water levels and the ideal curves provided per stakeholder.

 M  =  | - ^| 

  [1, 60]

 This gives us per-stakeholder errors {1, . . . } for  stakeholders. Note that higher values of
 correspond to a lower level of stakholder 's satisfaction. Keeping this in mind, we can reassign

weights for the next time horizon with
 1 =   
 That is, the new stakeholder weights are their normalized net errors from the last time period.

This allows stakeholders that had greater deviations to be prioritized during the next time period.

2The chosen time horizon is left up to the practitioner, but we recommend choosing one of the same order. I.e. 2-6 months

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2.5 Data-Based Model Of The Hydrosphere

Modeling the movement of water in the Great Lakes is difficult and requires a number of

assumptions to make the problem tractable. To ground our model in reality we used a data-based

approach to determine time lag, flow rates out of lakes, and other sources of water outside of the

flow between the lakes. We first needed to approximate the time lags between the lakes for our LP to create dam
 schedules. For instance, we needed to know that it takes roughly two months for the effect of a flow
in Lake Michigan to be observed in Lake Ontario. To identify the latencies, we used time-lagged
 cross correlation on the provided water level time series to find where the correlation is maximized.

 For example, the time-lagged cross-correlation between the water levels in Lake Superior and Lake
Michigan is given by Figure 5.

Correlation Correlation

Time lagged cross-correlation between Lake Huron and Lake Erie
0.90

 0.85

 ls 0.80

e 0.75

 od 0.70

 0.65

m -10

-5

0

5

10

Time lag (months)

Time lagged cross-correlation between Lake Erie and Lake Ontario
0. 7

0. 6

0. 5

0. 4

0. 3

0. 2

0. 1

0. 0

- 0.1

- 10

-5

0

5

10

Time lag (months)

 H Figure 5: Time lagged cross-correlation examples for between Lake Huron and Erie and between  T Lake Erie and Ontario computed from the dataset given for Problem D in [1].

A The correlation is maximized at a time lag of 1. Thus, we can interpret that a flow in Lake  Superior will take approximately a month to propagate in Lake Michigan. The full matrix of these M time-lagged correlations is given by Table 1.

Sup. Mich + Huron Erie Ont.

Sup.

0

1

33

 Mich. + Huron -1

0

12

Erie Ont.

-3 -3

-2 -2

00 00

Table 1: Time lag matrix between the Great Lakes
Additionally, we used the datasets to run a linear regression, pictured in figure 6, to obtain the

flow rates as a function of the height of the lake, which we will use to model the flow in and out of

the lakes. We suspect this to be true because if we approximate flow using Bernoulli's principle in

fluid mechanics, we find that by expanding for small oscillations we will have locally linear results.

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Niagara Riverflow vs Lake Erie height

Detroit River flow vs Lake Michigan/Huron height

8000 7500

7500

7000

Niagara river flow ( m3s- 1) Detroit River flow (m3s- 1)

7000 6500

6500 6000



6000 5500

5500 5000



5000

4500 4000



173.8 174.0 174.2 174.4 174.6 174.8 175.0 175.2 Lake Erie height ( m )

175.50 175.75 176.00 176.25 176.50 176.75 177.00 177.25 177.50 Lake Michigan/Huron height ( m)

 Figure 6: Flow along the Niagara River and Detroit River as a function of the height (relative to
sea level) of the lakes they flow from. The regression formulae are given by:

 ls Niagara = 2088.917Erie - 357993.824 with (R2 = 0.928) e Detroit = 1611.581Huron - 278618.674 with (R2 = 0.850)

 od Lastly, we also approximated other aspects of the hydrosphere (precipitation, groundwater,  evaporation, and surface runoff) by modeling the inflow of water based on how much it rained m because according to the biohydrological database for the Great Lakes makes up 20 percent of the
water flowing into the Great Lakes [3]. Using the NOAA data set of rainfall, evaporation, and
 H runoff into the Great Lakes [17], we determined the total inflow that would occur each month based
on historical data. We visualize the historical rainfall that occurred during each month in Figure 7.

 T Lake Superior net exogenous flow

A 25p

50p

 0.15

75p

0. 4

Lake Ontario net exogenous flow
25p 50p 75p

M 0.10

0. 3

Marginal height (m) Marginal height (m)

0. 05



0. 2

0. 00 - 0.05



0. 1

 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month

Figure 7: 1st (in blue), 2nd (in red), and 3rd (in orange) quartiles of exogenous flow into Lakes

Superior and Ontario

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2.6 Additional Considerations
Aside from the data-driven parameters we obtained, we considered other parameters to help make our model more realistic:

 Seasonal Wave Attenuation: To consider the seasonality of the Great Lakes we also add an attenuation factor that reduces the flow between the lakes. We added this factor because a study by Peng Bai showed that ice attenuates the waves in the Great Lakes and thus should

 reduce their flow from their turbulent effects [4].
Fuzzing The Model: Since exogenous inflow is highly random, we added noise to historical inflow

data to increase the robustness of our model. This makes the model more robust by removing biases towards certain shores at different water levels and helps us accurately assess the
 drought and flooding risk factors.

3 Microschedule Model
 One aspect that our Macroschedule model does not incorporate are the proximal effects of  ls opening the Moses-Saunders Dam. As noted by the problem statement, even small variances in the
water level can have drastic implications for stakeholders around Lake Ontario and the St. Lawrence
e River. As such, it is not enough to know how to release water in a day. It is also important to time  d outflows when it is generally most beneficial for these stakeholders. We develop a linear program
that schedules intra-day outflows to maximize community benefit.
 o Unlike the macroscheduling problem, there is no substantial time lag on flows between nodes in m the microscheduling problem. Thus, we can exert more immediate control over the water level and
flows in Lake Ontario and the St. Lawrence River. We use a similarly formulated linear program
 H to represent these flows, and the objective of maximizing proximity to ideal water levels and flows.
The variables of the problems are defined in the second table of Appendix A
 T And the LP is formulated as follows:

 A minimize M subject to

 1 | 

-

^  |

+ 2| 

-

^ |

 
 = -1 +  -   =  -1 +  (1 - )     = ^

        

 The first two constraints implement height and flow rate updates in Lake Ontario and the St.
Lawrence river respectively. Notice that the second constraint models the river flow rate as a convex

 combination of inflow from the dam and its prior flow rate. This is influenced by sound physical
intuition, but it is difficult to accurately model this parameter based on available data. The third constraint limits total outflow from the dam, and the fourth enforces that a certain amount of water

has left Lake Ontario by the end of the day. This ensures that the microscheduler adheres to the

daily requirements set forth by the larger schedule. Our proposed synthesis of the macroscheduler

and microscheduler is show in Figure 8

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Due to lower lag times, the model can satisfy daily requirements relatively well. This is important because there are likely days when dam usage must be limited to coordinate with monthly schedules. However, the water level can still be modulated to a level where stakeholders can still be largely satisfied.

Set Time Horizon





Macroscheduler

 Microscheduler

Manage Stakeholder Expectations
 Fit Macro  ls Parameters  e Generate Schedule

Approve
Test & Determine Robustness

Find Ideal Curves
Fit Micro Parameters
Generate Schedule

 mod Collect Data

Improve Model

 H Figure 8: The pipeline for using our control algorithms.

 AT 4 Model Insights  M Here we outline a brief overview of the main insights of our model:

  Effectively managing Lake Ontario means effectively Managing Lake Superior. Lake Superior is able to influence Lake Ontario after a significant time delay which makes proactive scheduling at Lake Superior able to create desirable water levels at Lake Ontario.

  Our model successfully achieves the desired water levels in most lakes even with varying degrees of rainfall.

  After determining optimal water levels through our macro-schedule algorithm, our microschedule algorithm is able to determine realistic and effective schedules for scheduling a dam

during any particular day.

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4.1 Dynamic Control Of The System

Most dams use a heuristic approach for dam scheduling called rule curves. That is, dams are

given fixed schedules depending on water level and precipitation thresholds. While our model is

amenable to producting rule curves, we believe that the dynamic nature of scheduling makes it

 superior to this approach. One of the most critical aspects of the model is that it can make decisions by "looking ahead".

 That is, given a range of forecasts, the model can schedule flows that are both robust to climate
events and still satisfy the needs of stakeholders. This way, action can be taken to mitigate potential crises well before they actually occur.

 After we "look ahead" we can then use a combination of the macroscheduler and microscheduler
model to forecast both the four-month time horizon and then locally on the daily horizon. By

modeling both of these time horizons, we gain valuable insights on the importance of factoring in the time lags to forecasting the dam schedules. The efficacy of our model is shown in Figure 9:

 Microscheduler flow error rate

Error (%)

 els 1.4  d 1.2 o 1.0  m 0.8  H 0.6  T 0.4 A 0.2  M 0.0  0

5

10

15

20

Hour

 Figure 9: Plot of the percent error between ideal flow and the flow achieved by the microscheduler

 after the optimal water levels were determined by the macroscheduling algorithm.



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4.2 Robustness to Severe Climate Events

We tested the schedules produced by our

control algorithm using past rain/runoff/evaporation data collected from [17]. We deter-

Quartile 1
Superior

mined the average amount of total rain, runoff,
183.4
and evaporation for each month from 1980 to 2020. Due to the fact that our model uses 183.2 a time-step of one day and the data available 183.0 form [17] tracks how much rain/runoff/evap- 182.8 oration occurred each month, we determined 182.6 how much it rained per day by running a cubic spline through the historical averages and 174.5

Water Height

176.5 176.4 176.3

Huron/Michigan

176.2
176.1 176.0 175.9

Erie

 175.8 175.7

Ontario

then dividing the value for total rain, runoff,
and evaporation by 30 to account for the fact
 that the month data was the total over a month
(roughly 30 days). This allows us to capture the
natural patterns of rain, runoff, and evaporation
 ls while making the data usable in our model. We
ran a similar procedure to determine the 25th
e and 75th percentile amount of rain, runoff, and  d evaporation that occurred on each day. o After obtaining these rain values, we ran our  linear program using the historic rain level (eim ther the 25th, 50th or 75th percentile) at body of  H water  during time  for the , parameter. The
time horizon for these experiments was set to a
 T year and the ideal water was determined using
the techniques in Section 2.2. The water levels
A achieved by the model are shown in Figures 10
and 11.
 M We observe that the Lake Ontario water lev els are very consistent across different levels of

Water Height

Water Height

Water Height

174.4 174.3 174.2 174.1 174.0
0
183.4 183.3 183.2 183.1 183.0 182.9
174.50 174.45 174.40 174.35 174.30 174.25 174.20

75.0

74.8

74.6

74.4

74.2

50 100 150 200 250 300 350
Days

0 50 100 150 200 250 300 350
Days

Qaurtile 2

Superior

Huron/Michigan
176.5

176.4

176.3

176.2

176.1

176.0

175.9

175.8

Erie

Ontario

75.1 75.0 74.9 74.8 74.7 74.6

 total rain, runoff, and evaporation while Lake
Superior's levels are not. This demonstrates that effectively scheduling Lake Superior is cru-

174.15

0

50

100 150 200 250 300 350

Days

74.5 74.4

0

50

100 150 200 250 300 350

Days

 cial to having desirable water levels at Lake On- Figure 10: Plots of the 1st and 2nd quartiles of the
tario. The types of schedules that are required in water levels in each of the Great Lakes drawn from

 Lake Superior to create desirable water levels at the historic rain data. The water levels achieved
Lake Ontario are non-robust to severe changes by the model are shown with dotted blue lines and in total inflow to the Great Lakes, but the water the ideal water levels with solid orange lines.

levels in the rest of the Great Lakes are indeed

robust to climate events.

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Quartile 3

Superior

Huron/Michigan

183.45

176.5

183.40

176.4

Water Height

183.35 183.30

176.3 176.2



183.25 183.20

176.1 176.0



Water Height

Erie
174.6

 174.5

 174.4

174.3

 ls 174.2  de 0

50 100 150 200 250 300 350
Days

Ontario

75.1 75.0 74.9 74.8 74.7 74.6

0

50 100 150 200 250 300 350

Days

o Figure 11: Plots of achieved water levels using the third quartile rain data (extreme rain). The water  levels achieved by the model are shown with dotted blue lines and the ideal water levels are shown m with solid orange lines.

 H 5 Sensitivity Analysis  T Since our model is primarily mechanistic, it relies heavily on parameters that represent real-life A values such as flow rates, flow decay, and attenuation. As such, it is reasonable to observe how the
model's output changes as we modulate some of these parameters. Specifically, we vary the St.
 M Lawrence river flow decay rate.  Moreover, we determine that it is possible to avoid flooding and dangerously low water levels

 while achieving desirable water levels in all lakes even with varying degrees of total inflow to
the lakes. This demonstrates the value of our linear programming approach as a valuable tool to determine dam schedules.

 The value that was used in the final model was 0.8. This was chosen based off observing flows in
the St. Lawrence River and how they change with respect to the height of Lake Ontario. However,

 since there isn't data on the status of the dam at these times, this link is admittedly tenuous. Thus,
we decided to observe the effects of modulating this parameter on the net flow error in the St. Lawrence River. That is, with different decay rates, we measure how hard is it for our control

algorithm to match the desired flow rate in the St. Lawrence River.

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We uniformly vary the parameter  in the range [0.7, 0.825] and observe the net error rate. We see that:
Yearly average flow error based on 

1000 980



Net flow error

960



940



920

0.70

0.72

0.74

0.76

0.78

0.80

0.82



 Figure 12: Flow in the St. Lawrence River is sensitive to 

 ls The error here varies quite significantly. In some sense, this is not surprising. Complex systems e such as the Great Lakes are very sensitive to slight physical changes. To maximize the effectiveness  d of the model, it is therefore important to be able to measure these parameters accurately, and provide
them to the model with high-resolution.
 o In conclusion, the model is largely robust to changes in climate forecasting. However, due m to its mechanistic nature, it is quite sensitive to parameters describing natural and artificial flow
processes.
 H 6 Conclusions  T  Our model demonstrates that to effectively manage Lake Ontario, the C must also effectively A manage Lake Superior. Lake Superior has the capacity to send large amounts of water to
Lake Ontario. While the time it takes for Lake Ontario to feel the effects of actions taken at
 M Lake Superior, by proactively increasing the water sent from Lake Superior, Lake Ontario  can more effectively manage its water levels.

 Dam schedules will consistently make tradeoffs betweeen flood/drought security and stakeholder satisfaction. It is important to manage stakeholder expectations and use equitable
 prioritization methods to ensure overall contentedness.
 Testing schedules against different severities of exogenous inflow yield outcomes that are
 significantly more robust to climate and hydrologic events. Regardless of the scheduling
algorithm, plans should always be tested against real historical inflows as well as randomly
 simulated climate events.

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6.1 Strengths
1. Due the speed of solving linear programs, our model can be employed to develop schedules for a large number of potential forecasts and ideal water levels. This allows practitioners to be more prepared for extreme weather events by being able to prep schedules for best-case,
 worst-case, and average-case weather events. If it becomes evident that the environmental
data that the dam control schedule was created from is poor, a new schedule can be quickly substituted in to mitigate damage.
 2. Our control algorithms can effectively meet stakeholder demands while the water levels remain close to historical averages. The Great Lakes water system needs to flow naturally while  still accounting for the demands of stakeholders, so our model must meet this benchmark. 3. Due to the constrained nature of our control algorithms, practitioners will always be aware
of the maximum and minimum water levels that are produced by the schedule. Since actions taken at Lake Superior affect Lake Ontario with a severe delay, there is an associated risk
 with increasing or decreasing the flow through the Compensating Works dam. Our control
algorithms allow for practitioners to predict the effect of the schedules they implement.
 ls 4. Our model is extremely flood and drought-resistant. In our analysis using realistic precipitation data our model didn't experience any droughts or floods.
 ode 6.2 Weaknesses  Practitioners need to be aware of the flaws of our model so that unexpected results from our m control algorithms can be mitigated. The weaknesses of our model are as follows:  H  Due to the mechanistic nature of our model, the schedule of the model is noticeably sensitive  T to changes in parameters related to natural processes. For example, if the rate of flow decay
in the St. Lawrence River is poorly measured, the model will produce a schedule that differs
A slightly, but noticeably, from the true ideal schedule. Practitioners should be aware of this
flaw and apply the model several times with varying values for parameters like the decay rate
 M and rain schedule. While the schedule will be sensitive to poor measurements, it will provide  valuable insights into how to best manage the dams.
 Our model assumes a linear relationship between the height of a body of water and the total outflow which may change over time as the landscape of the Great Lakes changes. To account
 for this, practitioners should routinely track the relationship between the height of each lake
and the observed flows of each lake's distributaries.
  Due to the constraints placed on flooding and dangerously low water levels, our model will occasionally determine that an adequate schedule is possible. If practitioners encounter this  issue when using our proposed control algorithms, they must choose the location and extent of flooding that will necessarily occur. We believe that no model can completely avoid flooding, so we believe that our model's ability to have controlled and predictable flooding is sufficiently safe.

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7 Future Exploration
To address some of the weaknesses we identified, we believe that our model could be expanded in the following ways:
 Pull from more refined data: By pulling from more refined data, our model would have more accurate information hydrological information. As a result, our model could improve the
 mechanistic nature of our model.
 Safety Optimizer: Another extension of our model could be using the results from running
 many fuzzy trials to determine which dam schedule is optimal against random weather
conditions. This would allow practitioners to understand the risk assosiated with the schedules
 they select.
 Create an algorithm to determine which stakeholders to prioritize. The time of year has a large effect on the possibility of pleasing any particular stakeholder. A future direction of
MATHmodels this work would be to implement the time of year into our weighting of the stakeholders. 
 

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8 Memo To The C
Dear C Leadership,
In light of the successes and failures experienced by Plan 2014 in regards to controlling the Moses-Saunders dam at Lake Ontario, we developed two linear programming-based control algorithms to effectively determine how to jointly manage the Compensating Works Dam at Lake Superior and the Moses-Saunders Dam at Lake Ontario to achieve ideal water levels throughout the
 Great Lakes. We believe that our algorithms build off of the adaptive control approach that Plan
2014 has successfully implemented while avoiding the limitations of Plan 2014 that have allowed
 for flooding in recent years.  In our model, we employ linear programming and a series of algorithms that forecast the needs
of the stakeholders to determine how much we need to open dams at Lake Superior and Ontario. Because we are using a linear program, our model can simulate hundreds of thousands of scenarios within a few minutes, this would allow for the C to have a more holistic approach to assessing
 the risks of running certain dam schedules for their stakeholders. Our algorithm for determining
the optimal levels of stakeholders is based on their desires outlined in the C report and draws from historical water levels [10]. Our model also utilizes historical data to simulate natural processes
 ls in the hydrosphere in the following ways:  e  We account for the time it takes water to flow between lakes. We were able to correlate the d water level data between different lakes to estimate the time it taken for flows to propagate o through the system.  m  We use a data-driven approach to determine flow rates of naturally flowing sources. More-
over, using the data available to us, we found a strong linear relationship between height and
 H the rate and which water flows out.  T  Our model accounts for various levels of rainfall and other hydrological processes. We utilized
a combination of precipitation from NOAA and preexisting models to get randomized, but
A representative rain data to run our situations [17] [22].  M To show the resilience of our model we ran it under various extreme climate events and found
it was effective in preventing floods and droughts. Specifically, the way our algorithms achieve
 near-optimal water levels even with severe climate events is by creatively managing Lake Superior.
By incorporating the relationship between Lake Superior and Lake Ontario into our model, we are
 able to achieve schedules that allow Lake Ontario to remain safe and beneficial to the stakeholders.  We believe that our model provides the C with valuable insights about how to best manage
the Compensating Works and Sault. St. Marie dams. Our model will allow the C to use the
 most recent sets of data being gathered at the monitoring points along the Great Lakes and adapt
an optimal dam schedule based on current conditions and predictive forecasts.
Thank you for your consideration,
Team #2429211

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A Notation

Name Type

Description.



Set

Set of time-steps.



Set

 Variable

 Variable

 Variable

, Variable

 Set of Bodies of Water.
Height of water let through the Sault St. Marie Dam at time . Height of water let through the Moses-Saunders Dam at time .
 Flow at River Lawrence at time .  Height of water in body of water  at time .

 Parameter

Weighting term for the objective function. High 

, Parameter The total height added by environmental factors in body of water  at time .

  Parameter

The surface area of body of water .

 ls  Parameter

Determined slope from regression.

e  Parameter

Determined bias from regression.

 d  Parameter

Lag time between specified bodies of water.

o  Parameter

Height of water required for a flood at body of water .

  Parameter

Height required for dangerously low water at body of water .

m 1 Parameter Maximum height of water releasable from Lake Superior during a time-step.

 H 2 Parameter Maximum height of water releasable from Lake Ontario during a time-step

 T  Parameter Influence of the Mosses-Saunders Dam on the flow of the St. Lawrence River

A Table 2: Description of variables involed in the mathematical program used to determine dam  schedules. All heights are in meters, all flows are in cubic meters per second, and all areas are M given meters squared, all lag times are given in days







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Name Type

Description.



Set Bodies of water (Ontario, St. Lawrence)

 Param

Time Horizon

 Param ^  Param

Hourly inflow into Lake Ontario Ideal water levels



^

Param

 Param

Ideal flow levels Flow conversion coefficient





Param

Initial water and flow levels

  Param Weights for stakeholder importance

 Param

Hourly flow decay coefficient

  Param
^ Param

Dam outflow limit Final water height

 ls  Variable

Water level

 e  Variable

Flow level

od  Variable

Moses-Saunders Dam outflow

 Table 3: Microschedule LP formulation. All water levels are heights in meters, all times are in  MATHm hours, and all flows are in cubic meters per second







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References
[1] 2024MCM/ICM Problems. https://www.immchallenge.org/mcm/index.html.
[2] International Joint Commission (2014). Lake Ontario St. Lawrence River Plan 201: Protecting against extreme water levels, restoring wetlands and preparing for climate change, 2014.
 [3] US army Corps of Engineers. John glenn great lakes basin program biohydrological information base, 2007.
 [4] Peng Bai, Jia Wang, Philip Chu, Nathan Hawley, Ayumi Fujisaki-Manome, James Kessler, Brent M Lofgren, Dmitry Beletsky, Eric J Anderson, and Yaru Li. Modeling the ice-attenuated waves in the great lakes. Ocean Dynamics, 70:9911003, 2020.
[5] Chiranjib Bhowmik, Sumit Bhowmik, Amitava Ray, and Krishna Murari Pandey. Optimal green energy planning for sustainable development: A review. Renewable and Sustainable
 Energy Reviews, 71:796813, 2017.
[6] Thomas Blsius, Adrian Feilhauer, and Jannik Westenfelder. Dynamic flows with timedependent capacities, 2023.
 [7] Murray Clamen and Daniel Macfarlane. Plan 2014: The historical evolution of lake ontario
st. lawrence river regulation. Canadian Water Resources Journal/Revue canadienne des
 ls ressources hydriques, 43(4):416431, 2018. e [8] Great Lakes Commission.  d [9] International Joint Commission. Options for managing lake ontario and st. lawrence river
water levels and flows (final report), 2006.
 o [10] International Joint Commission. Section 5: Impacts on various interests, 2023. m [11] George B Dantzig. Linear programming. History of mathematical programming, pages  1931, 1991. H [12] Jean-Luc Desgranges, Joel Ingram, Bruno Drolet, Jean Morin, Caroline Savage, and Daniel  T Borcard. Modelling wetland bird response to water level changes in the lake ontariost.
lawrence river hydrosystem. Environmental monitoring and assessment, 113:329365, 2006.
A [13] Google Earth. Great lakes, no clouds, 2010.  M [14] Lester Randolph Ford. Flows in networks. 2015.
[15] Inga Kanoshina, Urmas Lips, and Juha-Markku Leppnen. The influence of weather con-
 ditions (temperature and wind) on cyanobacterial bloom development in the gulf of finland
(baltic sea). Harmful Algae, 2(1):2941, 2003.
 [16] Frank Millerd. The economic impact of climate change on canadian commercial navigation on the great lake. Canadian Water Resources Journal, 30(4):269280, 2005.
 [17] NOAA. Noaa glerl data, 2021.  [18] National Oceanic and Atmospheric Administration. Great lakes ecoregion - national oceanic
and atmospheric administration, 2019.
[19] Frank J Rahel and Julian D Olden. Assessing the effects of climate change on aquatic invasive species. Conservation biology, 22(3):521533, 2008.

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[20] Charles S. Revelle, Daniel P. Loucks, and Walter R. Lynn. Linear programming applied to water quality management. Water Resources Research, 4(1):19, 1968.
[21] Michael H Veatch. Linear and convex optimization: A Mathematical Approach. John Wiley & Sons, 2020.
 [22] Daniel S Wilks and Robert L Wilby. The weather generation game: a review of stochastic weather models. Progress in physical geography, 23(3):329357, 1999.
  MATHmodels   

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Dear ICM Judges,

Report on Use of AI

No AI was used in the writing, code, or development of this project.

We wish you the best.







MATHmodels







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