Optimizing Dam Removals: A Balancing Act in Lake Erie

Rivers and lakes are complex ecosystems. For centuries, humans have built dams for various purposes like hydropower, flood control, and water supply. However, dams also fragment rivers, blocking fish migration and altering natural flow patterns. In recent decades, especially as dams age or become obsolete, there's growing interest in removing them to restore river health.

But deciding *which* dams to remove isn't simple. The paper "Optimizing multiple dam removals under multiple objectives: Linking tributary habitat and the Lake Erie ecosystem" by Zheng, Hobbs, and Koonce tackles this challenge for the U.S. watersheds feeding into Lake Erie. Removing a dam can be good for native fish like walleye, but it costs money and might unintentionally help invasive species like the destructive sea lamprey. Furthermore, the effects of removing one dam can depend on whether others upstream or downstream are also removed.

This post explores the core ideas of the paper, focusing on how they use mathematical optimization to find the best *portfolio* of dam removals, balancing conflicting goals in a complex system.

The Fundamental Trade-off: Benefits vs. Costs and Risks

Imagine you're a manager deciding on dam removals. You have a limited budget. Each potential removal has:

A cost to demolish the dam.
A potential benefit by opening up upstream habitat for desirable native fish (like walleye).
A potential risk by also opening up habitat for undesirable invasive species (like sea lamprey), which might require costly control measures.

Let's visualize this basic conflict. Use the slider below to simulate deciding how many dams to remove conceptually. Notice how benefits, costs, and risks often move together.

Conceptual Number of Removals:

This is a simplified illustration. In reality, *which* dams are removed matters more than just the total number.

The paper moves beyond this simple view by looking at specific dams and quantifying these effects more precisely.

Quantifying the Goals: Ecological Health and Economic Cost

To make informed decisions, we need to measure our objectives. The paper considers two main high-level goals:

Maximize Ecosystem Health: This isn't just about one fish species. It's a combined measure reflecting the overall state of the Lake Erie fish community.
Minimize Economic Cost: This includes the cost of removing the dams *and* the cost of controlling sea lamprey in newly opened habitats.

Measuring Ecosystem Health: More Than Just Walleye

How do you put a number on "ecosystem health"? The researchers used a technique called Multi-Criteria Value Analysis (MVA). They identified several specific indicators important to managers and stakeholders, grouped under broader categories:

Ecosystem Health Components (Simplified Hierarchy):

Community Balance

Walleye-to-other-predator ratio ($x_1$)
Predator-to-prey fish biomass ratio ($x_2$)
Total fish productivity ($x_3$)

Native Species

Native species biomass ratio ($x_4$)

Socio-Economic (Fisheries)

Lake-wide walleye biomass ($x_5$)
Walleye sport harvest ($x_6$)
Yellow perch commercial harvest ($x_7$)
Smelt commercial harvest ($x_8$)

Each indicator ($x_i$) gets a score based on its level, and these scores are combined using weights ($W_i$) reflecting their relative importance, into a single overall health index ($V$):

V(x_1, ..., x_8) = \sum_{i=1}^{8} W_i V_i(x_i)

Where $V_i(x_i)$ is a value function scaling the $i$-th indicator from 0 (worst) to 1 (best). The paper uses a sophisticated computer simulation model of Lake Erie (called LEEM) to predict how these indicators ($x_1$ to $x_8$) change when dam removals alter fish habitat.

We won't dive into the complex LEEM model itself, but the key idea is that it provides the link between habitat changes in tributaries and the overall lake ecosystem health index $V$.

Measuring Costs: Removal and Control

The economic side ($x_9$ in the paper's notation, though we'll just call it 'Cost') has two parts:

Dam Removal Cost ($C^{dam}$): How much does it cost to physically remove dam $j$? This depends on its size (height, length), type (earthen, concrete), and whether it served an important purpose (lost services add to cost). The paper uses a regression model based on past removals.
Sea Lamprey Control Cost ($C^{lamprey}$): If removing dam $j$ opens up habitat suitable for sea lamprey larvae, controlling them costs money (using chemicals called lampricides). This cost depends on the *length* of suitable habitat opened.

Total Cost = Sum of $C^{dam}_j$ for removed dams + Sum of $C^{lamprey}_j$ for necessary control.

Interactive Dam Removal Cost Estimation

Based on the paper's regression model (Eq. 14), explore how different factors influence the estimated removal cost for a single dam.

Dam Height (feet): 5 Dam Length (feet): 50 Serves Important Purpose? (e.g., water supply) Dam Type:

Estimated Removal Cost: $...

$$ \ln(C^{dam}_j) \approx 7.79 + 0.80 \ln(\text{Height}_j) + 0.33 \ln(\text{Length}_j) + 1.49 (\text{FuncW}_j) - 0.44 (\text{TypeE}_j) $$ Where $\text{FuncW}_j=1$ if important purpose, $0$ otherwise. $\text{TypeE}_j=1$ if earthen, $0$ otherwise. Costs are in 2006 dollars.

Linking Dams to Ecology: Habitat Matters

How does removing a dam actually affect the fish community indicators ($x_1..x_8$)? The crucial link is habitat.

Removing a dam potentially opens up upstream river segments.
We need to know if this new habitat is *suitable* for spawning and nursery stages of key species (walleye and sea lamprey).
The paper uses Habitat Suitability Index (HSI) models. HSI scores range from 0 (unsuitable) to 1 (perfectly suitable) based on physical characteristics like water depth, velocity, and substrate (river bottom material).
For walleye, the *area* of suitable spawning habitat matters.
For sea lamprey, the *length* of suitable nursery habitat (for larvae called ammocoetes) matters for control costs.

Visualizing Habitat Accessibility

Click on dams (circles) in this simplified river network to simulate removing them. Observe how upstream segments become potentially accessible (thicker green lines). Dams can only be removed if the one immediately downstream is also removed (or if it's the furthest downstream).

Accessible Walleye Habitat Area: 0 km²
Accessible Lamprey Habitat Length: 0 km

This visualization shows potential accessibility. The actual ecological impact depends on the HSI (suitability) of these segments and how habitat changes translate to fish populations via the LEEM model. Note the upstream/downstream constraint in action.

The change in suitable habitat, particularly for walleye young-of-year (YOY), is then fed into the LEEM ecosystem model. The paper uses "Impulse Response" functions – essentially, estimating how much each ecosystem indicator ($x_1..x_8$) changes for every unit increase in walleye YOY recruitment resulting from habitat improvements.

Putting it Together: The Optimization Model

Now we have the pieces: candidate dams, ecological benefits (via the health index $V$), costs ($C^{dam} + C^{lamprey}$), and the habitat link. The goal is to select the *best set* of dams to remove.

This is formulated as a Mixed Integer Linear Program (MILP). Don't worry about the full mathematical complexity, the core idea is:

Optimization Goal (Simplified):

Maximize: Aggregate Ecological Health Index ($z_1$, derived from $V(x_1..x_8)$)

Subject to:

Total Cost $\le$ Budget ($B$)
Upstream/Downstream Constraint: Dam $n$ upstream of $j$ can only be removed ($d_n=1$) if dam $j$ is also removed ($d_j=1$). That is, $d_n \le d_j$.
Decision Variables: For each dam $j$, decide $d_j = 1$ (remove) or $d_j = 0$ (keep).

The actual MILP links the $d_j$ variables to changes in habitat, then through LEEM's impulse responses to the $x_i$ indicators, and finally to the objective $z_1$. It also calculates the lamprey control cost based on opened habitat.

Interactive Portfolio Selection

Let's simulate the challenge. Below is a set of candidate dams, each with estimated costs and potential habitat gains. Try to manually select a portfolio of dams to remove that maximizes the "Ecological Score" (a simplified proxy for $z_1$) without exceeding the budget. Pay attention to the upstream/downstream constraints!

Budget ($ Million): 15 M

Click dams (dots) to select/deselect for removal. Red stroke = selected. Orange = constraint violation.

Selected Dams: 0 | Total Cost: $0 M | Total Walleye Gain: 0 km² | Total Lamprey Gain: 0 km | Eco Score: 0 %

Optimal Solution: (Click button)

Green stroke indicates dams in the optimal computer-generated solution for the current budget. The "Eco Score" here is a simplified linear combination of walleye gain minus lamprey gain, scaled 0-100. The MILP uses the more complex MVA score.

Results: The Efficient Frontier

Running the MILP for different budget levels ($B$) doesn't give just one answer. It reveals a fundamental trade-off between cost and ecological benefit. This is often visualized as an efficient frontier (or Pareto front).

Each point on the curve represents an "efficient" portfolio – meaning you can't get a better ecological score for the same or lower cost, and you can't get a lower cost for the same or better ecological score. Points *below* the curve are suboptimal; points *above* are infeasible with current technology/options.

Trade-off Curve: Ecological Health vs. Cost

This chart shows the efficient portfolios found by the MILP for different budgets (based on Table 2 in the paper). Hover over points to see details. Higher budgets allow for better ecological outcomes, but with diminishing returns.

Each point represents an optimal portfolio of dam removals for a specific budget.

What if We REALLY Dislike Sea Lamprey? Sensitivity Analysis

The baseline analysis balances multiple factors. But what if stakeholders are particularly worried about the invasive sea lamprey? The MVA weights could be changed, or, as the paper explores, we can add a specific objective to *minimize* the potential for lamprey habitat increase.

They introduce a weight, $WL$, representing the importance placed on avoiding lamprey habitat expansion (relative to maximizing the general ecological health score). $WL=0$ is the baseline case. As $WL$ increases, the optimization prioritizes removing dams that offer little or no lamprey habitat, even if they offer less walleye benefit or cost more.

Impact of Lamprey Aversion (Sensitivity Analysis)

Set a budget (e.g., $15M) and adjust the "Lamprey Aversion" weight ($WL$). See how the optimal portfolio and resulting outcomes change (based on Table 3). The map below shows the dams selected in the optimal portfolio for the chosen $WL$ and budget.

Budget ($ Million): 15 M Lamprey Aversion Weight (WL): 0.0

Optimal Portfolio for Budget = $15M and WL = 0.0:

As lamprey aversion ($WL$) increases, the model selects different dams (often more numerous, smaller ones with no lamprey risk) resulting in much lower lamprey habitat gain but also a lower general ecological score. This highlights how stakeholder priorities influence optimal strategy.

Conclusion: A Tool for Complex Decisions

The research by Zheng, Hobbs, and Koonce provides a powerful framework for tackling the complex problem of multiple dam removals. By integrating ecological models (HSI, LEEM), economic cost estimations, and multiobjective optimization (MILP), their approach allows decision-makers to:

Systematically evaluate a vast number of potential removal combinations.
Explicitly consider multiple, often conflicting, objectives.
Understand the trade-offs between ecological restoration goals and economic costs.
Explore how different priorities (like aversion to invasive species) change the optimal strategy.

While the model provides valuable insights for screening potential projects, the authors caution that site-specific studies are still essential before any actual dam removal occurs. Models like this are tools to guide decision-making in complex environmental management scenarios, making the process more transparent and informed.