Optimizing Dam Removals in Lake Erie: An Interactive Exploration

Based on: Zheng, P. Q., B. F. Hobbs, and J. F. Koonce (2009), Optimizing multiple dam removals under multiple objectives: Linking tributary habitat and the Lake Erie ecosystem, Water Resour. Res., 45, W12417, doi:10.1029/2008WR007589. [Link to Paper]

Lake Erie, the shallowest and most biologically productive of the Great Lakes, faces numerous environmental challenges. One significant issue is the impact of thousands of dams built on its tributary rivers. Many of these dams are old, serve no current economic purpose, and block fish migration routes, harming native species like the popular walleye.

Removing these dams sounds like a straightforward solution, right? Restore the rivers, help the fish! However, it's not that simple. Dam removal is expensive, and paradoxically, it can also help invasive species like the destructive sea lamprey by giving them access to new spawning grounds. Furthermore, removing one dam can affect the situation at others upstream or downstream.

So, managers face a complex puzzle: With a limited budget, which dams should be removed to achieve the *best overall outcome* for the Lake Erie ecosystem, balancing the benefits for desirable fish like walleye against the costs and the risks of helping invasive species like sea lamprey?

This article explores the approach taken by Zheng, Hobbs, and Koonce (2009) to tackle this problem using multi-objective optimization. We'll break down their model and use interactive visualizations to understand the core concepts.

The Balancing Act: Benefits, Costs, and Trade-offs

At its heart, the decision involves trade-offs:

• Benefit: Removing dams opens up upstream habitat, potentially increasing native fish populations (like walleye).
• Cost 1 (Economic): Removing dams costs money (demolition, lost services if any).
• Cost 2 (Ecological): Removing dams might also open up habitat for invasive sea lamprey, which requires costly control measures (lampricide treatment).

Imagine you have several dams you *could* remove. Each has a different cost, a different potential benefit for walleye, and a different risk for lamprey. How do you choose?

Quantifying the Factors

To make informed decisions, we need ways to measure these benefits and costs.

1. Measuring Ecosystem Health: The MVA Score

The researchers didn't just look at walleye. They considered the overall health of the Lake Erie ecosystem using multiple criteria, combined into a single score using Multi-Criteria Value Analysis (MVA). They identified key indicators representing ecological structure, function, and societal value (like fishing).

These indicators included:

• Walleye biomass relative to other predators
• Ratio of predator fish to prey fish
• Total fish productivity
• Ratio of native species to total species
• Total walleye biomass (good for recreation)
• Walleye caught by sport fishing
• Yellow perch caught commercially
• Smelt caught commercially

Each indicator ($x_i$) is assigned a weight ($W_i$) reflecting its importance, and its current level is converted to a value ($V_i(x_i)$) on a 0-1 scale (0=worst, 1=best). The overall ecosystem health score ($V$) is the weighted sum:

The key challenge is that removing dams changes these underlying criteria ($x_i$). The researchers used a complex simulation model of Lake Erie (LEEM - Lake Erie Ecological Model) to predict how changes in tributary habitat (specifically, walleye young-of-year survival) would ripple through the food web and affect these lake-wide indicators.

2. Estimating Costs

The model also needs cost estimates:

• Dam Removal Cost ($C_j^{dam}$): This depends on the dam's size (height, length), type (earthen, concrete), and original purpose (if it provided services that need replacing). The researchers developed a statistical model (regression) based on past removals:
  $$ \ln(C_j^{dam}) \approx 7.79 + 0.80 \ln(\text{Height}_j) + 0.33 \ln(\text{Length}_j) + 1.49 (\text{IsImportantPurpose}_j) - 0.44 (\text{IsEarthen}_j) $$

  Larger dams and those with important functions cost more; earthen dams cost less.

• Lamprey Control Cost ($C_j^{lamprey}$): If removing dam $j$ opens up suitable habitat for sea lamprey ($H_j^{lamprey}$), this habitat needs treatment with lampricide (TFM). The cost depends on the length of stream habitat opened up and the cost of TFM treatment over time (discounted to present value).
  $$ C_j^{lamprey} \approx \left( \frac{C^{TFM} \times \text{TFM Usage Rate}}{1 - (1+r)^{-m}} \right) \times H_j^{lamprey} $$

  Cost depends on TFM unit cost ($C^{TFM}$), usage rate per km, interest rate ($r$), and treatment cycle length ($m$). $H_j^{lamprey}$ is the length (km) of lamprey habitat opened by removing dam $j$.

The total cost ($x_9$) for a portfolio of removed dams is the sum of removal costs and necessary lamprey control costs for all selected dams.

The Optimization Puzzle: Selecting the Best Portfolio

Now we have the pieces: a way to measure overall ecosystem health (the MVA score) and ways to estimate the costs. The goal is to select a set of dams for removal that maximizes the ecosystem health score, without exceeding a given budget ($B$).

This is formulated as a Mixed Integer Linear Program (MILP). The core decision is binary for each potential dam $j$: either remove it ($d_j = 1$) or keep it ($d_j = 0$).

The objective is to Maximize $z_1$ (the MVA score):

Subject to:

1. Budget Constraint: The total cost ($x_9$) cannot exceed the budget ($B$).
$$ x_9 = \sum_{j \in J} (C_j^{dam} + C_j^{lamprey}) d_j \leq B $$
2. Ecosystem Linkage: The ecosystem criteria ($x_i$) depend on the baseline level ($X_i^{base}$) plus the cumulative effect of removed dams on walleye recruitment ($\Delta YOY^{walleye}$), modeled via impulse responses ($IR_{x_i}$).
$$ x_i = X_i^{base} + IR_{x_i} \Delta YOY^{walleye} \quad \text{where} \quad \Delta YOY^{walleye} = \sum_{j \in J} (\Delta YOY_j^{walleye}) d_j $$
3. Habitat Logic: A dam ($n$) cannot be removed if a dam ($j$) immediately downstream of it is *not* removed (ensures connectivity).
$$ d_n \leq d_j \quad \text{for all } n \text{ directly upstream of } j $$
4. Binary Decision: Each $d_j$ must be 0 or 1.
$$ d_j \in \{0, 1\} \quad \text{for all } j \in J $$

Seeing the Trade-offs: The Efficient Frontier

Solving the MILP for a single budget gives one optimal portfolio. But what if the budget changes? Or what if we want to see the *range* of possibilities?

By running the optimization model many times with different budget levels ($B$), the researchers traced out an efficient frontier (also called a Pareto frontier). This curve shows the maximum possible ecosystem health score achievable for any given level of spending.

Points on the curve represent "efficient" portfolios – you can't improve the ecosystem score further without spending more money, and you can't spend less money without accepting a lower ecosystem score.

What if Priorities Change? Sensitivity Analysis

The MVA score depends on the weights ($W_i$) assigned to each criterion. What if stakeholders are particularly concerned about the sea lamprey invasion risk? The researchers explored this by adding an objective to *minimize* potential lamprey habitat, giving this objective a weight ($WL$).

Maximize: $(1 - WL) \times (\text{Ecosystem Health } z_1) - WL \times (\text{Lamprey Habitat } z_3)$

Increasing the weight ($WL$) on minimizing lamprey habitat fundamentally changes which dams are selected in the optimal portfolio, even for the same budget.

Conclusion

The work by Zheng, Hobbs, and Koonce provides a powerful framework for tackling the complex problem of multiple dam removals. Key takeaways include:

• Systematic Approach:** Optimization models like MILP can systematically evaluate a vast number of possibilities (selecting from 139 dams means $2^{139}$ potential portfolios!) to find efficient solutions.
• Quantifying Trade-offs:** The model explicitly calculates the trade-offs between competing objectives (ecosystem health vs. cost) and visualizes them through the efficient frontier.
• Linking Scales:** It connects local actions (dam removal affecting tributary habitat) to system-wide consequences (Lake Erie ecosystem health) using ecological models (HSI, LEEM).
• Value Judgments Matter:** The "best" portfolio depends on the priorities (weights) assigned to different objectives, like the importance of walleye versus the risk of sea lamprey. Sensitivity analysis helps understand these impacts.