An example of a decision-making problem

Because the case studies of Hydro-Québec and Manitoba Hydro contain sensitive information and cannot be shared publicly, we created a fictive case study to showcase how the Robust Decision Making framework can be applied in the hydropower sector. This case study looks at different reservoir and generating station options to evaluate how their production and cost/benefit ratio is affected by long term changes in streamflow. The problem is defined through the lens of the XMLR framework, a framework designed to help decision-makers lay out the variables influencing their decisions. As discussed in the About page, it includes four components: Uncertainties (X), Metrics (M), Levers (L) and Relations (R).


The uncertainties that are considered in this example are:

  • dQ: changes in average streamflow;
  • dT: changes in temperature;
  • dP: future electricity prices;
  • dR: discount rate.

Changes in streamflow

Future streamflow values are modeled by multiplying the observed streamflow record by a factor varying linearly in time. This factor starts at 100% in 2016, and evolves to different values for dQ in 2050 ranging between -20% to +30% of the historical values: \(Q'=Q \times (1+dQ(t))\). This linear scaling is of course a rather crude simplification, as climate projections suggest changes to the annual cycle of streamflow, for example higher winter flows and lower summer flows.

Changes in temperature

The change in temperature is used to drive changes in the electrical demand for heating and cooling. With each degree C of warming, power demand shifts from a pattern typical for southern Québec to a pattern that resembles more that of New-England. The figure below shows the range of demand patterns that are explored.

Demand pattern
Reference and future demand power pattern varying with projected temperature.

Future electricity price

The price at which electricity will be sold in 2050 (dP) spans a large range of values, going from 10$/MWh to 110$/MWh. For comparison, the figure below shows the mean annual price in the Vermont market. Prices fluctuates from the annual scale to the hourly scale, but in this exercise we only account for annual changes and the mean monthly cycle. It's likely that with raising temperatures the pressure on winter prices would decrease, but this feedback is not included here.

Mean annual price
Mean annual electricity price in the Vermont market (left) and mean annual monthly cycle (right). Vertical bars describe the standard deviation of daily prices around the mean.

Discount rate

The discount rate (dR) describes the expected returns brought in by an investment in the stock market with a similar risk profile as the hydropower investment under consideration. This comparison is the basis on which decision-makers decide whether to simply invest their capital in the stock market or build a new infrastructure. Since interests accrue over time, revenues that occur ten years into the future have a smaller net present value than the same revenues in five years. Here, we explore a range of discount rates going from 0% to 12%.


Numerous metrics can be used to evaluate the performance of an hydropower investment, each with a relative importance that varies depending from person to person. Since the objective of decision-aiding tools is not to replace decision-makers but to provide them with easily accessible information, here we include seven different performance metrics that qualify the investment.

  • Energy: The mean annual energy (GWh) generated by the power station;
  • Firm Energy: The firm, or guaranteed, energy – that is, the minimum annual energy generated over the simulation period;
  • Spill: The mean monthly spill flow, that is, the flow that could not be turbined due to capacity constraints;
  • Drawdown: The mean annual reservoir fluctuations – that is, the mean of the difference between the annual maximum and minimum reservoir levels over the entire period;
  • Flooded Area: The area flooded by the reservoir at its maximum operating level;
  • Internal Rate of Return: The interest rate at which the net present value of the investment (revenues-costs) equals zero;
  • Net Present Value: The current value of the investment, including costs and discounted future revenues.

Among those metrics, energy would be used to plan resource adequacy, that is, making sure that future generation assets will be able to meet future estimated demand. Firm energy is a reliability metric that generators use to convince buyers and regulators that they are able to meet demand and honor the terms of sales contracts. Spill flow measures the amount of water that cannot be used to generate power, which engineers try to minimize either by increasing capacity or reservoir storage. This storage volume varies depending on flows to the reservoir and the energy generating outflows. The level of the reservoir thus fluctuates daily and seasonally according to the balance between water inflows and energy demand. When these fluctuations are large, they can have impacts on the shore erosion, biogeochemistry and habitats. Similarly, as the reservoir volume increases, so does the inundated area. Flooded area implies displacement of communities, loss of habitats and historical sites and conversion of forest carbon sinks into aquatic carbon sources. It should be noted however that these impacts are not necessarily all proportional to the inundated area. Internal rate of return and net present value are two economic measures of the value of an investment over its amortizement period.


This case study compares four different options for an hydroelectric generating station located in a northern region. These four options constitute the levers (L) of the XMLR-E framework and combine different reservoir volumes and turbine capacity. They are denoted as the Small, Medium/Energy, Medium/Capacity and Large options respectively, referencing the reservoir volume and whether the focus is on generating energy or responding to peak demand with high capacity.

  • Small: a reservoir of 4,000 hm³ with a medium capacity (~1,000 m³/s),
  • Medium/Energy: a reservoir of 17,000 hm³ with a small capacity (~900 m³/s),
  • Medium/Capacity: a reservoir of 12,000 hm³ with a large capacity (~1200 m³/s), and
  • Large: a reservoir of 47,000 hm³ with a large capacity (~1200 m³/s).

These four different options perform of course differently with respect to the metrics described above, and our objective here is to be able to display and compare rapidly the results obtained from these options in different future conditions. In each one of these four cases, we assume that construction starts in 2020, that the power station is in service in 2025 and its costs amortized over the next 50 years, that is, until 2065. Approximate costs have been set-up as defaults, but can be changed within the application interface.


The objective here being to display an example application of this decision-aiding tool, considerable simplifications are made to model the relations between climate change, hydropower generation and the different metrics. For one, the energy production model is extremely simple and based on the maximization of the firm energy. This energy model works based on the assumption that the monthly energy production has to meet at all times the monthly demand pattern multiplied by the annual firm energy. The optimization algorithm then finds the highest firm energy that can be reached without breaching reservoir minimum and maximum operating levels. Real production models are vastly more complex, taking into account instantaneous energy prices, flow forecasts, network congestion and many other operational constraints. This optimization of the production is done for all values of future flow (dQ), future temperature (dT) and levers (L). The results in terms of energy production, spilled flow and reservoir fluctuations are saved to compute the different metrics over the amortizement period.

Expert forecasts

For decision-makers, it's often useful to consider the opinion of various experts on complex topics. This application includes information about future runoff and temperature changes from climate models included in the CMIP5 ensemble used for the fifth IPCC report, it also includes information about discount rates used by different institutions, as well as energy price forecasts for New-England from the U.S. Energy Information Agency. By clicking on the item name in the left hand side menu, points are displayed on the graphic and slider axes, their position indicating the forecasted values.

How to use the application

The objective of this application is not to automatically find the best investment option, but rather to provide enough information for decision-makers to explore the consequences of each option so that they can make informed decisions. Try the application by selecting the metrics that you find the most relevant and comparing the different options over a range of futures that you think is plausible. Play with the sliders to evaluate the sensitivity of your decisions to changes in the uncertain variables. You can compare the metric values of multiple options over one dimension by clicking on Leverlines. You can also assess the regret of choosing one option, and not another, by clicking on the RegretMap button. Detailed explanations can be found by clicking on the "?" marker in the top menu of the application.