Stochastic Programming with Recourse - a practical example

Summary

This video demonstrates the application of stochastic programming with recourse, focusing on a city's transition to green energy. The city plans to build two wind farms, determining the optimal number of windmills (x1 and x2) based on uncertain power generation, represented by a normal distribution. The objective is to minimize costs while meeting the city's power demand. If demand isn't met, old generators are used as a recourse action, with a penalty for failing to meet green energy requirements. The model incorporates simulation and evaluates different scenarios to find the most cost-effective solution, balancing demand, cost, and recourse actions.

Takeaways

• 😀 The city plans to transition to green energy by building two wind farms.
• 😀 The number of windmills to be constructed in each wind farm is represented by integer variables x1 and x2.
• 😀 The power delivered by each windmill is stochastic and follows a normal distribution, dependent on wind conditions.
• 😀 The power demand of the city is denoted by the parameter d, which must be met by the wind farms' output.
• 😀 Each windmill has a setup cost, represented by c, and this cost is independent of the wind farm it belongs to.
• 😀 Windmills have an expected lifetime, and c represents the per-year cost of a windmill.
• 😀 The objective is to minimize the total costs, including the setup costs of windmills and the potential use of old generators.
• 😀 If the demand is not fully met by the windmills, old generators can be used as a recourse, but this incurs additional costs.
• 😀 The continuous variable x3 corresponds to the electricity generated by old generators, with a cost associated with their use (denoted by o).
• 😀 The problem is modeled as a two-stage stochastic programming model, with one stage for determining the number of windmills and another for using old generators if necessary.
• 😀 The simulation approach generates scenarios based on the probability distribution of the windmill power output, with each scenario equally likely.
• 😀 The model's objective is to minimize costs while ensuring demand is satisfied in all scenarios, with a recourse action if necessary.
• 😀 The model assumes that using a large number of scenarios will provide more robust solutions, but feasibility is always ensured with the recourse action.

Q & A

What is the primary objective of the city's energy transformation project?

The primary objective is to transform the city's energy consumption to green energy by constructing two new wind farms.

How are the windmill quantities in each wind farm determined?

The number of windmills in each wind farm is determined by integer variables x1 and x2, which represent the number of windmills in wind farm 1 and wind farm 2, respectively.

Why is the power delivered by each windmill considered stochastic?

The power delivered by each windmill is stochastic because it depends on the direction and strength of the wind, which is uncertain and follows a normal distribution.

What role do the parameters w1 and w2 play in the model?

Parameters w1 and w2 represent the average amount of power generated by a windmill in wind farm 1 and wind farm 2, respectively.

What is the purpose of the variable 'c' in the model?

The variable 'c' represents the per-year cost of purchasing and setting up a windmill. It takes into account the expected lifetime of the windmill.

What does the city need to minimize in this energy transformation project?

The city aims to minimize the total costs, including the cost of windmills and the use of old generators if necessary.

What is the role of the old generators in the project?

Old generators are used as a recourse action if the demand is not fully met by the windmills. However, using these generators incurs additional costs and penalties.

How is the electricity generated by the old generators represented in the model?

The electricity generated by the old generators is represented by the continuous variable x3, which corresponds to the amount of electricity they generate.

What is the structure of the stochastic programming model with recourse?

The model consists of two stages: the first stage determines the number of windmills to be constructed, and the second stage determines the amount of electricity to be generated by the old generators to meet the demand.

How are scenarios handled in the stochastic programming model?

Scenarios are generated using simulation based on the probability distributions of w1 and w2. Each scenario is assumed to be equally likely, and the objective function weighs these scenarios equally to optimize the solution.

What is the key weakness of the simulation-based approach?

The key weakness is the randomness of the simulation. A high number of scenarios may need to be simulated to ensure robust solutions, though this is not a critical issue for this project.