BEE 4750 Environmental Systems Analysis - Homework 3: Linear Programming Models
Homework 3
BEE 4750/5750
Due: Oct 20, 2022 by 9:00pm ET
1 Overview
1.1 Learning Objectives
Homework 3 is focused on assessing your ability to formulate and implement linear programming models.
1.2 Background Information
You have been asked to develop a generating capacity expansion plan for the utility in Riley, NY, which currently has no existing electrical generation infrastructure. The utility can build any of the following plant types: geothermal, coal, natural gas combined cycle gas turbine (CCGT), natural gas combustion turbine (CT), solar, and wind.
Each of these plant types has the following generation data (disclaimer: these are not real), given in Table 1.
Assume that all 365 days in a year are equivalent in terms of demand and wind and solar capacity factors (disclaimer: this is obviously stylized and not reflective of reality, but it makes the problem simpler). These are given in Table 2.
The utility will also penalize any non-served demand at the cost of non-served energy, which is set at $1000/MWh.
Table 1: Generation data for the available plant types.
Hour Demand (MW)
Table 2: Demand (MW) and solar and wind capacity factors by hour for the representative day.
2 Problems
2.1 Problem 1: Minimizing Cost
Your first task is to find a capacity expansion plan which minimizes total costs of investment and operation.
2.1.1 Problem 1.1: Identify Decision Variables
Identify and define all relevant decision variables and their notation.
2.1.2 Problem 1.2: Formulate Objective
Formulate the objective function. Make sure to include any needed derivations or justifica- tions for your equation(s).
2.1.3 Problem 1.3: Formulate Constraints
Derive all relevant constraints (you don’t need to write them all out, but they should all be represented through your notation). Make sure to include any needed justifications or derivations. Why is your set of constraints complete?
2.1.4 Problem 1.4: Implement in JuMP
Implement your optimization problem in JuMP. For this sub-problem, you only need to
provide a code block with the problem formulation.
2.1.5 Problem 1.5: Find the Solution
Find the optimal solution. How much should the utility build of each type of generating plant? What will the total cost be? How much energy will be non-served?
2.1.6 Problem 1.6: Visualize the Solution
Plot how much electricity will be generated under this plan each hour by each type of plant. Make both a line plot for the raw amounts (using plot) and a stacked area plot to show relative contributions to the total demand (using areaplot). What are your take-aways?
2.2 Problem 2: CO2 Limits
The NY state legislature is considering enacting an annual CO2 limit, which for the utility would limit the emissions in its footprint to 1.5 MtCO2/yr.
2.2.1 Problem 2.1: Identify Changes from Problem 1
What changes are needed to your linear program from Problem 1? Formulate any different variables, objectives, and/or constraints.
2.2.2 Problem 2.2: Implement in JuMP
Implement your optimization problem in JuMP. For this sub-problem, you only need to
provide a code block with the problem formulation.
2.2.3 Problem 2.3: Find the Solution
Find the optimal solution. How much should the utility build of each type of generating plant? What is different from your plan from Problem 1?
2.2.4 Problem 2.4: Visualize the Solution
Plot how much electricity will be generated under this plan each hour by each type of plant. Make both a line plot for the raw amounts (using plot) and a stacked area plot to show relative contributions to the total demand (using areaplot). What is different from your plan in Problem 1?
2.2.5 Problem 2.5: Value of Relaxing the Constraint
What would the value to the utility be of allowing it to emit an additional 1000 tCO2/yr?
2.3 References
List any external resources consulted, including classmates.
