MAE119HW3SP24 (pdf) - CliffsNotes (2024)

MAE 119 Introduction to Renewable EnergySpring 2024 P. Hidalgo-Gonzalez 05/15/2024 Homework #3 Due on Monday, May 27th before 11:59 P.M. PST. One submission per group through Gradescope. Groups of 1-2 students.Include all your work and code for full score. Exercise 1 (Capacity expansion and dispatch)Use the .m file provided to answer this exercise. Suppose you are an investor who needs to provide electricity to a town for 20 years. You have the choice of how much capacity,Ci in MW (decision variable), to build for five possible generators (two coal plants, two gas plants and a PV plant, as in lecture). Each possible generator has a maximum capacity allowed to be built due to space and materials constraints (Pmaxi input data provided in the .m file). Ci ≤Pmax i ∀i= 1...5 The minimum capacity you can build for each generatoriis zero: 0≤Ci ∀i= 1...5 Similarly to what we did in lecture, you need to optimize how to operate the plants (decide their hourly power generationpi,t ) at every hour to meet the town's demand during 20 years. To simplify this, assume the same daily profile for the electricity demand provided in the .m for all 365 days in all 20 years. The sum of power generation at timethas to be greater or equal to the demand at timet(i.e., curtailment of energy is allowed): 5 X i=1 pi,t≥dt ∀t= 1...24 wherepi,tis the power generation in MWh from power plantiat hourtanddt is the demand at timet. The operational constraints that need to be modeled are minimum and maximum allowed power output for each thermal power plant given their capacity,Ci , and ramping constraints for every power plant: 0≤pi,t≤Ci ∀t= 1...24,∀i= 1...4 1

|pi,t+1−pi,t| ≤Ri ∀t= 1...23,∀i= 1...4 whereRi is the maximum ramp (or change) in power output for power plantiin MWh. The solar power output should be calculated in a similar manner as we did in lecture (i.e., its capacity factor multiplied by the installed capacity, which is this case is the decision variable C5 ). Your objective is to minimize the net present value (NPV) of the total cost using a discount rate of 4% (provided in the .m file). Each generator has two cost components: investment cost and fuel cost. Assume the investment happens at the beginning of the simulation and the construction of the plants are immediate (i.e., you do not need to discount the investment cost). For each technology, the .m provides investment, or capital, costsMi in dollars per MW. Mathematically, the total investment cost is given by 5 X i=1 MiC i Each future year of the simulation will have associated fuel costs that will be given by 3654Xi=124 Xt=1 cip i,t whereci is the fuel cost in$/MWh associated to power planti(provided in the .m file) and the factor 365 is used to calculate the annual cost based on the daily cost. Notice the solar plant has zero fuel cost, hence the solar plant does not contribute to the annual fuel cost calculation. This total annual fuel cost needs to be discounted to calculate its NPV. Therefore, the objective function to be minimized is the sum of the NPV of the total fuel costs and the initial investment cost. And as described, the decision variables are the installed capacitiesCiof each of the five plants, and their power outputspi,t through the day. Part I: (1) write the optimization problem using CVX in Matlab (include your code as part of your PDF submission), (2) calculate total NPV of the cost, (3) report the optimal installed capacity for each of the five plants (coal plants, gas plants and solar plant), (4) make the following three plots: 1. Hourly demand over time 2. Hourly power generationpi,t for each power plant over time 3. Hourly total power generation and demand over time 2

and (5) calculate the LCOE of each plant. For this calculation, you should use: the invest- ment cost, the NPV of the fuel costs, and the NPV of the optimal dispatched electricity. In the same way as the optimization was set up, maintenance costs are assumed to be zero. With these metrics and figures, answer the following: 1. Describe the hourly dispatch obtained and why it is optimal 2. What are the driving factors that affect the installed capacities obtained from the optimization? 3. Which of the maximum capacities allowed to be builtPmaxi resulted in active con- straints?Explain why some of these constraints may or may not be active (be as specific as possible) 4. Rank and analyze the LCOE calculated for each of the five plants and compare it to their respective installed capacities. Did the optimization choose to install larger capacities for plants that show a cheaper or more expensive LCOE? Explain why this is the case. Part II: Now you are an environmentally conscious investor. You want to understand what is the cost and impact of decarbonizing the town. In this setting there are no constraints on how much solar power capacity you can install, and you can also decide to install as much wind power as optimal (capacity factor and capital cost are provided in the .m file). To analyze the cost and impact of decarbonization, write the optimization problem formulation using CVX in Matlab (include your code as part of your PDF submission) where you will include a constraint that imposes an upper bound on the amount of CO2 emitted during the 20 years (i.e., a CO2 cap). Your decision variables will be the same as in Part I, but also considering installed wind capacity and its dispatch. Solve this optimization problem for the following cases (constraints): 1. Unconstrained CO2 emissions (Consider this as the baseline) 2. Up to 80% of the optimal CO2 emissions from the baseline 3. Up to 60% of the optimal CO2 emissions from the baseline 4. Up to 40% of the optimal CO2 emissions from the baseline 5. Up to 20% of the optimal CO2 emissions from the baseline 6. Zero CO2 emissions 3

Because this is a premium document. Subscribe to unlock this document and more.