MATH6161: Deterministic OR Methods for Data Scientists
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Coursework No. 2 MATH6161 — Mathematical Programming
MATH6161: Deterministic OR Methods for Data Scientists
2021/2022
Power Generation
Power generators of four different types are available to satisfy daily electricity demands (in mega-watts, MW) summarized in the following table. We consider a sliding time horizon: the period 6pm-12pm of day d is followed by the period 0am-6am of day d + 1. We therefore need to model and optimise over only one day.
The power generators of the same type have a maximum capacity and may be connected to the network starting from a certain minimal power output. They have a start-up cost, a fixed hourly cost for working at minimal power, and an hourly cost per additional megawatt for anything beyond the minimal output. These data are given in the following table. (Min. output and max. capacity are given in megawatt, costs are given in $.)
A power generator can only be started or stopped at the beginning of a time period. As opposed to the start, stopping a power plant does not cost anything. At any moment, the working power generators must be able to cope with an increase by 15% of the demand forecast. This increase would have to be accomplished by adjusting the output
of generators already operating within their permitted limits.
Your first set of tasks is as follows:
• The operator of the power generators would like to minimize the total production cost while maintaining that demand is met, all technological constraints are met etc. Formulate an integer programming (IP) model to help decide which generators should be working in which periods of the day to minimize total cost. Implement and solve the model using Xpress-IVE. Report the best power generation strategy.
When formulating and implementing this model, recall that there are many power production problems with similar properties but different dimensions (e.g. more generators) and different technological characteristics. The aim is to formulate and implement a generalised model so that it can be sold to the operator for use in other situations. The IT staff of the operator are not experts in mathematical programming or XPress-IVE, but are able to change data and constants in an XPress-IVE model.
• What tariffs should be charged for the electricity produced?
• What would be the saving of lowering the 15% reserve output guarantee; i.e. what does this security of supply guarantee cost?
The operator of the power plants would now like to include the use of hydro power in the model. In addition to the generators above, a water reservoir powers two hydro generators, A and B. When a hydro generator is running, it operates at a fixed level and the depth of the reservoir decreases. The costs associated with each hydro generator are a fixed start-up cost and a running cost per hour. All data can be found in the following table. (Operating levels are given in megawatt, costs are given in $.)
For environmental reasons, the reservoir must be maintained at a depth between 15 and 20 meters. At midnight each night, the reservoir must be 16 meters deep. Energy from the 27 thermal generators (see above) can be used to pump water into the reservoir. To increase the level of the reservoir by 1 meter requires 3000 mega-watts of electricity. Rainfall does not affect the reservoir level.
At any time, it must be possible to meet an increase in demand for electricity by 15%. This can be achieved by an combination of the following.
• switching on a hydro generator, even if this would cause the reservoir depth to fall below 15 meters;
• using the output of a thermal generator that is used for pumping water into the reservoir;
• increasing the operating level of a thermal generator to its maximum.
Thermal generators cannot be switched on instantaneously to increased demand, but hydro generators can be.
Your second set of tasks are as follows:
• The operator of all power generators would like to minimize the total production cost while maintaining that demand is met, all technological constraints are met etc. Extend your integer programming (IP) model to help decide which generators should be working in which periods of the day to minimize total cost. Implement and solve the model using Xpress-IVE. Report the best power generation strategy.
2022-01-12