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Unformatted text preview: 6.079/6.975 S. Boyd & P. Parrilo December 1011, 2009. Final exam This is a 24 hour take-home final exam. Please turn it in to Professor Stephen Boyd, (Stata Center), on Friday December 11, at 5PM (or before). You may use any books, notes, or computer programs ( e.g. , Matlab, CVX), but you may not discuss the exam with anyone until December 11 after 5PM, after everyone has taken the exam. The only exception is that you can ask us for clarification, via email. Please address your emails to both professors and the TA . Please make a copy of your exam before handing it in. When a problem involves computation you must give all of the following: a clear discussion and justification of exactly what you did, the Matlab source code that produces the result, and the final numerical results or plots. Matlab files containing problem data are available on Stellar. All problems have equal weight. Some are easier than they might appear at first glance. And others are harder than they might appear at first glance. Be sure to check your email and the course web site on Stellar often during the exam, just in case we need to send out an important announcement. And one technical comment. For problems that require you to work out a numerical solution, you are welcome to use a solution method that involves solving more than just a single convex optimization problem. (Of course, only when this is necessary.) 1 1. Optimal generator dispatch. In the generator dispatch problem , we schedule the elec- trical output power of a set of generators over some time interval, to minimize the total cost of generation while exactly meeting the (assumed known) electrical demand. One challenge in this problem is that the generators have dynamic constraints, which couple their output powers over time. For example, every generator has a maximum rate at which its power can be increased or decreased. We label the generators i = 1 ,...,n , and the time periods t = 1 ,...,T . We let p i,t denote the (nonnegative) power output of generator i at time interval t . The (positive) electrical demand in period t is d t . The total generated power in each period must equal the demand: n p i,t = d t , t = 1 ,...,T. i =1 Each generator has a minimum and maximum allowed output power: P i min p i,t P i max , i = 1 ,...,n, t = 1 ,...,T. The cost of operating generator i at power output u is i ( u ), where i is an increasing strictly convex function. (Assuming the cost is mostly fuel cost, convexity of i says that the thermal eciency of the generator decreases as its output power increases.) We will assume these cost functions are quadratic: i ( u ) = i u + i u 2 , with i and i positive....
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