Math 560 Homework 07: Shadow Prices 1. (this question has little to do with optimization; it's just about the electric power industry). Visit https://www.misoenergy.org/MarketsOperations/RealTimeMarketData/Pages/LMPConsolidatedTable.aspx It takes a little while to load up the whole list of locations. Look for the 5-minute real-time "LMP" (Locational Marginal Price) at the location DECO.MONROE3 That's the large coal-fired power plant in Monroe, MI. You don't have to use DECO.MONROE3 ; you could use DECO.FERMI2, the nuclear plant south of Detroit, or you could use MICHIGAN.HUB, just south of Pontiac. Or anything, really. You can also hover your cursor over the map, at the westernmost tip of Lake Erie--that's DECO.MONROE3 The LMP is essentially the shadow price for the flow-balance constraint at that one node in a large NLP; it is also the wholesale price of electricity at that point. I believe the units are $/MWh. Report the value you find, and the date/time you looked. Also, convert it to cents per kWh. The household retail price for electricity (including distribution charges, taxes, etc.) is around 11 cents per kWh, just for comparison. Also, consider buying an 20-pack of AA batteries for $10.86 (I got the price from walmart.com). Each AA battery contains, let's say, 2500 milli-Amp-hours, which when multiplied by 1.5 volts gives 3.75 Watt-hours. How much are you paying per kWh? Pretty easy homework question, right? PS: You might also like this map, though it's not needed for the homework: https://www.misoenergy.org/LMPContourMap/MISO_All.html --------------------------------------------------------- #3: Read at least pages 461 to 464 of this paper on electric power markets: http://people.emich.edu/aross15/e/ramp-constraints.pdf Here is how the auction is run: in each period (peak and offpeak), you need to satisfy demand with equality rather than inequality. The generators' offers are "at most this many GW, at this price per MWh" (there are 1000 MW in one GW). The objective is to minimize social cost, which is the sum of (price per GWh)*(# of GWh bought) for each generator in each period. For example, if you buy 0.5 GW at a price of $10/MWh for 12 hours, you pay 0.5 * 1000 * $10 * 12. Formulate the LP that turns the data in Table 1 into the results in Table 2, and solve it using whatever you like (Excel, Python, Octave, etc.) Then, impose the ramp constraint that generator B specifies and re-solve to get the results in table 3. Get the Sensitivity report so you can see the shadow prices. What is the shadow price on that ramp constraint? How much change would be required in the RHS of the ramp constraint to make the constraint unimportant? If you multiply the shadow price by the required amount of change and penalize that generator the resulting amount, would that bring our net social cost back to the original amount before the ramp constraint was imposed? Also, be sure to read the "Visualization" section, and write down in your homework any questions that occur to you about it. --------------------------------------------------------- 3. "Transportation Problems" are a subclass of network flow problems. You have a set of source cities and a set of demand cities, the amount of supply in each source city, the amount of demand in each demand city, and the shipping costs for each (source,demand) city pair. You must exactly meet demand in each demand city, and each source city must ship out exactly what it produces (no more or less). a) Formulate and solve a transportation problem as follows: Source cities: San Francisco and Los Angeles (SF and LA) Demand cities: Las Vegas and Phoenix (LV and Ph) Costs: to: LV Ph from SF 50 300 LA 320 60 (costs are per item shipped). Supply: 5 in SF, 10 in LA Demand: 7 in LV, 8 in Ph Find the optimal shipping amounts and the total cost. When Solver says it has a solution, ask it for the Sensitivity report as well, but don't look at it yet. b) Add one unit of supply in SF and one unit of demand in Ph, re-solve, and report the new optimal decisions and total cost. Comment as appropriate. c) Go back to the sensitivity report you generated in part (a) and look at the shadow prices, and comment on them in relation to part (b). d) In both (a) and (b), total supply and total demand matched each other perfectly. What would you do if supply exceeded demand? e) What would you do if demand exceeded supply? --------------------------------------------------------- Optional: Consider a chain with N links hanging from (a,b) to (c,d). Link #i has a weight of w_i and a length of L_i. Formulate an NLP that will find the shape that the chain settles into, by minimizing the total gravitational potential energy. Part of the fun of this problem is seeing what solutions the solver settles on that are physically unlikely--if you see any of those, please include a graph and comment on it. a) Solve for N=20, with all links having the same weight and same length (1 gram and sqrt(2) cm). The left anchor point is at (-10,+10) and the right anchor point is at (+10,+10). Show your formulation and a graph of the resulting chain. b) Re-solve for N=40 links, each half as long as in part (a). Do this chain correspond with the chain in part (a) at every second link junction? c) Let's put a floor at y=5, that the chain may not go below. Re-solve part (a). Comment if anything interesting happens in the final chain. d) (optional) does either chain (a) or (b) correspond with the continuum limit of the catenary curve, which is related to hyperbolic-cosine? e) (optional) Figure out some other cool thing to try, like changing the weights or lengths of individual links, or other sorts of floor constraints, or whatever else you dream up. newer URL: As I said in class, if you don't want to be studying duality with electric power stuff, feel free to ask for or find an alternative paper.