Homework 5: Shadow Prices

1. (this question has little to do with optimization; it's just about
the electric power industry). Visit
https://www.midwestiso.org/MarketsOperations/RealTimeMarketData/Pages/LMPContourMap.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 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?
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#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, 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.

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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?

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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.