Homework 5

Due at 11:59pm at Nov 15th. Submit to https://canvas.uw.edu/courses/1828831/assignments/10827992.

Problem 1

Consider the following problem of supplying a load with two generators:

\[\begin{align} C(L)=\min\; & x_1^2+ 5 x_1 + 2 x_2^2+ 3 x_2 \\ \mbox{s.t. } & x_1+x_2 = L \\ \end{align}\]

where \(L>0\) is some load. Find \(C(L)\) and confirm that the price \(C'(L)\) is equal to the dual variable (Lagrange multiplier) associated with the constraint.

Problem 2

This problem illustrates that the cost could be nondifferentiable. Consider the following problem

\[\begin{align} C(L)=\min\; & x_1 + 3 x_2 \\ \mbox{s.t. } & x_1+x_2 = L \\ & 0 \leq x_1 \leq 1 \\ & x_2 \geq 0 \end{align}\]

Find \(C(L)\) and plot it. It would not be differentiable at \(L=1\). What is the price of power at \(L=1\)?

Problem 3

Consider the DC power flow model for a three bus fully connected network (e.g., a triangle). Suppose that the line between bus 1 and bus 2 has a capacity of 1 and the other lines have infinite capacity. All lines ahve equal susceptance values (\(b_{ij}\) are all equal). Bus 1 is the generator and buses 2 and 3 are load buses.

a) Suppose that the load at bus 2 is 1 and the load at bus 3 is 0.5. Find the power flow on the lines, if they exist, that satisfies these loads and the line capacity constraint.

b) Suppose that the load at bus 2 is 2 and the load at bus 3 is 4. Find the power flow on the lines, if they exist, that satisfies these loads and the line capacity constraint.

Problem 4

a) Same setup as the previous problem. Plot the feasible region of the loads at bus \(2\) and bus \(3\) that satisfies the line capacity constraint.

b) We remove the line between bus $1$ and bus $2$. Find the feasible region of the loads at bus \(2\) and bus \(3\). Is it bigger or smaller than the region in problem a)?

Problem 5

Consider the 3 bus network below with data given in Tables below.

Generator	Capacity (MW)	Marginal Cost ($/MWh)
A	150	12
B	200	15
C	150	10
D	400	8

Table: Generation data for problem 5.

Branch	Reactance	Capacity (MW)
1-2	0.2	250
1-3	0.3	250
2-3	0.3	250

Table: Branch data for problem 5.

a) Find the least cost generating solution satisfying all the loads and line flow capacities. You should use a computer.

b) Find the prices at each of the buses. Note, all convex optimization solvers can report the dual variables. For equality constraints, different solvers would interpret it in different ways. That is, \(Ax=b\) can be dualized as \(\lambda ^T (Ax-b)\) or \(\lambda^T (b-Ax)\). You may need to do some post processing to flip the sign of the dual variable.

Problem 6

Consider DC pwoer flow. There are some generators and loads in the system. Suppose the generator costs are \(c_i x_i\). If all the \(c_i\)’s are positive, could the price at any of the load buses ever be negative? If yes, find an example. If not, prove it. Hint: consider the impact of line constraints. Also, ChatGPT gives a very meandering answer that is partly correct, partly wrong, and partly irrelevant.