Homework 4

Due Oct 25th at 11:59pm. Submit to CANVAS at https://canvas.uw.edu/courses/1828831/assignments/10790845.

Problem 1

Consider the following tree network with paramters \(y_{01}=0.81-j0.91, y_{12}=0.13-j 0.92, y_{23}=0.64-j0.91, y_{24}=0.28-j0.55, y_{45}=0.96-j0.96\). Assume that all the voltage magnitudes are one. The active power loads are \(0.07, 0.04, 0.03, 0.02, 0.02\) at buses 1, 2, 3, 4, 5, respectfully (note these are loads, so if we write them as injections, there is a minus sign). Find the power flow solution (since the voltage magnitudes are fixed, this is equivalent to finding the angles). Note because \(g\)’s are not zero, it is slightly more complicated than the case covered in class. But the same strategy works, start at the leafs of the tree and go backward.

Problem 2

For the optimization problem below, you can use direct substitution to solve them (write \(x_2\) in terms of \(x_1\) and differentiate the objective function).

a) Let \(C_1(x_1)= 0.5 x_1^2+x_1\), \(C_2(x_2)= x_2^2 + 2 x_2\), \(L=1\), solve

\[\begin{align} \min_{x_1,x_2} \; & C_1(x_1)+C_2(x_2) \\ & x_1+x_2 = L \end{align}\]

b) Let \(C_1(x_1)= 2x_1^2+0.5 x_1\), \(C_2(x_2)= 0.1 x_2^2 + 2 x_2\), \(L=2\), solve

\[\begin{align} \min_{x_1,x_2} \; & C_1(x_1)+C_2(x_2) \\ & x_1+x_2 = L \end{align}\]

c) let \(x_1^*\) and \(x_2^*\) be the optimal solutions. Evaluate \(C_1'(x_1^*)\) and \(C_2'(x_2^*)\) for both cases. What you do observe? This is a general principle that guides of much of power system market design.

Problem 3

a) Show that if \(f\) and \(g\) are both convex, then \(f+g\) is convex.

b) Does there exist \(f\) and \(g\) both convex, and \(f-g\) is also convex.

c) Consider \(f(x,y)=xy\), with domain \(x>0\) and \(y>0\). Is this function convex?

Problem 4

Let \(A\) and \(B\) be two convex sets.

a) Show that \(A \cap B\) is convex.

b) Is \(A \cup B\) convex?

Problem 5

Consider the function \(f(x_1, x_2)=2x_1^3+x_1^2+2x_1x_2+\frac{1}{2}x_2^2-8x_1-2x_2-10\). Find the range of values of \((x_1,x_2)\) for whith \(f\) is convex, if any.

Problem 6

This is to get started on solving convex problem using a solver. Pick whatever language you’re comfortable with, learn how to call a convex optimization solver. A popular one is cvxpy, or its equivalent in Matlab or Julia, but there are many different ones out there and any of them would work. Solve the following problems, provide the optimal objective values and optimal solutions in your homework solutions.

a) Solve

\[\begin{align} \min_{x_1,x_2} \; & (x_1-2x_2)^2+x_1 \\ \mbox{s.t. } & 0 \leq x_2 \leq 1 \\ & x_1 \geq 0 \end{align}\]

b) Solve

\[\begin{align} \min_{x_1,x_2,x_3} \; & \vert x_1 \vert +e^{x_2+x_3} - \log (x_1+x_3) \\ \mbox{s.t. } & x_1 + x_3 \geq 2 \\ & 0 \leq x_2 \leq 3 \\ & x_1 \leq 1 \end{align}\]