Homework 2

Due Oct 23rd at 11:59pm. Submit to CANVAS at https://canvas.uw.edu/courses/1917157/assignments/11849576.

Problem 1

Consider a \(n\) by \(n\) matrix \(A\). We say \(A\) is diagonalizable if there exist some invertible matrix \(P\), where \(A=P \Lambda P^{-1}\), where \(\Lambda\) is a diagonal matrix.

a) Suppose \(A\) has \(n\) eigenvalues. Proof that we can take \(\Lambda\) to the digaonal matrix with eigenvalues on the diagonal and \(P\) to be matrix where each column is the corresponding eigenvector.

b) Find a matrix that is not diagonalizable. Hint: there is a 2 by 2 matrix that is not diagonalizable.

Problem 2

Let \(A\) and \(B\) be real \(n\) by \(n\) matrices. In general, \(AB \neq BA\). Show that if \(A\) and \(B\) are both diagonalizable, and they have the same eigenvectors, then \(AB=BA\).

Problem 3

Consider a lossy power system. That is, each line adminttance \(g_{ik}-j b_{ik}\) has \(g_{ik}\geq 0\). Proof that the the thermal loss in the system is nonnegative, and is zero if and only if all the bus voltages are equal. Hint: Once the thermal loss is appriopriately defined, the proof is about two lines.

Problem 4

Consider the equation \(f(x)=ax^2-x\), and we want to solve \(f(x)=0\). Of course, we can compute the roots pretty easily, but let’s try the Gauss Algorithm. This algorithm views the equation as a fixed point equation. That is, solving for \(f(x)=0\) is the same as solving for \(x\) such that \(ax^2=x\). Then a simple algorithm is to perform fixed point iterates: \(x^{(k+1)}=f(x^{(k)})\), and see if it converges. We call \(\mathcal{X}=\{x^{(0)}:\text{fixed point iterates converges starting from } x^{(0)}\}\) the region of convergence.

a) Pick \(a=0.5\) and find the region of convergence.

b) Pick \(a=2\) and find the region of convergence.

Problem 5

Let \(f(x)\) the same as in the last problem. We now use the Newtow algorithm covered in class. The iterations are

\[x^{(k+1)} = x^{(k)} - \frac{f(x^{(k)})}{f'(x^{(k)})}.\]

How about the region of convergence for this algorithm? Hint: run some simulations to help you build some intutions.

Problem 6

Here we look at solving power flow equation for a two bus system. We are going to set bus 1 as the slack bus, and bus 2 have complex voltage \(V \angle \theta\). The line admittance is \(1-j\).

a) Write the power flow equations for this system.

b) Use Newton’s method to find the values of \(V\) and \(\theta\) when \(P_2=-0.11\) and \(Q_2=0.083\).

Problem 7

We stay with the system in the last problem. Let’s say we want to solve the power flow problem where \(P_2=5\) and \(Q_2=5\).

a) Before running the algorithm, look at the values, and decide whether there is a solution and explain why or why not.

b) What happens when we do try to run the Netwon algorithm anyways?