Homework 3

Due Oct 18th at 11:59pm. Submit to CANVAS at https://canvas.uw.edu/courses/1828831/assignments/10779006.

Problem 1

Consider the following system of nonlinear equations

\[\begin{align} x_1^2-x_1 x_2+x_2 &= 1 \\ x_2^3-x_1^2 x_2 &=6 \end{align}\]

Solve it using the Newton Raphson method. Plot the trajectory of the iterations. That is, start at your chosen initial point, and plot the subsequent values of \(x_1\) and \(x_2\) in each iteration until they converge.

Problem 2

Consider the following function \(f(\theta_1, \theta_2)=\begin{bmatrix} f_1(\theta_1,\theta_2) \\ f_2(\theta_1,\theta_2) \end{bmatrix} = \begin{bmatrix} \cos(\theta_1-\theta_2)+\sin(\theta_1) \\ \sin(\theta_1-\theta_2)+\cos(\theta_2) \end{bmatrix}.\)

a) Plot the all possible values of \(f_1\) and \(f_2\) by varying \(\theta_1\) and \(\theta_2\).

b) Say we want to solve

\[f=\begin{bmatrix} 3 \\ 1 \end{bmatrix}.\]

This equation doesn’t have a solution (adding \(\sin\) and \(\cos\) cannot be bigger than \(2\)). If we tried to use Newton Raphson, the algorithm would diverge or oscillate. But suppose we use the regularized version. That is, at each step \(k\), we solve (with some positive \(\lambda\))

\[\min_{\Delta \theta} \; \Vert f (\theta^{(k)}) + \nabla f (\theta^{(k)}) \Delta \theta \Vert ^2+\lambda \Vert \Delta \theta \Vert^2\]

and update the soluiton as \(\theta^{(k+1)}=\theta^{(k)}+\Delta \theta\).

Would this converge? If so, what does it converge to? If not, why not?

Problem 3

A fundamental problem in power system operations is that given a set of active and reactive injections, we want to know whether they are feasible. That is, whether they can be achieved by some complex voltages. Currently, we don’t really have an algorithm that’s better than just to solve power flow and see. Note that the feasibility problem looks to be an easier problem, since it’s just asking for a yes/no answer, but solving power flow returns the actual voltages. Later in the course will look at cases where answering yes/no quesitons is easier to finding the soluiton. For this problem, we look at a necessary condition on the active powers.

For a \(n+1\)-bus system where bus \(0\) is the slack bus. Show that if an active power injection vector \((P_0,\dots,P_n)\) is feasible, then it must satisfy \(\sum_{i=0}^{n+1} P_i \geq 0\).

Problem 4

It is often useful to have a softer measure of how “feasible” a power injection is. More precisely, we say a power injection is more feasible if after a large perturbation, it remains feasible. Conversely, we say a power injection is on the feasibility boundary if it is feasible, but can become infeasible after a very small perturbation. Find a metric that quantifies how feasible a soultion is. Hint: this is related to question 2b).

Problem 5

Consider following system. For simplicity, we fix all the voltages to be \(1\) and only consider active power. The feasible set of all possible \((P_1,P_2)\) that is achievable by varying the angles at buses 1 and 2. Show the condition you derived in problem 4 makes sense for this example. That is, if your condition decided a point is on the boundary, is it actually on the boundary? You don’t need to prove things analytically and can do this through plotting.