Homework 3

Due Nov 6th at 11:59pm. Submit to CANVAS at https://canvas.uw.edu/courses/1917157/assignments/11872111.

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. Try a few different starting points and see if you can observe qualitatively different behaviors. 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

onsider the following tree network with paramters \(y_{01}=0.81-j0.91, y_{12}=0.13-j 0.92, y_{23}=0.64-j0.91, y_{24}=0.28-j0.55, y_{45}=0.96-j0.96\), where bus \(0\) is the slack bus. 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).

a) Draw the network.

b) Write out the power flow equations for this network. Note that since we fix all voltages to be one, there variables are the angles at buses 1,2,3,4,5. And there are five equations, one for the active power at each bus.

c) Use Newton method to solve for the angles that would satisfy the active powers.

Problem 3

Use the solution you obtained from the last problem.

a) Compute the reactive power injections at each bus.

b) Write the power flow equations for where both \(V\) and \(\theta\) are the variables. So there are ten equations: active and reactive power at each bus.

c) Use Newton method to solve for the angle and voltages.

Problem 4

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

b) 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 5

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

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

You can do this by hand by substituting \(x_2\) with \(x_1\).

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

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

You can do this by hand by substituting \(x_2\) with \(x_1\).

c) let \(x_1^*\) and \(x_2^*\) be the optimal solutions. Evaluate \(C_1'(x_1^*)\) and \(C_2'(x_2^*)\) for both a) and b). What you do observe?

d) Without using a computer, solve the following problem. Use the observation you made in part c). The load at a bus is 10 MW, and there are four generators with costs

\[C_1(x_1)=0.5 x_1^2+2 x_1, \; C_2(x_2)=x_2^2+x_2, \; C_3(x_3)=0.1x_3^2 + 5 x_3, \; C_4 (x_4)=5 x_4+6 x_4\]

Find the optimal, that is, the least cost generation that meets the load.

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

\[\begin{align} \min_{x_1,x_2} \; & (x_1-2x_2)^2+x_1 \\ \mbox{s.t. } & 2 \leq x_2 \leq 4 \\ & x_1 \geq 0 \end{align}\]

b) Solve

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