Consider the three$-$bus power system in the Fig$\mathrm{.}1.$ Table $1$ shows the data about the generators

connected to this system. Table $2$ gives the branch data for the three$-$bus power system.

Assume Bus $1$ is the slack bus, what is the distribution factor matrix? $(10$ pt $)$

Calculate the unconstrained economic dispatch and the nodal prices for the loading conditions

shown in the figure $(10$ pt $)$

Formulate the security constraint economic dispatch. $(10$ pt$)$

Calculate the security constrained economic dispatch. $(20$ pt $)$

At the security constrained economic dispatch, calculate the nodal prices for the same loading

conditions. $(25$ pt $)$

Calculate the shadow price for the three branches. $(25$ pt $)$

Solutions:

The distribution factor matrix can be found as $($using current division concept from ECEC $214)$

$[[$F$\_(12)],[$F$\_(13)],[$F$\_(23)]]=[[(-0\mathrm{.}6)/(0\mathrm{.}6)-6\mathrm{.}2^(2),(-0\mathrm{.}3)/(0\mathrm{.}3-6\mathrm{.}5)],[(-6\mathrm{.}2)/(0\mathrm{.}6+0\mathrm{.}2),(-0\mathrm{.}5)/(0\mathrm{.}3+0\mathrm{.}5)],[(0\mathrm{.}2)/(0\mathrm{.}6+0\mathrm{.}2),(-0\mathrm{.}3)/(0\mathrm{.}3+0\mathrm{.}5)]][[$P$\_(2)],[$P$\_(3)]]$

Therefore, the distribution factor matrix is

$[[-(3)/(4),-(3)/(8)],[-(1)/(4),-(5)/(8)],[(1)/(4),-(3)/(8)]]$

specifically $-$ how did he get the answer? the matrix at the bottom is the correct answer $($from class$)$ how did he get those numbers?