For a four$-$bus power system, bus $1$ is the swing bus with$111\mathrm{.}0,0$o

V

$$

$==$ ; bus $2$ is a PV$-$bus

with$2$V and injected active power known; bus $3$ is a PQ$-$bus with real and reactive load

power known; and bus $4$ is a PQ$-$bus with real and reactive load power known. Line

resistances are neglected. The system bus admittance matrix Ybus is known with its element at

ith row and jth column denoted as$,(1$ ~ $4,1$ ~ $4)$ij ijY jB i j$===\mathrm{.}12$

$34$

$(1)$ If the power flow is written in polar form, please determine the dimension of the Jacobian

matrix for power flow analysis and illustrate what are the variables to be solved.

$(2)$ Please write the power flow equations for the real power at bus $2,$ real and reactive power

at bus $4.$

$(3)$ Please determine the following elements of Jacobian matrix:$24$

$43$

$,$

P P

V

$$

$$

$,$ and$4$

$4$

Q

$$

$$

$.$

$($present each derivation step$)$