$4.$ Power flow $(20$ Points$)$

a$.(5$ pts$)$ Draw the single line diagram $($SLD$,$ or "one$-$line" diagram$)$ for the following power system. Refer to lecture note example diagrams, and IEEE Std $315-1975$ available on the course website for symbol references.

$1.$ A Y$-$connected, neutral$-$grounded synchronous generator drives a local load impedance $\backslash ($ Z$\_\{1\}\backslash )$ in parallel with a $3-$phase $\backslash ($ Y $\backslash )-\backslash (\backslash$Delta $\backslash )$ transformer $($Bus $1$ is the generator output$).$

$2.$ The transformer secondary $($with primary on Bus $1)$ drives a $40$km $3-$phase transmission line, where it meets the primary of a ZigZag transformer.

$3.$ A second generator $(3-$phase Y$-$connected, neutral grounded$)$ drives a $3-$phase Y$-$Y transformer with grounded neutrals; the transformer secondary drives a $20$ km transmission line, the other end of which is connected to the secondary of the ZigZag transformer described in bullet $\text{'}2\text{'}.$

$4.$ In parallel with the secondary of the ZigZag transformer of bullet $\text{'}3\text{'},$ there is a load consuming $\backslash ($ P$=40\backslash$mathrm$\{$~kW$\}\backslash )$ of active power, and $\backslash ($ Q$=20\backslash )$ kVARs of reactive power. b$.$ Consider the single$-$line diagram below, and the table defining the diagram values. Download the Gauss$-$Seidel example code from the course website and modify it for this power system, then answer the following questions. Figure $5.$ Single$-$line diagram for Question $4,$ part b$.$ i$.(7$ pts$)$ Calculate all bus voltage phasors $($in magnitude$-$phase angle form$)$ using the Gauss$-$Seidel method.

ii$.(4$ pts$)$ What is the active and reactive power out of the slack bus $1?$ What is the active and reactive power into load $\backslash ($ Z$\_\{4\}^\{\backslash$text $\{$Load $\}\}\backslash )?$

iii. $(4$ pts$)$ Calculate the ratio of the bus $3$ voltage magnitude to the bus $1$ voltage magnitude. Reduce the slack bus voltage to $\backslash (15$ k V$\_\{$R M S$\}\backslash )$ and recalculate $-$ how has the fraction changed? Reason why that would be$.$