Branch

Line or off-nominal transformer between two buses.

Model Parameters

Symbol	Units	Data	Description	Note
\(R\)	[p.u.]	`R`	Series resistance	Default 0
\(X\)	[p.u.]	`X`	Series reactance	Default 0
\(G\)	[p.u.]	`G`	Total line shunt conductance, split between the ports	Default 0
\(B\)	[p.u.]	`B`	Total line shunt susceptance, split between the ports	Default 0
\(G_\mathrm{mag}\)	[p.u.]	`Gmag`	Magnetizing shunt conductance at bus 1	Default 0
\(B_\mathrm{mag}\)	[p.u.]	`Bmag`	Magnetizing shunt susceptance at bus 1	Default 0
\(\tau\)	[p.u.]	`tap`	Off-nominal tap magnitude on the bus-1 side	Default 1
\(\theta\)	[rad]	`phase`	Phase-shift angle	Default 0
\(S^{\max}\)	[p.u.]	`Smax`	Apparent power limit at each port	Default unbounded
\(u\)	[-]	`in_service`	Service status	Default true

Parameter Validation

\(R^2 + X^2 > 0\)
\(\tau > 0\)
\(S^{\max} > 0\)

Model Derived Parameters

\[\begin{split}\begin{aligned} g &= R / (R^2 + X^2) \\ b &= -X / (R^2 + X^2) \\ g_{11} &= u (-g / \tau^2 - G / 2 - G_\mathrm{mag}) \\ b_{11} &= u (-b / \tau^2 - B / 2 - B_\mathrm{mag}) \\ g_{12} &= u (g \cos\theta - b \sin\theta) / \tau \\ b_{12} &= u (b \cos\theta + g \sin\theta) / \tau \\ g_{21} &= u (g \cos\theta + b \sin\theta) / \tau \\ b_{21} &= u (b \cos\theta - g \sin\theta) / \tau \\ g_{22} &= u (-g - G / 2) \\ b_{22} &= u (-b - B / 2) \end{aligned}\end{split}\]

Model Ports

Name	Port	Description
`bus1`	Bus	Bus-1 port, the tapped side
`bus2`	Bus	Bus-2 port

Model Variables

Internal Variables

None.

External Variables

Symbol	Units	Description	Note
\(V_{\mathrm{r}1}\)	[p.u.]	Bus-1 real voltage	Owned by bus 1
\(V_{\mathrm{i}1}\)	[p.u.]	Bus-1 imaginary voltage	Owned by bus 1
\(V_{\mathrm{r}2}\)	[p.u.]	Bus-2 real voltage	Owned by bus 2
\(V_{\mathrm{i}2}\)	[p.u.]	Bus-2 imaginary voltage	Owned by bus 2

Model Equations

\[\begin{split}\begin{aligned} I_{\mathrm{r}1} &= g_{11} V_{\mathrm{r}1} - b_{11} V_{\mathrm{i}1} + g_{12} V_{\mathrm{r}2} - b_{12} V_{\mathrm{i}2} \\ I_{\mathrm{i}1} &= b_{11} V_{\mathrm{r}1} + g_{11} V_{\mathrm{i}1} + b_{12} V_{\mathrm{r}2} + g_{12} V_{\mathrm{i}2} \\ I_{\mathrm{r}2} &= g_{21} V_{\mathrm{r}1} - b_{21} V_{\mathrm{i}1} + g_{22} V_{\mathrm{r}2} - b_{22} V_{\mathrm{i}2} \\ I_{\mathrm{i}2} &= b_{21} V_{\mathrm{r}1} + g_{21} V_{\mathrm{i}1} + b_{22} V_{\mathrm{r}2} + g_{22} V_{\mathrm{i}2} \\ P_1 &= V_{\mathrm{r}1} I_{\mathrm{r}1} + V_{\mathrm{i}1} I_{\mathrm{i}1} \\ Q_1 &= V_{\mathrm{i}1} I_{\mathrm{r}1} - V_{\mathrm{r}1} I_{\mathrm{i}1} \\ P_2 &= V_{\mathrm{r}2} I_{\mathrm{r}2} + V_{\mathrm{i}2} I_{\mathrm{i}2} \\ Q_2 &= V_{\mathrm{i}2} I_{\mathrm{r}2} - V_{\mathrm{r}2} I_{\mathrm{i}2} \end{aligned}\end{split}\]

Objective

None.

Internal Constraints

Equality

None.

Inequality

With a finite rating \(S^{\max}\):

\[\begin{split}\begin{aligned} P_1^2 + Q_1^2 &\le (S^{\max})^2 \\ P_2^2 + Q_2^2 &\le (S^{\max})^2 \end{aligned}\end{split}\]

External Constraints

Power into each bus, added to its balance:

\[\begin{split}\begin{aligned} \Delta P_1^\mathrm{bus} &\mathrel{+}= P_1 \\ \Delta Q_1^\mathrm{bus} &\mathrel{+}= Q_1 \\ \Delta P_2^\mathrm{bus} &\mathrel{+}= P_2 \\ \Delta Q_2^\mathrm{bus} &\mathrel{+}= Q_2 \end{aligned}\end{split}\]

Initialization

None.