@@ -7,7 +7,9 @@ contributions are oriented entering the adjacent buses.
77Notes:
88- Setting $\tau = 1$ and $\theta = 0$ gives the ordinary symmetric
99 transmission-line $\pi$ model.
10- - $G$ and $B$ are total branch shunt values split equally between terminals.
10+ - The total line shunt $G + jB$ is added outside the $\mathbf{M}$ transformation,
11+ while the total magnetizing shunt $G_ \text{mag} + jB_ \text{mag}$ is transformed
12+ by $\mathbf{M}$; both total shunts are split equally between terminals.
1113- The branch has no solver-owned variables; it contributes current residuals
1214 directly to the connected buses.
1315
@@ -23,40 +25,43 @@ Figure 1: Branch equivalent circuit
2325
2426## Model Parameters
2527
26- Symbol | Units | Description | Typical Value | Note
27- ------------|---------|--------------------------------------------------|---------------|------
28- $R$ | [ p.u.] | Branch series resistance | |
29- $X$ | [ p.u.] | Branch series reactance | |
30- $G$ | [ p.u.] | Total branch shunt conductance | 0 |
31- $B$ | [ p.u.] | Total branch shunt susceptance | 0 |
32- $\tau$ | [ p.u.] | Off-nominal tap magnitude on bus-1 side | 1 | Parameter name: ` tap `
33- $\theta$ | [ rad] | Phase-shift angle | 0 | Parameter name: ` phase `
28+ Symbol | Units | JSON | Description | Typical Value | Note
29+ ---------------------|--------|---------|-----------------------------------------|---------------|------
30+ $R$ | [ p.u.] | ` R ` | Branch series resistance | |
31+ $X$ | [ p.u.] | ` X ` | Branch series reactance | |
32+ $G$ | [ p.u.] | ` G ` | Total line shunt conductance | 0.0 |
33+ $B$ | [ p.u.] | ` B ` | Total line shunt susceptance | 0.0 |
34+ $G_ \text{mag}$ | [ p.u.] | ` Gmag ` | Total magnetizing shunt conductance | 0.0 |
35+ $B_ \text{mag}$ | [ p.u.] | ` Bmag ` | Total magnetizing shunt susceptance | 0.0 |
36+ $\tau$ | [ p.u.] | ` tap ` | Off-nominal tap magnitude on bus-1 side | 1.0 |
37+ $\theta$ | [ rad] | ` phase ` | Phase-shift angle | 0.0 |
3438
3539### Parameter Validation
3640
3741Invalid Branch parameter sets are rejected by the following checks:
3842
3943``` math
4044\begin{aligned}
41- &R, X, G, B, \tau, \theta \in \mathbb{R}\ \text{and finite} \\
45+ &R, X, G, B, G_\text{mag}, B_\text{mag}, \tau, \theta
46+ \in \mathbb{R}\ \text{and finite} \\
4247 &R^2 + X^2 > 0 \\
4348 &\tau > 0
4449\end{aligned}
4550```
4651
4752### Model Derived Parameters
4853
49- The series and shunt admittances are:
54+ The series, magnetizing shunt, and line shunt admittances are:
5055
5156``` math
5257\begin{aligned}
53- Y_{\mathrm{br}} &= \dfrac{1}{R + jX} \\
54- Y_{\mathrm{sh}} &= G + jB
58+ Y_{\mathrm{br}} &= \dfrac{1}{R + jX} \\
59+ Y_{\mathrm{mag}} &= G_\text{mag} + jB_\text{mag} \\
60+ Y_{\mathrm{sh}} &= G + jB
5561\end{aligned}
5662```
5763
58- The nominal $\pi$-branch admittance matrix is the sum of the series and shunt
59- admittance contributions:
64+ The series, magnetizing shunt, and line shunt admittance matrices are:
6065
6166``` math
6267\begin{aligned}
@@ -67,7 +72,17 @@ admittance contributions:
6772 Y_{\mathrm{br}}
6873 & -Y_{\mathrm{br}}
6974 \end{bmatrix}
70- +
75+ \\
76+ \mathbf{Y}_\text{mag}
77+ &=
78+ \dfrac{1}{2}
79+ \begin{bmatrix}
80+ -Y_{\mathrm{mag}} & 0 \\
81+ 0 & -Y_{\mathrm{mag}}
82+ \end{bmatrix}
83+ \\
84+ \mathbf{Y}_\text{sh}
85+ &=
7186 \dfrac{1}{2}
7287 \begin{bmatrix}
7388 -Y_{\mathrm{sh}} & 0 \\
@@ -85,8 +100,21 @@ The off-nominal transformer transformation uses bus 1 as the tap side:
85100 \begin{bmatrix}
86101 \tau^{-1} & 0 \\
87102 0 & e^{j\theta}
88- \end{bmatrix} \\
89- \mathbf{Y} &= \mathbf{M}^{\dagger}\mathbf{Y}_0\mathbf{M}
103+ \end{bmatrix}
104+ \end{aligned}
105+ ```
106+
107+ The line shunt is added outside the transformation:
108+
109+ ``` math
110+ \begin{aligned}
111+ \mathbf{Y}
112+ &=
113+ \mathbf{M}^{\dagger}
114+ \left(\mathbf{Y}_0 + \mathbf{Y}_\text{mag}\right)
115+ \mathbf{M}
116+ +
117+ \mathbf{Y}_\text{sh}
90118\end{aligned}
91119```
92120
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