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- Parameter estimation example - Power flow model composer with test example - Improved CMake build - Updated documentation
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| # Branch Model | ||
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| Transmission lines and different types of transformers (traditional, Load Tap-Changing transformers (LTC) and Phase Angle Regulators (PARs)) can be modeled with a common branch model. | ||
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| ## Transmission Line Model | ||
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| The most common circuit that is used to represent the transmission line model is $`\pi`$ circuit as shown in Figure 1. The nominal flow direction is from sending bus _s_ to receiving bus _r_. | ||
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| <div align="center"> | ||
| <img align="center" src="../../Documentation/Figures/TL.jpg"> | ||
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| Figure 1: Transmission line $`\pi`$ equivalent circuit | ||
| </div> | ||
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| Here | ||
| ``` math | ||
| Z'=R+jX | ||
| ``` | ||
| and | ||
| ``` math | ||
| Y'=G+jB, | ||
| ``` | ||
| where $`R`$ is line series resistance, $`X`$ is line series reactance, $`B`$ is line shunt charging, and $`G`$ is line shunt conductance. As can be seen from Figure 1 total $`B`$ and $`G`$ are separated between two buses. | ||
| The current leaving the sending bus can be obtained from Kirchhoff's current law as | ||
| ```math | ||
| I_s = y(V_s - V_r) + \frac{Y'}{2} V_s, | ||
| ``` | ||
| where $`V_s`$ and $`V_r`$ are voltages on sending and receiving bus, respectively, and | ||
| ```math | ||
| y = \frac{1}{Z'} = \frac{R}{R^2+X^2} + j\frac{-X}{R^2+X^2} = g + jb. | ||
| ``` | ||
| Similarly, current leaving receiving bus is given as | ||
| ```math | ||
| -I_R = y(V_r - V_s) + \frac{Y'}{2} V_r. | ||
| ``` | ||
| These equations can be written in a compact form as: | ||
| ```math | ||
| \begin{bmatrix} | ||
| I_{s}\\ | ||
| -I_{r} | ||
| \end{bmatrix} | ||
| = \mathbf{Y}_{TL} | ||
| \begin{bmatrix} | ||
| V_{s}\\ | ||
| V_{r} | ||
| \end{bmatrix} | ||
| ``` | ||
| where: | ||
| ```math | ||
| \mathbf{Y}_{TL}=\begin{bmatrix} | ||
| g + jb + \dfrac{G+jB}{2} & -(g + jb) \\ | ||
| -(g + jb) & g + jb + \dfrac{G+jB}{2} | ||
| \end{bmatrix} | ||
| ``` | ||
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| ### Branch contributions to residuals for sending and receiving bus | ||
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| Complex power leaving sending and receiving bus is computed as | ||
| ```math | ||
| \begin{bmatrix} | ||
| S_{s}\\ | ||
| S_{r} | ||
| \end{bmatrix} | ||
| = | ||
| \begin{bmatrix} | ||
| V_{s}\\ | ||
| V_{r} | ||
| \end{bmatrix} | ||
| \begin{bmatrix} | ||
| I_{s}\\ | ||
| -I_{r} | ||
| \end{bmatrix}^* | ||
| = | ||
| \begin{bmatrix} | ||
| V_{s}\\ | ||
| V_{r} | ||
| \end{bmatrix} | ||
| \mathbf{Y}_{TL}^* | ||
| \begin{bmatrix} | ||
| V_{s}\\ | ||
| V_{r} | ||
| \end{bmatrix}^* | ||
| ``` | ||
| After some algebra, one obtains expressions for active and reactive power that the branch takes from adjacent buses: | ||
| ```math | ||
| P_{s} = \left(g + \frac{G}{2}\right) |V_{s}|^2 + [-g \cos(\theta_s - \theta_r) - b \sin(\theta_s - \theta_r)] |V_{s}| |V_{r}| | ||
| ``` | ||
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| ```math | ||
| Q_{s} = -\left(b + \frac{B}{2}\right) |V_{s}|^2 + [-g \sin(\theta_s - \theta_r) + b \cos(\theta_s - \theta_r)] |V_{s}| |V_{r}| | ||
| ``` | ||
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| ```math | ||
| P_{r} = \left(g + \frac{G}{2}\right) |V_{r}|^2 + [-g \cos(\theta_s - \theta_r) + b \sin(\theta_s - \theta_r)] |V_{s}| |V_{r}| | ||
| ``` | ||
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| ```math | ||
| Q_{r} = -\left(b + \frac{B}{2}\right) |V_{r}|^2 + [ g \sin(\theta_s - \theta_r) + b \cos(\theta_s - \theta_r)] |V_{s}| |V_{r}| | ||
| ``` | ||
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| These quantities are treated as _loads_ and are substracted from $`P`$ and $`Q`$ residuals computed on the respective buses. | ||
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| ## Branch Model | ||
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| **Note: Transformer model not yet implemented** | ||
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| The branch model can be created by adding the ideal transformer in series with the $`\pi`$ circuit as shown in Figure 2 where $`\tau`$ is a tap ratio magnitude and $`\theta_{shift}`$is the phase shift angle. | ||
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| <div align="center"> | ||
| <img align="center" src="../../Documentation/Figures/branch.jpg"> | ||
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| Figure 2: Branch equivalent circuit | ||
| </div> | ||
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| The branch admitance matrix is then: | ||
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| ```math | ||
| \mathbf{Y}_{BR}= | ||
| \begin{bmatrix} | ||
| \left(g + jb + \dfrac{G+jB}{2} \right)\dfrac{1}{\tau^2} & -(g + jb)\dfrac{1}{\tau e^{-j\theta_{shift}}}\\ | ||
| &\\ | ||
| -(g + jb)\dfrac{1}{\tau e^{j\theta_{shift}}}. & g + jb + \dfrac{G+jB}{2} | ||
| \end{bmatrix} | ||
| ``` | ||
| ### Branch contribution to residuals for sending and receiving bus | ||
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| The power flow contribution for the transformer model are obtained in a similar manner as for the $`\pi`$-model. |
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