# Forming Bus Admittance Matrix Using Singular Transformation Given: - Reference node: ** 6** ## Step 1: List the Bus and Line Data Assume the system has **n** buses (nodes), and **m** branches (lines). Since you haven't provided the actual data, I'll describe the generic process and notation. Suppose the network has nodes: **1, 2, 3, 4, 5, 6** and Node 6 is the reference. ## Step 2: Primitive Admittance Matrix (\( [y] \)) - List the admittance (\( y_k \)) of each branch \( k \). - Form a diagonal matrix \( [y] \) of size \( m \times m \) with these values. For example, if there are 7 branches: \[ [y] = \begin{bmatrix} y_1 & & & & & & \\ & y_2 & & & & & \\ & & y_3 & & & & \\ & & & y_4 & & & \\ & & & & y_5 & & \\ & & & & & y_6 & \\ & & & & & & y_7 \end{bmatrix} \] ## Step 3: Bus Incidence Matrix (\( [A] \)) - Exclude the reference node (node 6). - \( [A] \) is an \( (n-1) \times m \) matrix. Each row: corresponds to a non-reference bus. Each column: corresponds to a branch. - Entry is **+1** if branch leaves the bus - **-1** if it enters - **** otherwise **Example:** Suppose branch 1 is from bus 1 to 2: | | 1 | 2 | 3 | ... | |-----|---|---|---|-----| | **1** | 1 | | | ... | | **2** | -1 | 1 | | ... | | **3** | | -1 | 1 | ... | | **4** | | | -1 | ... | | ... | ... | ... | ... | ... | Fill out based on actual network. ## Step 4: Bus Admittance Matrix (\( Y_{bus} \)) by Singular Transformation \[ Y_{bus} = A \cdot y \cdot A^T \] - \( A \): Bus incidence matrix (excluding reference node) - \( y \): Primitive admittance matrix ### Steps: 1. Compute \( A \cdot y \): Multiply the incidence matrix by the primitive admittance matrix. 2. Compute \( (A \cdot y) \cdot A^T \): Multiply the result by the transpose of the incidence matrix. This yields \( Y_{bus} \), an \( (n-1) \times (n-1) \) matrix (since the reference node is excluded). ## Step 5: Example (with hypothetical values) Suppose we have: - 6 nodes (node 6 is reference) - 7 branches with admittances \( y_1, y_2, ..., y_7 \) - Bus incidence matrix \( [A] \) is \( 5 \times 7 \) **Calculate:** \[ Y_{bus} = A \cdot y \cdot A^T \] This will give a \( 5 \times 5 \) matrix representing the bus admittance matrix for nodes 1-5, with node 6 as reference. ## Step 6: Final Matrix The final \( Y_{bus} \) matrix can be used for power flow, fault analysis, etc. --- **Note:** If you provide the actual line data (which nodes each branch connects, and their admittances), I can compute the explicit \( Y_{bus} \) matrix for your network. --- **References:** - [Power System Analysis by Hadi Saadat](https://www.amazon.com/Power-System-Analysis-Hadi-Saadat/dp/0073545513) - [Stevenson & Grainger: Power System Analysis] --- **Diagram Example (bus network):**  *Alt text: Example of a 3-bus network diagram (for illustration only)* --- If you provide the actual data (branch connections and admittances), I can fill in all matrices explicitly.