Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalPower System

Option B

$-y_{12}$

Quick Summary:

In the direct inspection method for forming the bus admittance matrix ( $Y_{bus}$ ), the off-diagonal elements $Y_{ij}$ (where $i \neq j$ ) are defined as the negative of the sum of all admittances connected directly between buses $i$ and $j$ . Therefore, $Y_{12}$ is equal to the negative of the admittance value $y_{12}$ connected between bus 1 and bus 2.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$Y_{ii} = \sum_{j=0, j \neq i}^{n} y_{ij}$ — Sum of admittances connected to bus $i$

$Y_{ij} = -y_{ij}$ — Negative of admittance between bus $i$ and $j$

⚙️ Working Principle

The bus admittance matrix is constructed by applying Kirchhoff's Current Law at each node. Diagonal elements $Y_{ii}$ represent the sum of all admittances connected to bus $i$ , while off-diagonal elements $Y_{ij}$ represent the mutual connection between buses. When current flows between node $i$ and $j$ via admittance $y_{ij}$ , the contribution to the nodal current equation introduces a negative sign, leading to $Y_{ij} = -y_{ij}$ .

📌 Key Points

▸
The $Y_{bus}$ matrix is symmetric if the system is linear and bilateral.
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The direct inspection method is only applicable for systems without mutual coupling between transmission lines.
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The size of the $Y_{bus}$ matrix is $n \times n$ , where $n$ is the number of buses.

✅ Advantages

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Computationally efficient for simple systems.
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Easy to implement in software for sparse matrices.

❌ Disadvantages / Limitations

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Cannot account for mutual coupling directly.
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Requires inversion of full matrix if impedance form ( $Z_{bus}$ ) is needed.

🛠️ Applications / Uses

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Load flow studies (Newton-Raphson method).
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Short circuit analysis.

📄 Additional Information

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In the $Y_{bus}$ matrix, $y_{ij}$ represents the shunt admittance if it exists between the buses, or the series admittance of the branch.
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If no direct path exists between bus $i$ and $j$ , $Y_{ij} = 0$ .

📊 Diagram / Illustration

✅

B is correct — By definition of the direct inspection method, the off-diagonal elements $Y_{ij}$ of the admittance matrix are given by the negative sum of all admittances connected directly between buses $i$ and $j$ .

Core Concepts Used

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Bus Admittance Matrix ($Y_{bus}$) Direct Inspection Method Power System Network Topology

💡 EXAM TIP

Always remember that $Y_{bus}$ is a sparse matrix; for large power systems, most off-diagonal elements are zero, which is critical for efficient computation in load flow studies.

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