Correct Answer

⚙️ TE • Technical Concept & PrincipleElectricalPower System

Option B

$i_{i} + j_{i} = y_{i} \times v_{i}$

Quick Summary:

In power system network analysis, the performance of a branch 'i' is represented by the relationship between the current flowing through it, the source current (if any), the branch voltage, and the branch admittance. This is derived from Ohm's law applied in the context of a network node or loop equation where the branch current injected into the network plus the source current equals the product of branch admittance and nodal voltage difference.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$i_i + j_i = y_i \times v_i$ — Performance equation for branch i

$Y_{bus} = A^{T} \cdot Y_{branch} \cdot A$ — Relation to Bus Admittance Matrix construction

⚙️ Working Principle

For a transmission line or branch represented by an equivalent $\pi$ -model, the total current $i_i$ flowing into the branch from a node plus any external source current $j_i$ injected at that node must equal the net current determined by the admittance of the branch $y_i$ multiplied by the voltage across it $v_i$ . This equation $i_i + j_i = y_i \times v_i$ forms the fundamental basis for constructing the Bus Admittance Matrix ( $Y_{bus}$ ) used in load flow studies.

📌 Key Points

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The equation represents the nodal current balance in admittance form.
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$y_i$ is the complex admittance, defined as $y_i = \frac{1}{z_i} = g_i + jb_i$ .
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It is the foundation for the Gauss-Seidel and Newton-Raphson load flow methods.
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The term $j_i$ accounts for shunt currents or current sources connected at the branch terminals.

✅ Advantages

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Simplifies the formulation of linear algebraic equations for power systems.
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Admittance matrix is sparse, reducing computational complexity for large networks.

❌ Disadvantages / Limitations

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Limited to linear modeling of network components.
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Requires inversion of the matrix if solving for voltages directly.

🛠️ Applications / Uses

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Load flow analysis in power systems.
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Short circuit analysis (fault calculation).
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Network reduction and equivalence studies.

📄 Additional Information

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Standard power system analysis utilizes the $Y_{bus}$ matrix constructed from these branch equations.
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Option A is incomplete as it ignores the source/injection current term $j_i$ often present in complex networks.
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The variables $i_i$ , $j_i$ , and $v_i$ are usually represented as phasors in steady-state AC analysis.

📊 Diagram / Illustration

✅

B is correct — The equation $i_i + j_i = y_i \times v_i$ correctly balances the current injected into the branch with the product of the branch admittance and voltage.

Core Concepts Used

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Admittance Matrix Nodal Analysis Power System Modeling

💡 EXAM TIP

Always verify sign conventions for current injections ( $j_i$ ) in nodal equations, as standard practice often dictates that current entering a node is positive.

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