--- title: "Ybus, Zbus and 4-bus Ybus formulation" subject: "Power System Analysis II" chapter: 1 key: "semester_5/power_system_analysis_ii:ybus-zbus-four-bus" source: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-zbus-four-bus site: BCT Engineering --- # Ybus, Zbus and 4-bus Ybus formulation Power System Analysis II · Chapter 1: Interconnected Power System - Asked in: 2081 Bhadra, 2+4+2 marks - Video: [The Ybus Admittance Matrix for Solving Power Flow Equations Part 1](https://www.youtube.com/watch?v=8jNuSwnL7HE) — GeneralPAC: Power System Tutorials - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/power_system_analysis_ii/ybus-zbus-four-bus - Whole chapter: https://www.bctengineer.com/md/semester_5/power_system_analysis_ii/chapter-1 ## Question What are bus admittance matrix and bus impedance matrix? Formulate the bus admittance matrix for 4-bus system and state the properties of the matrix including general equations of their elements. ## Short answer Ybus relates injected bus currents to voltages, I = Ybus V; Zbus is its inverse V = Zbus I. For a 4-bus system, diagonal elements Y_ii = y_i0 + sum of all line admittances connected to bus i, off-diagonals Y_ik = –y_ik (zero if no line). Ybus is symmetric, sparse, and diagonally dominant. ## Answer #### Bus admittance and bus impedance matrices - **Bus admittance matrix** $Y_{bus}$: relates injected bus currents to bus voltages, $\mathbf{I} = \mathbf{Y}_{bus}\mathbf{V}$. It is sparse and formed by inspection. - **Bus impedance matrix** $Z_{bus}$: the inverse, $\mathbf{V} = \mathbf{Z}_{bus}\mathbf{I}$, with $Z_{bus} = Y_{bus}^{-1}$. It is a full matrix, formed by the building algorithm or inversion, and used mainly in fault analysis. #### Formulation for a 4-bus system Assume lines 1-2, 1-3, 2-3, 2-4 and 3-4 with series admittances $y_{ik}$, and shunt admittances $y_{i0}$ at each bus (half line charging). ```text (1)---------y12---------(2) | / | | / | y13 y23 y24 | / | | / | (3)---------y34---------(4) No line 1-4. Each bus i also has y_i0 to ground. ``` KCL at each bus: $$ \begin{aligned} I_1 &= y_{10}V_1 + y_{12}(V_1 - V_2) + y_{13}(V_1 - V_3) \\ I_2 &= y_{20}V_2 + y_{12}(V_2 - V_1) + y_{23}(V_2 - V_3) + y_{24}(V_2 - V_4) \\ I_3 &= y_{30}V_3 + y_{13}(V_3 - V_1) + y_{23}(V_3 - V_2) + y_{34}(V_3 - V_4) \\ I_4 &= y_{40}V_4 + y_{24}(V_4 - V_2) + y_{34}(V_4 - V_3) \end{aligned} $$ In matrix form: $$ \begin{bmatrix} I_1 \\ I_2 \\ I_3 \\ I_4 \end{bmatrix} = \begin{bmatrix} Y_{11} & -y_{12} & -y_{13} & 0 \\ -y_{12} & Y_{22} & -y_{23} & -y_{24} \\ -y_{13} & -y_{23} & Y_{33} & -y_{34} \\ 0 & -y_{24} & -y_{34} & Y_{44} \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \\ V_3 \\ V_4 \end{bmatrix} $$ with $Y_{11} = y_{10} + y_{12} + y_{13}$, $Y_{22} = y_{20} + y_{12} + y_{23} + y_{24}$, $Y_{33} = y_{30} + y_{13} + y_{23} + y_{34}$, $Y_{44} = y_{40} + y_{24} + y_{34}$. #### General equations of elements $$ Y_{ii} = y_{i0} + \sum_{k \ne i} y_{ik},\qquad Y_{ik} = Y_{ki} = -y_{ik} $$ #### Properties of Ybus 1. **Symmetric** about the leading diagonal (no phase shifters). 2. **Sparse:** $Y_{ik} = 0$ when no line joins $i$ and $k$ (here $Y_{14} = 0$). 3. **Generally diagonally dominant:** $|Y_{ii}| \ge \sum_{k\ne i} |Y_{ik}|$ when the shunt admittances are small or absent. 4. **Non-singular** only when shunt elements to ground exist. 5. Easy to form by inspection and to modify for adding or removing a line. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is Zbus related to Ybus? A. Zbus is the transpose of Ybus B. Zbus = inverse of Ybus, V = Zbus I C. Zbus is the diagonal of Ybus D. Zbus = Ybus times I Answer: B. Zbus = Ybus^-1 and gives voltages from injected currents. (Key point: Zbus definition) 2. For lines 1-2, 1-3, 2-3, 2-4, 3-4, which expression gives Y44? A. y40 + y24 + y34 B. y40 + y14 + y24 + y34 C. -y24 - y34 D. y40 + y12 + y13 Answer: A. Bus 4 connects only through lines 2-4 and 3-4 plus its shunt. (Key point: Y44 for the 4-bus network) 3. Why is Y14 zero in this network? A. Bus 1 is the slack bus B. No line joins buses 1 and 4 C. Shunt elements are neglected D. Ybus is diagonally dominant Answer: B. Off-diagonal Y_ik is non-zero only when a line connects i and k. (Key point: Sparsity) 4. Ybus is non-singular only when: A. It is symmetric B. Shunt elements to ground exist C. All lines have equal impedance D. There are an even number of buses Answer: B. Without ground paths the rows sum to zero and the matrix is singular. 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