Network Reduction Theory
This document explains the theory and mathematics behind network reduction techniques in power systems.
Why Network Reduction?
Power system networks can be very large, with thousands of buses and branches. Network reduction techniques simplify these networks while preserving essential characteristics for analysis.
Benefits of Reduction:
- Computational Efficiency: Smaller matrices are faster to compute and invert
- Focus: Reduces complexity to focus on areas of interest
- Scalability: Makes large-scale studies tractable
- Clarity: Simplifies visualization and understanding
Preservation Goals:
Network reduction aims to preserve:
- Power flow relationships at retained buses
- Impedance relationships between key points
- Stability characteristics (when applicable)
- Essential topology features
The type of reduction (e.g. RadialReduction, DegreeTwoReduction, WardReduction) determines the extent to which these characteristics are retained
Radial Branch Reduction
What is a Radial Branch?
A radial branch is one that connects to a bus that has only one connection to the rest of the network. Think of it as a "dead-end" in the network.
Main Network --- Bus A --- Bus B (radial)
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Bus C (radial)In this example, buses B and C are radial - they each connect to only one other bus.
Mathematical Basis
Radial buses can be reduced because power flow to/from them is uniquely determined by the connection bus. The reduction involves:
- Identifying radial buses: Buses with degree = 1
- Transferring loads: Moving load/generation to the connection point
- Removing the branch: Eliminating the radial connection
Why Radial Reduction Works
For a radial bus $k$ connected to bus $j$:
\[P_k = \frac{V_j V_k}{X_{jk}} \sin(\theta_j - \theta_k)\]
Under DC approximation with $V_j \approx V_k \approx 1$:
\[P_k \approx \frac{\theta_j - \theta_k}{X_{jk}}\]
Since $P_k$ is determined by the load at bus $k$, the angle $\theta_k$ is uniquely determined:
\[\theta_k = \theta_j - P_k X_{jk}\]
The radial bus angle is completely determined by its parent bus, so it can be eliminated by transferring the load.
Implications of Radial Reduction
Preserved:
- Power flow at non-radial buses
- Voltage angles at non-radial buses
- System topology at the core network
Changed:
- Number of buses and branches (reduced)
- Detailed behavior at radial locations
- Local voltage profiles
When to Use:
- Radial connections are not of interest for analysis
- Focusing on transmission backbone
- Computational efficiency is important
When to Avoid:
- Studying distribution feeders (often radial)
- Detailed voltage analysis needed
- Radial buses have critical measurements
Degree Two Reduction (Kron Reduction)
What is a Degree Two Bus?
A degree two bus connects exactly two other buses, acting as a "pass-through" point.
Bus A --- Bus B (degree 2) --- Bus CBus B has degree 2 - it connects A and C but has no other connections.
Mathematical Basis: Kron Reduction
Kron reduction is a systematic method to eliminate buses from the admittance matrix. Degree two reduction is a simple case of Kron reduction for eliminating degree two nodes.
Given admittance matrix:
\[\begin{bmatrix} I_r \\ I_e \end{bmatrix} = \begin{bmatrix} Y_{rr} & Y_{re} \\ Y_{er} & Y_{ee} \end{bmatrix} \begin{bmatrix} V_r \\ V_e \end{bmatrix}\]
where subscript $r$ denotes retained buses and $e$ denotes eliminated buses.
If $I_e = 0$ (no injection at eliminated buses):
\[Y_{ee} V_e = -Y_{er} V_r\]
\[V_e = -Y_{ee}^{-1} Y_{er} V_r\]
Substituting back:
\[I_r = (Y_{rr} - Y_{re} Y_{ee}^{-1} Y_{er}) V_r = Y_{reduced} V_r\]
The reduced admittance matrix is:
\[Y_{reduced} = Y_{rr} - Y_{re} Y_{ee}^{-1} Y_{er}\]
For a Single Degree Two Bus
Consider bus $k$ connecting buses $i$ and $j$:
Bus i ----(y_ik)---- Bus k ----(y_kj)---- Bus jThe equivalent admittance directly between $i$ and $j$ after eliminating $k$ is:
\[y_{ij}^{new} = y_{ij}^{old} + \frac{y_{ik} \cdot y_{kj}}{y_{ik} + y_{kj} + y_{kk}}\]
If there's no shunt at bus $k$ ($y_{kk} = 0$):
\[y_{ij}^{new} = y_{ij}^{old} + \frac{y_{ik} \cdot y_{kj}}{y_{ik} + y_{kj}}\]
This is analogous to combining series impedances in circuit theory.
Physical Interpretation
Eliminating a degree two bus:
- Combines the series impedances of the two connecting branches
- Creates an equivalent direct connection
- Preserves the overall impedance between endpoints
Why Degree Two Reduction Works
The key insight is that a degree two bus with no injection ($P_k = 0$) serves only to pass power through. Its voltage angle is determined by:
\[P_k = \frac{\theta_i - \theta_k}{X_{ik}} + \frac{\theta_j - \theta_k}{X_{kj}} = 0\]
Solving for $\theta_k$:
\[\theta_k = \frac{X_{kj} \theta_i + X_{ik} \theta_j}{X_{ik} + X_{kj}}\]
The angle is a weighted average of its neighbors, so eliminating it and creating a direct equivalent connection preserves the flow relationship.
Implications of Degree Two Reduction
Preserved:
- Power flows at retained buses
- Voltage angles at retained buses
- Overall impedance relationships
- Equivalence for power flow studies
Changed:
- Number of buses (reduced)
- Detailed behavior at eliminated bus
- Branch topology (new equivalent branches created)
When to Use:
- Pass-through buses are not of analytical interest
- Focusing on injection/load buses
- Simplifying large transmission systems
- Reducing computational burden
When to Avoid:
- The degree two bus has measurements or controls
- Detailed branch flows needed at that location
- Bus has significant shunt elements
- Studying protection or relay settings
Combining Reductions
Multiple reductions can be applied sequentially. In these cases, the order of the reduction matters as after each reduction, new candidate buses may appear. In general, it is recommended that radial reduction is applied before degree two reduction as applying radial reduction first may expose new degree two buses.
Practical Considerations
Load and Generation at Reduced Buses
Eliminating buses without any connected injection components can be done cleanly. For eliminated buses that have load or generation attached, those devices must be mapped to retained buses. For buses with shunt admittances, this mapping process can affect the equivalent admittance matrix. RadialReduction and DegreeTwoReduction have options for specifying buses that should not be eliminated (e.g. due to the presence of connected injectors).
Validation
Always verify reductions:
- Compare power flows before and after
- Check that key quantities are preserved
- Validate on a small test system first
Limitations
What Reduction Cannot Do:
- Preserve detailed voltage profiles everywhere
- Capture local dynamics at eliminated buses
- Represent phenomena requiring those buses (e.g., local stability)
- Maintain exact AC power flow at eliminated locations
When Full Network is Required:
- Detailed state estimation
- Protection coordination
- Local voltage studies
- Distributed generation integration at eliminated buses
Further Reading
For practical application, see: