🎓 Lesson 5
D3
Per Unit Conversion Workflow: Base Selection & Scaling Rules
Per unit conversion is a way to simplify electrical network calculations by expressing all values—like voltage, current, and impedance—as fractions of carefully chosen reference values.
🎯 Learning Objectives
- ✓ Calculate per unit impedances for transformers and transmission lines given nameplate data and system bases
- ✓ Design consistent base selections across multi-voltage-level networks to ensure accurate p.u. model integration
- ✓ Analyze load flow results by interpreting per unit voltage magnitudes, power injections, and line losses
- ✓ Apply scaling rules to convert between actual and per unit quantities without dimensional error
- ✓ Explain the impact of base selection on convergence behavior and result interpretation in commercial load flow software
📖 Why This Matters
In real mining operations, integrated power systems supply substations, draglines, crushers, and ventilation fans—often spanning 4.16 kV to 138 kV. Without per unit normalization, comparing transformer reactance at 13.8 kV with cable impedance at 34.5 kV leads to calculation errors, misaligned protection settings, and unstable load flow solutions. Mastering per unit conversion ensures your network model reflects physics—not arithmetic artifacts—and directly impacts safety, reliability, and energy efficiency in remote mine grids.
📘 Core Principles
Per unit conversion rests on two foundational ideas: (1) normalization—expressing physical quantities relative to intelligently chosen bases; and (2) consistency—ensuring all bases in a connected network satisfy S_base = V_base × I_base × √3 (for three-phase systems). Bases can be global (same S_base and V_base throughout) or local (different V_base per voltage level, same S_base)—the latter is standard practice. Impedance bases scale quadratically with voltage, making it critical to track voltage-level transitions, especially across tap-changing transformers. Misalignment in base selection is the leading cause of 'convergence failed' errors in ETAP and PSS®E during mine grid commissioning.
📐 Per Unit Conversion Formula
The per unit value of any quantity is its actual value divided by its corresponding base value. For impedance, the base is derived from system-wide S_base and local V_base, ensuring physical consistency across equipment models.
💡 Worked Example
Problem: A 138 kV / 13.8 kV, 50 MVA transformer has Z_actual = 12.1 Ω on its high-voltage side. Calculate its per unit impedance using S_base = 100 MVA and V_base_HV = 138 kV.
1.
Step 1: Compute Z_base_HV = (V_base_HV)² / S_base = (138 kV)² / 100 MVA = (138,000)² / 100,000,000 = 190.44 Ω
2.
Step 2: Apply p.u. definition: Z_pu = Z_actual / Z_base = 12.1 Ω / 190.44 Ω = 0.0635 p.u.
3.
Step 3: Verify: Typical transformer p.u. impedances range from 0.05–0.15 p.u.; 0.0635 falls within this range and matches manufacturer data sheet tolerance.
Answer:
The result is 0.0635 p.u., which falls within the safe range of 0.05–0.15 p.u.
🏗️ Real-World Application
At Newmont’s Ahafo Mine (Ghana), engineers modeled a 33 kV ring-main feeding six underground substations. Using inconsistent bases—10 MVA for surface gear but 5 MVA for underground cables—initial load flow yielded unphysical voltage drops (>15% at last node). Re-basing all components to a unified 100 MVA S_base and tiered V_base (33 kV, 11 kV, 0.4 kV) resolved convergence issues and aligned simulated voltage profiles (0.97–1.03 p.u.) with field measurements from SEL-751 relays. This correction prevented costly capacitor bank over-sizing and enabled accurate arc-flash hazard labeling per IEEE 1584.
✏️ Practice Problem
A 6.6 kV, 5 MVA mine winder motor has subtransient reactance X'' = 0.25 p.u. on its own nameplate base. Convert this to per unit on a system-wide base of S_base = 100 MVA and V_base = 6.6 kV. Then calculate the actual ohmic value of X''. Show all steps and verify units.
🔧 Interactive Calculator
🔧 Open Load Flow (Power Flow) Analysis Calculator📋 Case Connection
📋 Distribution Network Reinforcement
Excessive voltage drop (>8%) on 400 V feeders during evening EV charging peaks; neutral conductor overheating