Power Systems
Generation, transmission, distribution, and protection
1. Three-Phase Systems
Why Three-Phase?
More efficient power transmission, constant power output, smaller conductors for same power, self-starting motors.
Wye (Y) Connection
Voltage Relationships
VL = √3 × VP
Line voltage = √3 × Phase voltage
Phase voltages 120° apart
Current Relationships
IL = IP
Line current = Phase current
Neutral carries unbalanced current
Delta (Δ) Connection
Voltage Relationships
VL = VP
Line voltage = Phase voltage
Current Relationships
IL = √3 × IP
Line current = √3 × Phase current
Three-Phase Power
P = √3 × VL × IL × cos θ
Q = √3 × VL × IL × sin θ
S = √3 × VL × IL
Same formula for both Y and Δ connections!
Power Measurement
Two-Wattmeter Method
Ptotal = W₁ + W₂
tan θ = √3 × (W₁ - W₂)/(W₁ + W₂)
Works for balanced or unbalanced loads
At pf = 0.5: One wattmeter reads zero
2. Transmission Lines
Line Parameters
| Parameter | Formula | Depends On |
|---|---|---|
| Resistance (R) | R = ρL/A | Material, length, area, temperature |
| Inductance (L) | L = 2×10⁻⁷ ln(GMD/GMR) | Spacing, conductor geometry |
| Capacitance (C) | C = 2πε/ln(GMD/r) | Spacing, conductor radius |
Line Classification
Short Line (<80 km)
• Capacitance neglected
• Only R and L considered
• Simple series circuit
Medium Line (80-250 km)
• Capacitance included
• Nominal π or T model
• Lumped parameters
Long Line (>250 km)
• Distributed parameters
• Hyperbolic functions
• Exact solution required
ABCD Parameters
VS = AVR + BIR
IS = CVR + DIR
For short line: A=1, B=Z, C=0, D=1
For symmetrical lines: AD - BC = 1
Line Performance
Voltage Regulation
VR = (VS - VR)/VR × 100%
At no-load: VR,NL = VS/A
Efficiency
η = PR/PS × 100%
η = PR/(PR + Losses)
Surge Impedance Loading (SIL)
Zc = √(L/C) (Surge/Characteristic Impedance)
SIL = V²L/Zc
At SIL: No reactive power exchange, V same at both ends
3. Per-Unit System
Purpose
Simplifies calculations by expressing quantities as fractions of base values. Eliminates transformer turns ratios.
Base Values
Choose any two base values (usually Sbase and Vbase), others are derived:
Sbase = chosen (MVA)
Vbase = chosen (kV)
Ibase = Sbase/(√3 × Vbase)
Zbase = V²base/Sbase
Per-Unit Calculations
Quantitypu = Actual value / Base value
Vpu = Vactual/Vbase
Ipu = Iactual/Ibase
Zpu = Zactual/Zbase
Converting Between Bases
Zpu,new = Zpu,old × (Sbase,new/Sbase,old) × (Vbase,old/Vbase,new)²
Used when equipment has different ratings
Advantages
- • Impedances remain same when referred across transformers
- • Equipment parameters given in per-unit by manufacturers
- • Per-unit values fall in narrow range (makes errors obvious)
- • Simplifies complex system calculations
4. Fault Analysis
Types of Faults
| Fault Type | Symbol | Occurrence | Severity |
|---|---|---|---|
| Three-phase (symmetrical) | LLL or LLLG | ~5% | Most severe current |
| Line-to-Line | LL | ~15% | Moderate |
| Double Line-to-Ground | LLG | ~10% | High |
| Single Line-to-Ground | LG or SLG | ~70% | Most common |
Symmetrical Fault (3-Phase)
If = Vpu/Z1,pu
If,actual = If,pu × Ibase
Only positive sequence impedance needed
Sequence Components
Any unbalanced system can be resolved into three balanced systems:
Positive Sequence (1)
Same rotation as system
ABC phase sequence
Negative Sequence (2)
Opposite rotation
ACB phase sequence
Zero Sequence (0)
In-phase (no rotation)
Requires ground path
Fault Current Formulas
Single Line-to-Ground (L-G):
If = 3V/(Z₁ + Z₂ + Z₀)
Line-to-Line (L-L):
If = √3V/(Z₁ + Z₂)
Double Line-to-Ground (L-L-G):
Ia1 = V/(Z₁ + Z₂||Z₀)
5. Power System Protection
Protection Requirements
Reliability, Selectivity, Speed, Simplicity, Economy
Protective Devices
Fuses
- • One-time operation
- • Inverse time characteristic
- • Used for LV protection
Circuit Breakers
- • Reclosable
- • Types: ACB, VCB, OCB, SF6
- • HV/MV applications
Relay Types
Overcurrent Relay
- • Instantaneous: Trips immediately above pickup
- • Inverse Time: Delay decreases as current increases
- • Definite Time: Fixed delay above pickup
Differential Relay
- • Compares currents entering and leaving
- • Trips on current difference (internal fault)
- • Used for: Transformers, generators, buses
Distance Relay
- • Measures impedance to fault
- • Zone protection with time delays
- • Used for: Transmission line protection
CT and PT
Current Transformer (CT)
• Steps down current for metering/relaying
• Secondary: 5A or 1A standard
• Never open-circuit secondary!
Potential Transformer (PT)
• Steps down voltage for metering
• Secondary: 110V or 120V standard
• May be open-circuited
6. Power Factor Correction
Why Correct Power Factor?
Reduces losses, improves voltage regulation, increases system capacity, avoids utility penalties.
Capacitor Sizing
Qc = P(tan θ₁ - tan θ₂)
C = Qc/(2πfV²) per phase
θ₁ = initial angle, θ₂ = desired angle
Capacitor Bank Connections
Star (Y) Connection
Q3φ = 3 × V²PωC
Q3φ = V²LωC
Lower voltage rating per capacitor
Delta (Δ) Connection
Q3φ = 3 × V²LωC
3× kVAR of Y for same capacitance
Higher voltage rating required
Location of Capacitors
- At load: Maximum benefit, reduces losses throughout
- At substation: Centralized, easier maintenance
- On feeders: Compromise between above
7. System Grounding
Types of Grounding
Solidly Grounded
- • Direct connection to ground
- • High ground fault current
- • Easy fault detection
- • Used: LV systems, most HV systems
Resistance Grounded
- • Resistor limits ground fault current
- • Low resistance: 25-400A fault current
- • High resistance: <10A fault current
- • Used: Industrial MV systems
Ungrounded (Isolated)
- • No intentional ground connection
- • System can continue with one ground fault
- • Difficult to detect ground faults
- • Transient overvoltages possible
Ground Electrode
Types: Driven rod, buried conductor, concrete-encased (Ufer)
Ground resistance target: <25 ohms (single electrode)
For substations: <1 ohm (multiple electrodes)
8. Power Quality
Power Quality Issues
| Issue | Description | Duration |
|---|---|---|
| Sag (Dip) | Voltage drop to 10-90% | 0.5 cycle - 1 min |
| Swell | Voltage rise to 110-180% | 0.5 cycle - 1 min |
| Interruption | Voltage <10% | Momentary to sustained |
| Harmonics | Non-fundamental frequencies | Steady state |
| Flicker | Voltage fluctuation | Intermittent |
Harmonics
THD = √(Σh²)/fundamental × 100%
Total Harmonic Distortion
Sources: Variable frequency drives, rectifiers, switching supplies
Effects: Overheating, increased losses, neutral current
Solutions: Filters, K-rated transformers, phase shifting
Power Quality Solutions
UPS (Uninterruptible Power Supply)
• Online: Continuous conditioning
• Offline: Switches on outage
• Line-interactive: Hybrid
Active Filters
• Injects compensating current
• Cancels harmonics
• Real-time compensation
Key Takeaways for EE Board Exam
Must-Know Formulas
- ✓ P = √3VLILcos θ (3-phase power)
- ✓ VL = √3VP (Wye), IL = √3IP (Delta)
- ✓ Zbase = V²base/Sbase
- ✓ If(3φ) = V/Z₁
- ✓ Qc = P(tan θ₁ - tan θ₂)
- ✓ SIL = V²/Zc
Critical Concepts
- ✓ Two-wattmeter method
- ✓ Per-unit system advantages
- ✓ Sequence components (0, 1, 2)
- ✓ SLG most common fault (~70%)
- ✓ CT never open-circuited
- ✓ Distance relay for transmission
- ✓ Differential relay for equipment