Reactive Energy Conversion Formulas
Complete formulas for converting between different reactive energy units including VARh, kVARh, MVARh, and related AC electrical energy calculations.
๐ Basic Reactive Energy Unit Conversions
Standard Unit Conversions
VARh โ kVARh
Example: 45,000 VARh
45,000 VARh = 45 kVARh
kVARh โ MVARh
Example: 8,500 kVARh
8,500 kVARh = 8.5 MVARh
VARh โ MVARh
Example: 12,000,000 VARh
12,000,000 VARh = 12 MVARh
Reactive Energy Calculation
From Reactive Power and Time
25 kVAR for 8 hours
In VARh: 200 ร 1000 = 200,000 VARh
Reactive energy: 200 kVARh
Average Reactive Power
1,500 kVARh over 30 days
Time period: 30 ร 24 = 720 hours
Average reactive power: 2.08 kVAR
๐ Energy Triangle and Relationships
Energy Components
From Apparent and Real Energy
E_S = 1,000 kVAh, E_P = 800 kWh
Reactive energy: 600 kVARh
From Power Factor
E_P = 800 kWh, PF = 0.8
ฯ = arccos(0.8) = 36.87ยฐ
Reactive energy: 600 kVARh
Monthly Energy Calculations
Load Profile Integration
Variable reactive load over time
Day shift (8h): 50 kVAR average
Evening (4h): 30 kVAR average
Night (12h): 10 kVAR average
Daily: (50ร8) + (30ร4) + (10ร12) = 640 kVARh
Monthly: 640 ร 30 = 19,200 kVARh
Peak vs Off-Peak Energy
Time-of-use reactive energy
Peak hours (6h): 75 kVAR ร 6h ร 22 days = 9,900 kVARh
Off-peak (18h): 35 kVAR ร 18h ร 22 days = 13,860 kVARh
Weekends (48h): 25 kVAR ร 48h ร 8 days = 9,600 kVARh
Total monthly: 33,360 kVARh
๐ฐ Utility Billing and Reactive Energy
| Customer Type | kVARh/month | kWh/month | Power Factor | VAR/kW Ratio |
|---|---|---|---|---|
| Residential | 150 | 1,000 | 0.98 | 0.15 |
| Small Commercial | 2,400 | 8,000 | 0.95 | 0.30 |
| Office Building | 18,000 | 50,000 | 0.94 | 0.36 |
| Manufacturing | 120,000 | 200,000 | 0.85 | 0.60 |
| Steel Mill | 2,500,000 | 4,000,000 | 0.78 | 0.63 |
| Data Center | 45,000 | 150,000 | 0.95 | 0.30 |
๐ Reactive Energy Metering and Measurement
Electronic Energy Meters
Four-Quadrant Metering
Directional reactive energy measurement
Q1: +kWh, +kVARh (motoring, inductive)
Q2: -kWh, +kVARh (generating, inductive)
Q3: -kWh, -kVARh (generating, capacitive)
Q4: +kWh, -kVARh (motoring, capacitive)
Import/export tracking for each quadrant
Instantaneous Reactive Power
Digital signal processing approach
Voltage: v(t) = 170 sin(ฯt)
Current: i(t) = 10 sin(ฯt - 36.87ยฐ)
Reactive power: q = 850 sinยฒ(ฯt) VAR
Average: Q = 425 VAR
Integration over billing period
Billing Calculations
Reactive Energy Charges
Industrial customer billing
Monthly kVARh: 85,000
Free allowance: 40% of kWh = 0.4 ร 150,000 = 60,000
Billable kVARh: 85,000 - 60,000 = 25,000
Rate: $0.008/kVARh
Monthly charge: $200
Power Factor Penalty Alternative
Some utilities use PF penalty instead
Peak demand: 500 kW
Power factor: 0.8
Penalty factor: (0.9/0.8 - 1) = 0.125
Penalty demand: 500 ร 0.125 = 62.5 kW
Extra charge: 62.5 ร $15 = $938/month
โฐ Time-of-Use Reactive Energy Calculations
Peak Period Analysis
Summer Peak Reactive Load
Peak hours: 1 PM - 6 PM weekdays
HVAC load dominates: 0.82 power factor
Average reactive power: 150 kVAR
Peak season (Jun-Sep): 4 months
Peak days: 22 days/month ร 4 = 88 days
Peak energy: 150 ร 5 ร 88 = 66,000 kVARh
Summer peak reactive energy: 66,000 kVARh
Off-Peak Optimization
Off-peak hours: 6 PM - 1 PM next day
Reduced HVAC load: 0.88 power factor
Process loads dominant: motors at 0.85 PF
Average reactive power: 85 kVAR
Off-peak energy: 85 ร 20 ร 88 = 149,600 kVARh
Opportunity for capacitor switching
Capacitor Control Strategy
Automatic Switching Schedule
Base capacitors: 50 kVAR (always on)
Step 1: 25 kVAR (8 AM - 10 PM)
Step 2: 25 kVAR (10 AM - 8 PM)
Step 3: 50 kVAR (12 PM - 6 PM peak)
Total available: 150 kVAR compensation
Reduces peak reactive demand by 80%
Energy Savings Calculation
Without capacitors: 66,000 kVARh peak
With 100 kVAR correction during peak:
Residual reactive: 50 kVAR
Peak energy with caps: 50 ร 5 ร 88 = 22,000 kVARh
Reduction: 66,000 - 22,000 = 44,000 kVARh
Savings: 67% reduction in peak reactive energy