To calculate the all-day efficiency of a transformer, you need to determine both the energy delivered and the energy lost over a 24-hour period. Here’s a step-by-step guide on how to perform the calculation.
### Given Data:
- **Transformer Rating**: 500 kVA
- **Copper Losses (Full Load)**: 5 kW
- **Iron Losses (No Load)**: 3 kW
### Load Profile:
1. **400 kW, Power Factor 0.8, Duration 6 hours**
2. **300 kW, Power Factor 0.75, Duration 12 hours**
3. **100 kW, Power Factor 0.8, Duration 3 hours**
4. **No Load, Duration 3 hours**
### Calculation Steps:
#### 1. **Calculate Total Energy Delivered**
Energy delivered is calculated as:
\[ \text{Energy Delivered (kWh)} = \text{Load (kW)} \times \text{Duration (hours)} \]
- For 400 kW load for 6 hours:
\[
\text{Energy}_{400} = 400 \text{ kW} \times 6 \text{ hours} = 2400 \text{ kWh}
\]
- For 300 kW load for 12 hours:
\[
\text{Energy}_{300} = 300 \text{ kW} \times 12 \text{ hours} = 3600 \text{ kWh}
\]
- For 100 kW load for 3 hours:
\[
\text{Energy}_{100} = 100 \text{ kW} \times 3 \text{ hours} = 300 \text{ kWh}
\]
- For No Load for 3 hours (No load means no power delivered, but losses still occur):
\[
\text{Energy}_{\text{No Load}} = 0 \text{ kWh}
\]
Total energy delivered:
\[
\text{Total Energy Delivered} = 2400 + 3600 + 300 = 6300 \text{ kWh}
\]
#### 2. **Calculate Total Losses**
Losses include both copper losses and iron losses. Copper losses vary with the load, while iron losses are constant.
- **Copper Losses**: These are proportional to the square of the load. To calculate average copper losses, we need to calculate them for each load condition.
**Load Conditions**:
- For 400 kW load:
\[
\text{Copper Loss}_{400} = \left(\frac{400 \text{ kW}}{500 \text{ kVA}}\right)^2 \times 5 \text{ kW} = (0.8)^2 \times 5 \text{ kW} = 0.64 \times 5 \text{ kW} = 3.2 \text{ kW}
\]
- For 300 kW load:
\[
\text{Copper Loss}_{300} = \left(\frac{300 \text{ kW}}{500 \text{ kVA}}\right)^2 \times 5 \text{ kW} = (0.6)^2 \times 5 \text{ kW} = 0.36 \times 5 \text{ kW} = 1.8 \text{ kW}
\]
- For 100 kW load:
\[
\text{Copper Loss}_{100} = \left(\frac{100 \text{ kW}}{500 \text{ kVA}}\right)^2 \times 5 \text{ kW} = (0.2)^2 \times 5 \text{ kW} = 0.04 \times 5 \text{ kW} = 0.2 \text{ kW}
\]
**Average Copper Losses** over 24 hours:
\[
\text{Average Copper Loss} = \frac{(3.2 \text{ kW} \times 6 \text{ hours}) + (1.8 \text{ kW} \times 12 \text{ hours}) + (0.2 \text{ kW} \times 3 \text{ hours})}{24 \text{ hours}}
\]
\[
= \frac{19.2 + 21.6 + 0.6}{24} = \frac{41.4}{24} \approx 1.725 \text{ kW}
\]
- **Iron Losses**: Constant at 3 kW for the entire 24 hours.
Total losses:
\[
\text{Total Losses} = (\text{Average Copper Loss} + \text{Iron Losses}) \times \text{Time}
\]
\[
= (1.725 \text{ kW} + 3 \text{ kW}) \times 24 \text{ hours}
\]
\[
= 4.725 \text{ kW} \times 24 = 113.4 \text{ kWh}
\]
#### 3. **Calculate All-Day Efficiency**
The all-day efficiency is the ratio of the energy delivered to the total energy input. The total energy input is the sum of the energy delivered and the losses.
\[
\text{All-Day Efficiency} (\%) = \left(\frac{\text{Energy Delivered}}{\text{Energy Delivered} + \text{Total Losses}}\right) \times 100
\]
\[
\text{All-Day Efficiency} = \left(\frac{6300}{6300 + 113.4}\right) \times 100
\]
\[
= \left(\frac{6300}{6413.4}\right) \times 100 \approx 98.2\%
\]
### Summary
The all-day efficiency of the transformer is approximately **98.2%**.