Question

An industrial consumer charged with the scheduled tariff of Rs.250 /kVA per month for maximum demand and 150 paisa per unit consumed for load factor of 60% and 80%. Find overall cost per unit at i) unity P.F. ii) 0.9 P.F. consider maximum demand of 50 kVA.

Answer 1

To determine the overall cost per unit for an industrial consumer based on the given data, we need to calculate the cost considering both the maximum demand and the energy consumption. Here's how we can approach the problem step by step.

### Given Data:
1. **Scheduled Tariff for Maximum Demand**: Rs. 250 per KVA per month
2. **Energy Charge**: 150 Paisa per unit (which is Rs. 1.50 per unit)
3. **Maximum Demand**: 50 KVA
4. **Load Factors**: 60% and 80%
5. **Power Factor (P.F.)**: 1.0 (Unity) and 0.9

### Steps to Calculate the Overall Cost Per Unit:

1. **Calculate the Maximum Demand Charge:**
   The maximum demand charge is fixed per KVA per month.

   \[
   \text{Maximum Demand Charge} = 50 \text{ KVA} \times 250 \text{ Rs/KVA} = 12500 \text{ Rs/month}
   \]

2. **Calculate the Energy Consumption:**
   The load factor represents the ratio of the average load to the maximum demand. Thus,

   \[
   \text{Average Load} = \text{Load Factor} \times \text{Maximum Demand}
   \]

   For a Load Factor of 60%:
   \[
   \text{Average Load} = 0.60 \times 50 \text{ KVA} = 30 \text{ KVA}
   \]

   For a Load Factor of 80%:
   \[
   \text{Average Load} = 0.80 \times 50 \text{ KVA} = 40 \text{ KVA}
   \]

3. **Calculate the Energy Consumption in kWh:**
   To find the energy consumption in kWh, we use the average load. Assuming 1 month = 30 days and 24 hours per day:

   \[
   \text{Energy Consumption (kWh)} = \text{Average Load (kW)} \times 24 \text{ hours/day} \times 30 \text{ days}
   \]

   For a Load Factor of 60%:
   \[
   \text{Energy Consumption} = 30 \text{ kW} \times 24 \times 30 = 21600 \text{ kWh}
   \]

   For a Load Factor of 80%:
   \[
   \text{Energy Consumption} = 40 \text{ kW} \times 24 \times 30 = 28800 \text{ kWh}
   \]

4. **Calculate the Total Energy Charges:**
   \[
   \text{Energy Charge} = \text{Energy Consumption} \times 1.50 \text{ Rs/kWh}
   \]

   For a Load Factor of 60%:
   \[
   \text{Energy Charge} = 21600 \text{ kWh} \times 1.50 = 32400 \text{ Rs}
   \]

   For a Load Factor of 80%:
   \[
   \text{Energy Charge} = 28800 \text{ kWh} \times 1.50 = 43200 \text{ Rs}
   \]

5. **Calculate the Total Monthly Cost:**
   \[
   \text{Total Monthly Cost} = \text{Maximum Demand Charge} + \text{Energy Charge}
   \]

   For a Load Factor of 60%:
   \[
   \text{Total Monthly Cost} = 12500 \text{ Rs} + 32400 \text{ Rs} = 44900 \text{ Rs}
   \]

   For a Load Factor of 80%:
   \[
   \text{Total Monthly Cost} = 12500 \text{ Rs} + 43200 \text{ Rs} = 55700 \text{ Rs}
   \]

6. **Calculate the Overall Cost Per Unit:**
   \[
   \text{Overall Cost Per Unit} = \frac{\text{Total Monthly Cost}}{\text{Energy Consumption}}
   \]

   For a Load Factor of 60%:
   \[
   \text{Overall Cost Per Unit} = \frac{44900}{21600} \approx 2.08 \text{ Rs/kWh}
   \]

   For a Load Factor of 80%:
   \[
   \text{Overall Cost Per Unit} = \frac{55700}{28800} \approx 1.93 \text{ Rs/kWh}
   \]

### Considering Power Factor:

The power factor (P.F.) does not affect the energy charge directly in this calculation because the tariff is based on KVA and kWh. However, it is useful to understand the load demand in terms of real power (kW).

1. **At Unity P.F. (1.0):**
   The calculations above assume a unity power factor, so no additional adjustments are needed.

2. **At 0.9 P.F.:**
   For a power factor of 0.9, the demand in kW is:

   \[
   \text{Real Power} = \text{Apparent Power} \times \text{P.F.} = 50 \text{ KVA} \times 0.9 = 45 \text{ kW}
   \]

   The energy consumption calculation remains the same as it’s based on the load factor, not the power factor. Therefore, the overall cost per unit remains as calculated above.

### Summary:
- **Overall Cost Per Unit at 60% Load Factor:**
  - Unity P.F.: Rs. 2.08/kWh
  - 0.9 P.F.: Rs. 2.08/kWh

- **Overall Cost Per Unit at 80% Load Factor:**
  - Unity P.F.: Rs. 1.93/kWh
  - 0.9 P.F.: Rs. 1.93/kWh

The power factor does not alter the overall cost per unit directly in this case.

Answer 2

To find the overall cost per unit of electricity for the industrial consumer with different power factors, we'll need to calculate the total cost and then divide it by the total energy consumed. Let's break it down step by step.

### Given Data:
- **Scheduled Tariff for Maximum Demand:** Rs. 250/KVA per month
- **Energy Cost Tariff:** 150 Paisa (or Rs. 1.50) per unit (kWh)
- **Load Factor:** 60% and 80%
- **Maximum Demand:** 50 KVA

### i) Unity Power Factor (1.0)

**1. Calculate the Maximum Demand Charge:**
\[ \text{Maximum Demand Charge} = \text{Maximum Demand} \times \text{Scheduled Tariff} \]
\[ \text{Maximum Demand Charge} = 50 \, \text{KVA} \times 250 \, \text{Rs/KVA} = 12500 \, \text{Rs/month} \]

**2. Calculate the Energy Consumed:**

- **Load Factor (LF) = 60%:**
\[ \text{Energy Consumed} = \text{Maximum Demand} \times \text{Number of Hours in a Month} \times \text{Load Factor} \]
Assuming a 30-day month, the number of hours in a month is \( 30 \times 24 = 720 \) hours.
\[ \text{Energy Consumed} = 50 \, \text{KVA} \times 720 \, \text{hours} \times 0.60 = 21600 \, \text{kWh} \]

- **Load Factor (LF) = 80%:**
\[ \text{Energy Consumed} = 50 \, \text{KVA} \times 720 \, \text{hours} \times 0.80 = 28800 \, \text{kWh} \]

**3. Calculate the Total Energy Cost:**
\[ \text{Energy Cost} = \text{Energy Consumed} \times \text{Energy Tariff} \]
\[ \text{Energy Cost} = 21600 \, \text{kWh} \times 1.50 \, \text{Rs/kWh} = 32400 \, \text{Rs} \]
\[ \text{Energy Cost} = 28800 \, \text{kWh} \times 1.50 \, \text{Rs/kWh} = 43200 \, \text{Rs} \]

**4. Calculate the Overall Cost per Unit:**

- **Load Factor 60%:**
\[ \text{Total Cost} = \text{Maximum Demand Charge} + \text{Energy Cost} \]
\[ \text{Total Cost} = 12500 \, \text{Rs} + 32400 \, \text{Rs} = 44900 \, \text{Rs} \]
\[ \text{Overall Cost per Unit} = \frac{44900 \, \text{Rs}}{21600 \, \text{kWh}} \approx 2.08 \, \text{Rs/kWh} \]

- **Load Factor 80%:**
\[ \text{Total Cost} = \text{Maximum Demand Charge} + \text{Energy Cost} \]
\[ \text{Total Cost} = 12500 \, \text{Rs} + 43200 \, \text{Rs} = 55700 \, \text{Rs} \]
\[ \text{Overall Cost per Unit} = \frac{55700 \, \text{Rs}}{28800 \, \text{kWh}} \approx 1.93 \, \text{Rs/kWh} \]

### ii) Power Factor of 0.9

Since the power factor does not directly affect the maximum demand charge but affects the energy consumption indirectly, the overall cost per unit remains the same as for unity power factor. The energy consumed remains the same because load factor and power factor are independent in this calculation.

So, the results are:

- **For Load Factor of 60%:**
  - Unity P.F.: \( \approx 2.08 \, \text{Rs/kWh} \)
  - P.F. of 0.9: \( \approx 2.08 \, \text{Rs/kWh} \)

- **For Load Factor of 80%:**
  - Unity P.F.: \( \approx 1.93 \, \text{Rs/kWh} \)
  - P.F. of 0.9: \( \approx 1.93 \, \text{Rs/kWh} \)