Question

For the tariff of 125/kVA of maximum demand and 3.00 per unit consumed; load factor = 50%, find overall cost/unit at i) unity power factor ii) 0.8 p.f consider maximum demand = 10 kVA.

Answer 1

To calculate the overall cost per unit (kWh) for the given tariff structure, we need to use the following steps:

### Given Data:
- **Tariff for maximum demand**: ₹125 per kVA of maximum demand
- **Tariff per unit consumed**: ₹3.00 per unit (kWh)
- **Load factor**: 50% (0.5)
- **Maximum demand**: 10 kVA
- **Power factors**: Unity (1.0) and 0.8

### Steps to solve:
1. **Determine the actual energy consumption**:
   Energy consumption (in kWh) is related to the maximum demand, load factor, and the hours in a billing period.

   The formula for energy consumption is:
   \[
   \text{Energy consumption (kWh)} = \text{Maximum demand (kW)} \times \text{Load factor} \times \text{Hours in the period}
   \]
   Typically, we assume 30 days in a month and 24 hours in a day, so:
   \[
   \text{Hours in a month} = 30 \times 24 = 720 \, \text{hours}
   \]
   
   The maximum demand in kW depends on the power factor:
   \[
   \text{Maximum demand (kW)} = \text{Maximum demand (kVA)} \times \text{Power factor (p.f.)}
   \]

2. **Calculate for Unity Power Factor (p.f. = 1.0)**:
   \[
   \text{Maximum demand (kW)} = 10 \, \text{kVA} \times 1.0 = 10 \, \text{kW}
   \]
   
   Now, calculate the energy consumption:
   \[
   \text{Energy consumption (kWh)} = 10 \, \text{kW} \times 0.5 \times 720 = 3600 \, \text{kWh}
   \]

   - **Cost for maximum demand**:
     \[
     \text{Cost for max demand} = 125 \, \text{₹/kVA} \times 10 \, \text{kVA} = ₹1250
     \]
   
   - **Cost for energy consumption**:
     \[
     \text{Cost for energy} = 3600 \, \text{kWh} \times ₹3.00 = ₹10800
     \]
   
   - **Total cost**:
     \[
     \text{Total cost} = ₹1250 + ₹10800 = ₹12050
     \]

   - **Overall cost per unit**:
     \[
     \text{Overall cost/unit} = \frac{₹12050}{3600 \, \text{kWh}} = ₹3.35 \, \text{per unit}
     \]

3. **Calculate for Power Factor of 0.8**:
   \[
   \text{Maximum demand (kW)} = 10 \, \text{kVA} \times 0.8 = 8 \, \text{kW}
   \]
   
   Now, calculate the energy consumption:
   \[
   \text{Energy consumption (kWh)} = 8 \, \text{kW} \times 0.5 \times 720 = 2880 \, \text{kWh}
   \]

   - **Cost for maximum demand**:
     \[
     \text{Cost for max demand} = 125 \, \text{₹/kVA} \times 10 \, \text{kVA} = ₹1250
     \]
   
   - **Cost for energy consumption**:
     \[
     \text{Cost for energy} = 2880 \, \text{kWh} \times ₹3.00 = ₹8640
     \]
   
   - **Total cost**:
     \[
     \text{Total cost} = ₹1250 + ₹8640 = ₹9890
     \]

   - **Overall cost per unit**:
     \[
     \text{Overall cost/unit} = \frac{₹9890}{2880 \, \text{kWh}} = ₹3.43 \, \text{per unit}
     \]

### Summary:
1. **At unity power factor (1.0 p.f.)**: Overall cost per unit = **₹3.35**
2. **At power factor 0.8**: Overall cost per unit = **₹3.43**

Answer 2

To find the overall cost per unit of electricity, we need to consider both the fixed costs associated with the maximum demand and the variable costs based on the energy consumed. Here's how to calculate it:

### Given Data:
- **Tariff:**
  - Maximum Demand Charge = ₹125 per KVA
  - Energy Charge = ₹3.00 per unit
- **Load Factor (LF)** = 50% (or 0.5)
- **Maximum Demand** = 10 KVA
- **Power Factor (P.F)**:
  - Case I: Unity Power Factor (P.F = 1)
  - Case II: 0.8 Power Factor (P.F = 0.8)

### Step 1: Calculate the Energy Consumption

The energy consumption can be calculated using the following formula:

\[
\text{Energy Consumption (kWh)} = \text{Maximum Demand (kW)} \times \text{Load Factor} \times \text{Hours in a Month}
\]

- Hours in a month = \( 24 \times 30 = 720 \) hours.

#### Case I: Unity Power Factor (P.F = 1)

Here, the maximum demand in kW is the same as the KVA:

\[
\text{Maximum Demand (kW)} = 10 \text{ KVA} \times 1 = 10 \text{ kW}
\]

So,

\[
\text{Energy Consumption} = 10 \times 0.5 \times 720 = 3600 \text{ kWh}
\]

#### Case II: 0.8 Power Factor (P.F = 0.8)

Here, the maximum demand in kW will be:

\[
\text{Maximum Demand (kW)} = 10 \text{ KVA} \times 0.8 = 8 \text{ kW}
\]

So,

\[
\text{Energy Consumption} = 8 \times 0.5 \times 720 = 2880 \text{ kWh}
\]

### Step 2: Calculate the Total Cost

The total cost consists of:
- **Fixed Cost** (based on maximum demand)
- **Variable Cost** (based on energy consumption)

#### Fixed Cost

\[
\text{Fixed Cost} = \text{Maximum Demand (KVA)} \times \text{Tariff per KVA}
\]

\[
\text{Fixed Cost} = 10 \times 125 = ₹1250
\]

#### Variable Cost

\[
\text{Variable Cost} = \text{Energy Consumption (kWh)} \times \text{Energy Charge per Unit}
\]

For each case:

**Case I (P.F = 1):**

\[
\text{Variable Cost} = 3600 \times 3 = ₹10800
\]

**Case II (P.F = 0.8):**

\[
\text{Variable Cost} = 2880 \times 3 = ₹8640
\]

### Step 3: Calculate Overall Cost per Unit

The overall cost per unit is given by:

\[
\text{Overall Cost per Unit} = \frac{\text{Total Cost}}{\text{Total Energy Consumption (kWh)}}
\]

Where,

\[
\text{Total Cost} = \text{Fixed Cost} + \text{Variable Cost}
\]

**Case I (P.F = 1):**

\[
\text{Total Cost} = ₹1250 + ₹10800 = ₹12050
\]

\[
\text{Overall Cost per Unit} = \frac{₹12050}{3600 \text{ kWh}} = ₹3.35 \text{ per unit}
\]

**Case II (P.F = 0.8):**

\[
\text{Total Cost} = ₹1250 + ₹8640 = ₹9890
\]

\[
\text{Overall Cost per Unit} = \frac{₹9890}{2880 \text{ kWh}} = ₹3.43 \text{ per unit}
\]

### Final Answers:
- **At Unity Power Factor (P.F = 1):** Overall Cost per Unit = ₹3.35 per unit
- **At 0.8 Power Factor (P.F = 0.8):** Overall Cost per Unit = ₹3.43 per unit