Question

A plant consists of two 10 MVA generators of reactance 18% each and two 5 MVA generators of 12% each. All are connected to bus bar to supply a load through three step up transformers of 8 MVA each having reactance of 8%. Determine fault MVA on HV side of any one transformer.

Answer 1

To determine the fault MVA on the high voltage (HV) side of one transformer, we need to find the total equivalent reactance of the system and then use it to calculate the fault MVA. Here's a step-by-step method to solve this problem:

### 1. Convert All Reactances to a Common Base

First, we need to convert the reactances of all generators and transformers to a common base. Typically, this base is chosen to be the rating of one of the transformers or generators, but for simplicity, we will use the base of the transformers (8 MVA) here.

#### Generators

**Base Power for Conversion**: 8 MVA (Transformers' rating)

**Generator 1 Reactances**:
- **10 MVA generators** with **18% reactance** (0.18 per unit on 10 MVA)

  \[
  X_{gen1} = 0.18 \times \frac{10 \text{ MVA}}{8 \text{ MVA}} = 0.225 \text{ per unit on 8 MVA base}
  \]

**Generator 2 Reactances**:
- **5 MVA generators** with **12% reactance** (0.12 per unit on 5 MVA)

  \[
  X_{gen2} = 0.12 \times \frac{5 \text{ MVA}}{8 \text{ MVA}} = 0.075 \text{ per unit on 8 MVA base}
  \]

#### Transformers

**Transformers** are already on the 8 MVA base:
- **Reactance**: 8%

  \[
  X_{tr} = 0.08 \text{ per unit on 8 MVA base}
  \]

### 2. Find the Total Equivalent Reactance

**Reactance of Generators**:

We have:
- Two 10 MVA generators: \( X_{gen1} = 0.225 \text{ per unit} \)
- Two 5 MVA generators: \( X_{gen2} = 0.075 \text{ per unit} \)

**Calculate Total Equivalent Reactance of Generators**:

For generators with equal rating, we can calculate their combined reactance as:

\[
\frac{1}{X_{gen_{total}}} = \frac{1}{X_{gen1}} + \frac{1}{X_{gen2}}
\]

where:
- Two 10 MVA generators: \( X_{gen1} = 0.225 \text{ per unit} \)
- Two 5 MVA generators: \( X_{gen2} = 0.075 \text{ per unit} \)

Combined reactance of the generators:
- Two 10 MVA generators in parallel: \( X_{gen_{10MVA}} = \frac{0.225}{2} = 0.1125 \text{ per unit} \)
- Two 5 MVA generators in parallel: \( X_{gen_{5MVA}} = \frac{0.075}{2} = 0.0375 \text{ per unit} \)

Total combined reactance of all generators:
\[
\frac{1}{X_{gen_{total}}} = \frac{1}{0.1125} + \frac{1}{0.0375} = 8.8889 + 26.6667 = 35.5556
\]
\[
X_{gen_{total}} = \frac{1}{35.5556} = 0.0281 \text{ per unit}
\]

**Total System Reactance**:

The total reactance seen at the bus bar, including the transformers:
- Reactance of each transformer: 0.08 per unit
- Three transformers in parallel:

  \[
  X_{tr_{total}} = \frac{0.08}{3} = 0.0267 \text{ per unit}
  \]

  Combined reactance of all transformers:
  \[
  X_{total} = X_{gen_{total}} + X_{tr_{total}} = 0.0281 + 0.0267 = 0.0548 \text{ per unit}
  \]

### 3. Determine the Fault MVA

To determine the fault MVA, use the formula:
\[
\text{Fault MVA} = \frac{\text{Base MVA}}{X_{total}}
\]

Here, the base MVA is 8 MVA (transformer base):
\[
\text{Fault MVA} = \frac{8}{0.0548} = 145.8 \text{ MVA}
\]

So, the fault MVA on the HV side of one transformer is approximately **145.8 MVA**.

Answer 2

To determine the fault MVA on the HV side of one transformer, follow these steps:

1. **Convert Generator and Transformer Reactances to a Common Base:**
   - Base MVA: 8 MVA (since the transformers are 8 MVA)

2. **Convert the Generators' Reactances to the Transformer Base:**

   - For the 10 MVA generators (Reactance = 18%):
     \[
     X_{10 \text{ MVA}} = \frac{18}{100} \times \frac{8}{10} = 0.144 \text{ pu}
     \]

   - For the 5 MVA generators (Reactance = 12%):
     \[
     X_{5 \text{ MVA}} = \frac{12}{100} \times \frac{8}{5} = 0.192 \text{ pu}
     \]

3. **Calculate the Total Reactance:**
   - Convert to equivalent reactances for the base MVA and sum them.
   - Assume parallel operation of generators:
     \[
     X_{\text{eq}} = \left(\frac{1}{X_1} + \frac{1}{X_2} + \frac{1}{X_3} + \frac{1}{X_4}\right)^{-1}
     \]
     Where:
     \[
     X_1 = 0.144 \text{ pu (10 MVA generator)}
     \]
     \[
     X_2 = 0.144 \text{ pu (10 MVA generator)}
     \]
     \[
     X_3 = 0.192 \text{ pu (5 MVA generator)}
     \]
     \[
     X_4 = 0.192 \text{ pu (5 MVA generator)}
     \]

     \[
     X_{\text{eq}} = \left(\frac{1}{0.144} + \frac{1}{0.144} + \frac{1}{0.192} + \frac{1}{0.192}\right)^{-1}
     \]
     \[
     X_{\text{eq}} = \left(6.94 + 6.94 + 5.21 + 5.21\right)^{-1} = 0.087 \text{ pu}
     \]

4. **Add Transformer Reactance:**
   - Each transformer has 8% reactance:
     \[
     X_{\text{total}} = X_{\text{eq}} + X_{\text{transformer}} = 0.087 + 0.08 = 0.167 \text{ pu}
     \]

5. **Calculate the Fault MVA:**
   \[
   \text{Fault MVA} = \frac{\text{Base MVA}}{X_{\text{total}}}
   \]
   \[
   \text{Fault MVA} = \frac{8}{0.167} \approx 47.9 \text{ MVA}
   \]

Thus, the fault MVA on the HV side of one transformer is approximately 47.9 MVA.