A power transformer in a substation must reliably handle load currents with- out overheating. The load current It and the transformer's current capacity Ic are both normally distributed. The load current It has a mean of 1200 A and a standard deviation of 100 A. The current capacity Io of the trans- former has a mean of 1500 A and a standard deviation of 150 A. The limit state function is given by: g(IC, IL) = IC - IL. Calculate the reliability index ẞ for the power transformer. Also, determine the design point or MPP in the original variable space. [Ans: ẞ ≈ 1.6641; (I, I₁) ≈ (1292.3, 1292.3)]

To solve this problem, we need to calculate the **reliability index (ẞ)** for the power transformer. The reliability index is a measure of how likely the system is to meet its performance requirements, i.e., ensuring that the load current \( I_L \) does not exceed the transformer's current capacity \( I_C \).

### Given Information:
- Load current \( I_L \) is normally distributed with:
  - Mean \( \mu_L = 1200 \, \text{A} \)
  - Standard deviation \( \sigma_L = 100 \, \text{A} \)

- Current capacity \( I_C \) is normally distributed with:
  - Mean \( \mu_C = 1500 \, \text{A} \)
  - Standard deviation \( \sigma_C = 150 \, \text{A} \)

### Limit State Function:
The limit state function is defined as:
\[
g(I_C, I_L) = I_C - I_L
\]
The transformer is safe when \( g(I_C, I_L) > 0 \), meaning the current capacity exceeds the load current.

### Step 1: Compute the Reliability Index \( ẞ \)

The reliability index \( ẞ \) is given by the formula:
\[
ẞ = \frac{\mu_C - \mu_L}{\sqrt{\sigma_C^2 + \sigma_L^2}}
\]

#### Calculate the combined standard deviation:

\[
\sigma_g = \sqrt{\sigma_C^2 + \sigma_L^2} = \sqrt{(150)^2 + (100)^2} = \sqrt{22500 + 10000} = \sqrt{32500} \approx 180.28
\]

#### Calculate the reliability index \( ẞ \):

\[
ẞ = \frac{\mu_C - \mu_L}{\sigma_g} = \frac{1500 - 1200}{180.28} \approx \frac{300}{180.28} \approx 1.664
\]

Thus, the reliability index \( ẞ \) is approximately **1.664**.

### Step 2: Determine the Design Point (Most Probable Point, MPP

A power transformer in a substation must reliably handle load currents without overheating. The load current It and the transformer's current capacity Ic are both normally distributed. The load current It has a mean of 1200 A and a standard deviation of 100 A. The current capacity Io of the transformer has a mean of 1500 A and a standard deviation of 150 A. The limit state function is given by: g(IC, IL) = IC - IL. Calculate the reliability index ẞ for the power transformer. Also, determine the design point or MPP in the original variable space. [Ans: ẞ ≈ 1.6641; (I, I₁) ≈ (1292.3, 1292.3)]

This problem involves calculating the reliability index (ẞ ≈ 1.6641) and the Most Probable Point (MPP ≈ 1292.3 A) for a power transformer with a normally distributed load current (mean 1200 A, std dev 100 A) and current capacity (mean 1500 A, std dev 150 A). The solution uses reliability theory and normal distribution formulas to ensure the transformer handles load without overheating.

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