Smart Grid Dynamic Power Clamping (Computer & Electrical Engineering) Solution

Problem Overview

We evaluate the continuity parameter $k$ for the dynamic power transfer function:

P(I) = \begin{cases} \frac{I^2 - 16}{I - 4} & : I \neq 4 \\ k I - 2 & : I = 4 \end{cases}

to ensure continuity across the calibration threshold

I = 4\text{ A}

Step-by-Step Mathematical Solution

Step 1: Evaluate the limit as $I \to 4$

Direct substitution of $I = 4$ into the open rational branch yields $\frac{4^2 - 16}{4 - 4} = \frac{0}{0}$ (indeterminate form).

Resolve the indeterminate form by factoring the difference of squares in the numerator (see Factor and Cancel):

\lim_{I \to 4} \frac{I^2 - 16}{I - 4} = \lim_{I \to 4} \frac{(I - 4)(I + 4)}{I - 4}

For all $I \neq 4$ , cancel the shared zero factor $(I - 4)$ :

\lim_{I \to 4} (I + 4) = 4 + 4 = 8\text{ kW}

Step 2: Evaluate the defined function value at $I = 4$

From the point definition branch at $I = 4$ :

P(4) = k(4) - 2 = 4k - 2

Step 3: Enforce Condition 3 of the Continuity Rubric

For $P(I)$ to be continuous at $I = 4$ , the approached limit must equal the assigned point value:

\lim_{I \to 4} P(I) = P(4)

8 = 4k - 2

10 = 4k \implies k = \frac{10}{4} = 2.5

Engineering Conclusion

Setting the microcontroller gain parameter to $k = 2.5$ forces the assigned power value $P(4) = 4(2.5) - 2 = 8\text{ kW}$ to match the limiting power trajectory exactly. This eliminates the removable discontinuity (hole) at $I = 4\text{ A}$ , ensuring smooth, uninterrupted power delivery and preventing false-positive circuit breaker trips caused by computational glitches.