Which of the following equations is used to estimate the trend term in the deterministic part of load?

Question

Examoogle · Accepted Answer

Option C: N∑ddσσσ=1N-∑ddσσ=1N∑σσ=1NN∑σ2σ=1N-∑σσ=1N2

Explanation & Concepts:
The deterministic part of electrical load is often modeled as a linear trend represented by the equation $d_d(\sigma) = a\sigma + b$. The coefficients $a$ and $b$ are determined using the method of least squares regression, where the slope $a$ is calculated to minimize the sum of squared residuals between the observed and predicted values.

Working Principle: In linear regression, to minimize $S = \sum [d_d(\sigma) - (a\sigma + b)]^2$, we take partial derivatives with respect to $a$ and $b$ and set them to zero. Solving the resulting normal equations yields the slope $a = \frac{N\sum(d_d(\sigma)\sigma) - \sum d_d(\sigma)\sum \sigma}{N\sum \sigma^2 - (\sum \sigma)^2}$. This provides the best-fit line for the long-term trend in load data.

Key Formulas: $d_d(\sigma) = a\sigma + b$ — Linear trend equation; $a = \frac{N\sum(d_d(\sigma)\sigma) - \sum d_d(\sigma)\sum \sigma}{N\sum \sigma^2 - (\sum \sigma)^2}$ — Slope of the deterministic trend

Key Points: The deterministic load component represents the predictable long-term growth or seasonal trends.; The least squares method ensures the trend line has the minimum total error.; The denominator represents the variance of the independent variable (time/sigma).

Advantages: Mathematically precise identification of long-term growth.; Reduces noise from random load fluctuations.

Disadvantages: Sensitive to outliers in historical data.; Assumes linear growth which may not hold over very long horizons.

Applications: Long-term power system planning.; Peak demand estimation for infrastructure investment.

Additional Info: The variable N denotes the total number of data points.. The numerator represents the covariance between the load and time, scaled by N.. Option A and B are incorrect due to incorrect signs in the denominator leading to divergence.

Answer: C is correct — The provided equation represents the standard slope calculation for the least squares regression method used to estimate the deterministic trend of load data.

Exam Tip: Remember that for any linear trend estimation $y=mx+c$ via least squares, the denominator must involve the variance of x, which always involves a subtraction: $N\sum x^2 - (\sum x)^2$.

Examoogle · Answer

Option A: N∑ddσσσ=1N+∑ddσσ=1N∑σσ=1NN∑σ2σ=1N+∑σσ=1N2

Examoogle · Answer

Option B: N∑ddσσσ=1N+∑ddσσ=1N∑σσ=1NN∑σ2σ=1N+∑σσ=1N2

Examoogle · Answer

Option D: None of above

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