The Power Demand Of Electric Customers Is Frequently Measure

The Power Demand Of Electric Customers Is Frequently Measured In Meg

The Power Demand Of Electric Customers Is Frequently Measured In Meg

The "power demand" of electric customers is frequently measured in Megawatts (abbreviated MW). The power company, “Just Charge It”, provides electric power to two different cities, City A and City B. The two cities require different power demands, but the amount of electric power each city needs varies in a sinusoidal fashion over the course of a typical day. Here are the details:

(a) City A: At midnight, City A demands 40 Megawatts of power. By noon, the city demand is at its maximum consumption of 90 Megawatts. By midnight, it once again needs 40 Megawatts. This pattern repeats every day. (i) Find a simple formula for

Paper For Above instruction

Introduction

Electric power demand exhibits daily cyclical patterns driven by human activity, environmental factors, and economic routines. Understanding and modeling these patterns is crucial for efficient power generation, distribution, and grid management. This paper explores the sinusoidal modeling of daily power demand, focusing specifically on the case of City A, which exhibits a predictable and consistent demand pattern over a 24-hour period.

Mathematical Modeling of Power Demand

The demand for City A varies sinusoidally, reaching a minimum of 40 MW at midnight, peaking at 90 MW at noon, and returning to 40 MW at the next midnight. Such a pattern can be effectively modeled using a sinusoidal function, specifically a cosine or sine function, which captures periodic phenomena. To derive an appropriate formula, key parameters such as amplitude, vertical shift, period, and phase shift must be considered.

Parameters of the Sinusoidal Model

1. Vertical Shift (Midline): This is the average demand across the cycle. Since the demand oscillates between 40 MW and 90 MW, the average (midline) is

Midline = (Maximum + Minimum) / 2 = (90 + 40) / 2 = 65 MW.

2. Amplitude: This reflects how much demand deviates from the midline, calculated as

Amplitude = (Maximum - Minimum) / 2 = (90 - 40) / 2 = 25 MW.

3. Period: The pattern repeats every 24 hours, so the period T = 24 hours.

4. Frequency: The angular frequency ω is related to the period by ω = 2π / T, thus

ω = 2π / 24 = π / 12 radians per hour.

Constructing the Formula

Choosing a cosine function for convenience, and assuming the maximum demand occurs at noon (t = 12 hours), the model is

d(t) = A * cos(ω(t - φ)) + D, where

  • A = 25 (amplitude)
  • ω = π / 12
  • D = 65 (midline)
  • φ = phase shift to align the maximum at noon (t = 12 hours)

Since cosine reaches its maximum at 0 radians, to make the maximum occur at t = 12, set

ω(12 - φ) = 0 ⇒ φ = 12 hours.

Therefore, the formula for City A's power demand is

d(t) = 25 cos(π/12 (t - 12)) + 65, where t is in hours, 0 ≤ t

Conclusion

This sinusoidal model captures the pattern of City A's power demand over a 24-hour day, with demand oscillating between 40 MW and 90 MW, peaking at noon, and returning to the minimum at midnight. Such models can be extended or refined for more accurate analyses, including other variables influencing power demand.

References

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