Original practice · fully worked
Original variant: choose a threshold for discrete renewable surplus
Daily renewable output is 200 megawatt-hours on one fifth of days, 500 on one half, and 900 on the remaining three tenths. A storage operator keeps only output above a threshold h between 200 and 500. Choose h so expected stored surplus is 224 megawatt-hours.
- A
300 MWh - B
340 MWh - C
350 MWh - D
370 MWh - E
400 MWh
Answer and calculation
For a threshold between 200 and 500, only the 500 and 900 outcomes contribute. Solving 0.5(500-h)+0.3(900-h)=224 gives h=370.
Setup
Setup
Because 200<h<500, the 200-MWh outcome contributes no stored surplus, while the 500- and 900-MWh outcomes contribute 500-h and 900-h.
X∈{200,500,900} 200<h<500 Model
Model
Weight the two positive surplus amounts by probabilities 0.5 and 0.3 and set their sum equal to the required mean of 224 MWh.
E[(X−h)+]=0.5(500−h)+0.3(900−h) Compute
Compute
The expectation simplifies to 520-0.8h. Solving 520-0.8h=224 gives h=370, which lies in the assumed interval.
520−0.8h=224 Answer
Answer
Thus the required storage threshold is 370 MWh, corresponding to choice D.
370 MWh(D) 