A renewable energy storage station manages a grid battery system with an initial energy re… — GMAT

A renewable energy storage station manages a grid battery system with an initial energy reserve of

E_0 = 100\text{ MWh}

. At the end of each operating interval

k

(for

k = 1, 2, 3, 4

), the updated energy reserve

E_k

(in MWh) transitions according to the rule:

E_k = 0.8 E_{k-1} + P_k

where

P_k

represents the net energy added during interval

k

. The net additions for the four intervals are

P_1 = 40\text{ MWh}

P_2 = 10\text{ MWh}

P_3 = 50\text{ MWh}

, and

P_4 = 20\text{ MWh}

Arrange the operating intervals in ascending order based on the battery energy reserve $E_k$ measured at the end of each interval, from the lowest energy reserve to the highest energy reserve.

1End of Interval 2 ( $E_2$ )
2End of Interval 1 ( $E_1$ )
3End of Interval 4 ( $E_4$ )
4End of Interval 3 ( $E_3$ )

Answer

The correct sequence from lowest to highest energy reserve is End of Interval 2 (106 MWh), End of Interval 1 (120 MWh), End of Interval 4 (127.84 MWh), and End of Interval 3 (134.8 MWh).

Evaluating each recursive state step by step yields

E_1 = 120\text{ MWh}

E_2 = 106\text{ MWh}

E_3 = 134.8\text{ MWh}

, and

E_4 = 127.84\text{ MWh}

. Arranging these quantities from smallest to largest results in the sequence: End of Interval 2, End of Interval 1, End of Interval 4, and End of Interval 3.

Step-by-Step Solution

Calculate the energy reserve at the end of Interval 1 (

E_1

)

E_1 = 0.8(100) + 40 = 80 + 40 = 120\text{ MWh}

Apply the state transition formula using initial state

E_0 = 100\text{ MWh}

and addition

P_1 = 40\text{ MWh}

Calculate the energy reserve at the end of Interval 2 (

E_2

)

E_2 = 0.8(120) + 10 = 96 + 10 = 106\text{ MWh}

Apply the state transition formula using state

E_1 = 120\text{ MWh}

and addition

P_2 = 10\text{ MWh}

Calculate the energy reserve at the end of Interval 3 (

E_3

)

E_3 = 0.8(106) + 50 = 84.8 + 50 = 134.8\text{ MWh}

Apply the state transition formula using state

E_2 = 106\text{ MWh}

and addition

P_3 = 50\text{ MWh}

Calculate the energy reserve at the end of Interval 4 (

E_4

)

E_4 = 0.8(134.8) + 20 = 107.84 + 20 = 127.84\text{ MWh}

Apply the state transition formula using state

E_3 = 134.8\text{ MWh}

and addition

P_4 = 20\text{ MWh}

Sort the computed values in ascending order

E_2 (106\text{ MWh}) < E_1 (120\text{ MWh}) < E_4 (127.84\text{ MWh}) < E_3 (134.8\text{ MWh})

Comparing the values gives the exact ordered sequence: Interval 2, Interval 1, Interval 4, Interval 3.

Key Concept

Sequential Processes and State Transitions

Estimated Time:2m 0s

Question

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