A smart battery storage unit operates over three consecutive cycles (k = 1, 2, 3). At the … — GMAT

A smart battery storage unit operates over three consecutive cycles ( $k = 1, 2, 3$ ). At the start of Cycle 1 ( $k = 0$ ), the stored energy ( $E_0$ ) is $50\text{ kWh}$ and the grid demand factor ( $D_0$ ) is $10$ .

For each cycle $k$ :
- Stored energy updates according to the demand factor of the previous cycle: $E_k = E_{k-1} + 20$ if $D_{k-1} \ge 10$ ; otherwise, $E_k = E_{k-1} - 10$ .
- Grid demand factor updates according to: $D_k = D_{k-1} - 2$ .

Which of the following statements regarding the system states must be true? (Select all that apply.)

At the end of Cycle 1 ( $k = 1$ ), the stored energy is $70\text{ kWh}$ .Answer
At the end of Cycle 3 ( $k = 3$ ), the grid demand factor is $4$ .Answer
C
At the end of Cycle 2 ( $k = 2$ ), the stored energy is $90\text{ kWh}$ .
D
At the end of Cycle 3 ( $k = 3$ ), the stored energy is $70\text{ kWh}$ .
E
At the end of Cycle 2 ( $k = 2$ ), the grid demand factor is $8$ .

Answer

The statements confirming that the stored energy at the end of Cycle 1 is 70 kWh and that the grid demand factor at the end of Cycle 3 is 4 are both correct.

Evaluating the transition steps sequentially: At Cycle 1 (

k=1

D_0 = 10 \ge 10

, so stored energy becomes

50 + 20 = 70\text{ kWh}

and demand factor becomes

10 - 2 = 8

. This confirms the statement regarding Cycle 1 stored energy. Continuing through Cycle 2 (

D_1 = 8 < 10 \Rightarrow E_2 = 60, D_2 = 6

) to Cycle 3 (

D_2 = 6 < 10 \Rightarrow E_3 = 50, D_3 = 4

), the grid demand factor reaches

4

, confirming the statement regarding Cycle 3 grid demand.

Step-by-Step Solution

Calculate state values at the end of Cycle 1 (

k = 1

)

E_1 = 50 + 20 = 70\text{ kWh}

because

D_0 = 10 \ge 10

D_1 = 10 - 2 = 8

Apply the conditional energy rule and the linear demand decrease rule using initial values.

Calculate state values at the end of Cycle 2 (

k = 2

)

E_2 = 70 - 10 = 60\text{ kWh}

because

D_1 = 8 < 10

D_2 = 8 - 2 = 6

Since

D_1 < 10

, the energy decreases by

10\text{ kWh}

Calculate state values at the end of Cycle 3 (

k = 3

)

E_3 = 60 - 10 = 50\text{ kWh}

because

D_2 = 6 < 10

D_3 = 6 - 2 = 4

Since

D_2 < 10

, the energy decreases by

10\text{ kWh}

and demand factor decreases by

2

Key Concept

Sequential Processes and State Transitions

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