Give step-by-step solution with explanation and final answer:
P3 (Refer to Bus Charging Station Version 2 Design) The efficiency nst depends on the efficiency of the battery storage (22) and the effic

Question

Give step-by-step solution with explanation and final answer:
P3 (Refer to Bus Charging Station Version 2 Design) The efficiency nst depends on the efficiency of the battery storage (22) and the efficiency of the charging station (24). Both efficiency curves are non-linear functions of the input power; as such, for the efficiency NSL it must be considered which power is transmitted by the grid and by PV. Determine the formula for the power loss S', and the combined efficiency NSL. For the power from the grid and PV, set a combined value Prow (row = rest of world).

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# Step-by-Step Solution

## 1. **Definitions**

- Let $ \eta_{st} $: **Total system efficiency** (NSL)
- Let $ \eta_{bat}(P) $: **Battery storage efficiency** as a function of input power $ P $
- Let $ \eta_{cs}(P) $: **Charging station efficiency** as a function of input power $ P $
- Let $ P_{row} $: **Total input power** from grid and PV (“rest of world”)
- Let $ S' $: **Power loss** in the system

## 2. **Power Flow Relationships**

The total input power $ P_{row} $ passes through two stages:
1. First through the charging station (efficiency $ \eta_{cs} $)
2. Then through the battery storage (efficiency $ \eta_{bat} $)

So, the output power delivered to the bus ($ P_{bus} $) is:

$$
P_{bus} = P_{row} \cdot \eta_{cs}(P_{row}) \cdot \eta_{bat}(P_{row})
$$

## 3. **Combined Efficiency Formula ($ \eta_{st} $)**

The combined system efficiency is the ratio of output to input power:

$$
\eta_{st} = \frac{P_{bus}}{P_{row}} = \eta_{cs}(P_{row}) \cdot \eta_{bat}(P_{row})
$$

**This formula accounts for both non-linear efficiency curves, evaluated at the total power $ P_{row} $.**

## 4. **Power Loss Formula ($ S' $)**

The power loss $ S' $ is the difference between the input and output power:

$$
S' = P_{row} - P_{bus}
$$

Substituting from above:

$$
S' = P_{row} - P_{row} \cdot \eta_{cs}(P_{row}) \cdot \eta_{bat}(P_{row})
$$

$$
S' = P_{row} \left[1 - \eta_{cs}(P_{row}) \cdot \eta_{bat}(P_{row})\right]
$$

## 5. **Summary of Final Answers**

### **Power Loss Formula**

$$
\boxed{
S' = P_{row} \left[1 - \eta_{cs}(P_{row}) \cdot \eta_{bat}(P_{row})\right]
}
$$

### **Combined Efficiency Formula**

$$
\boxed{
\eta_{st} = \eta_{cs}(P_{row}) \cdot \eta_{bat}(P_{row})
}
$$

## 6. **Explanation**

- **Both efficiency curves are non-linear:** You must use the actual values of $ \eta_{cs} $ and $ \eta_{bat} $ at the particular $ P_{row} $.
- **All grid and PV power is combined:** You use the sum as the input power ($ P_{row} $) for both efficiency functions.
- **The total system (NSL) efficiency is the product of the individual efficiencies** at the operating point.

---

**In summary:**  
- Power loss is input minus output, using the product of both nonlinear efficiency functions.  
- Combined efficiency is the product of the two efficiency functions, both evaluated at the total input power.

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