# Modeling the discharge efficency of grid connected energy storage systems

### Short form

A battery has a charge and discharge efficiency of $$\95\$$%. The initial SoC is $$\3 \text{ MWh}\$$. It charges for an hour with $$\1 \text{ MW}\$$, then discharges for an hour with $$\1 \text{ MW}\$$. What is the final SoC?

### Long form

Suppose we have a battery storage system connected to the grid, e.g. the Tesla Powerwall. The battery is rated at $$\1 \text{ MW}\$$ power output (charge and discharge) and has a capacity of $$\5 \text{ MWh}\$$. The charge and discharge efficiency is $$\95\$$%, so it has a round-trip efficiency of approx. $$\90\$$%.

The initial state of charge is $$\3 \text{ MWh}\$$. Then, the battery charges from the grid with the maximum rate for an hour. The SoC increases to $$\3.95 \text{ MWh}\$$. In the next hour, the battery discharges with the maximum rate to the grid for an hour.

Now, what is the SoC after that?

1. Is it $$\3 \text{ MWh}\$$? The battery charges with $$\1 \text{ MW}\$$ for an hour, so $$\1 \text{ MWh}\$$ is discharged from the battery, but only $$\0.95 \text{ MW}\$$ arrive at the grid due to efficiency losses. The calculation is $$3 \text{ MWh} = 3.95 \text{ MWh} - 1.0 \text{ MW} \cdot 0.95 \cdot 1 \text{h}$$
2. Is it $$\2.90 \text{ MWh}\$$? The battery discharges with $$\1.05 \text{ MW} = \frac{1.0 \text{ MW}}{0.95}\$$ for an hour, so $$\1.05 \text{ MWh}\$$ is discharged, but only $$\1 \text{ MW}\$$ arrive at the grid. $$2.90 \text{ MWh} = 3.95 \text{ MWh} - \frac{1.0 \text{ MW}}{0.95} \cdot 1 \text{h}$$

The question boils down to when is the discharge efficiency "applied"?

The second option seems inconsistent, as the "real" discharge rate of $$\1.05 \text{ MW}\$$ would exceed the maximum rated discharge rate of $$\1.0 \text{ MW}\$$. But, the second option is used in the majority of distributed energy resource scheduling models in the academic literature. I'm confused.

• If 1MW power output implies 1.05MW internal power consumption, and the battery is rated for 1MW power output, then it is rated for 1.05MW internal power consumption. Otherwise it is not honestly rated for 1MW power output. – Brian Drummond Oct 13 at 11:30
• The discharge is not linear, it's more like an acos(x^2). – a concerned citizen Oct 13 at 11:33
• Ok, so if a spec sheet specifies 1 MW, then that usually refers to the external power output? So option 2 is correct? – ktnr Oct 13 at 11:38