Storage Systems

For the solution of the differential equation of the storage system are three different methods considered. These are Forward Euler, Central Euler and Backward Euler. The Central Euler method is also called Crack-Nicolsen Approach.

The first Equation constraints states that the storage system must be smaller that its maximum capacity.

\[E_{st}^t \leq E_{st,ref}^{[\frac{t}{l_{yr}}]}\]

Forward Euler

\[E_{st}^t - E_{st}^{t-1} \cdot \eta_{self}^{t-1} = \sum (P^{t-1} \;or \; \dot{Q}^{t-1}) \cdot \delta_{t_h}^{t-1}\]

Startpoint

\[E_{st}^{t_{start}} = f_{SOC} \cdot E_{st,ref}^{[\frac{t}{l_{yr}}]}\]

If start- and endpoint are the same:

\[E_{st}^{t_{start}} - E_{st}^{t_{end}}\cdot \eta_{self}^{t_{end}} = \sum(P^{t_{end}}\;or\;\dot{Q}^{t_{end}}) \cdot \delta_{t_h}^{t_{end}}\]

Backward Euler

\[E_{st}^t - E_{st}^{t-1} \cdot \eta_{self}^{t-1} = \sum (P^{t} \;or \; \dot{Q}^{t}) \cdot \delta_{t_h}^{t}\]

Startpoint

\[E_{st}^{t_{start}} = f_{SOC} \cdot E_{st,ref}^{[\frac{t}{l_{yr}}]} \cdot \eta_{self}^{t_{start}} + \sum(P^{t_{start}}\;or\;\dot{Q}^{t_{start}}) \cdot \delta_{t_h}^{t_{start}}\]

If start- and endpoint are the same:

\[E_{st}^{t_{start}} - E_{st}^{t_{end}} \cdot \eta_{self}^{t_{end}} = \sum(P^{t_{start}}\;or\;\dot{Q}^{t_{start}}) \cdot \delta_{t_h}^{t_{start}}\]

Central Euler

\[E_{st}^t - E_{st}^{t-1} \cdot \eta_{self}^{t-1} = \sum (P^{t-1} \;or \; \dot{Q}^{t-1}) \cdot \frac{\delta_{t_h}^{t-1}}{2} + \sum (P^{t} \;or \; \dot{Q}^{t}) \cdot \frac{\delta_{t_h}^{t}}{2}\]

Startpoint

\[E_{st}^{t_{start}} = f_{SOC} \cdot E_{st,ref}^{[\frac{t}{l_{yr}}]} \cdot \eta_{self}^{t_{start}} + \sum(P^{t_{start}}\;or\;\dot{Q}^{t_{start}}) \cdot \frac{\delta_{t_h}^{t_{start}}}{2}\]

If start- and endpoint are the same:

\[E_{st}^{t_{start}} - E_{st}^{t_{end}} \cdot \eta_{self}^{t_{end}} = \sum(P^{t_{start}}\;or\;\dot{Q}^{t_{start}}) \cdot \frac{\delta_{t_h}^{t_{start}}}{2} + \sum(P^{t_{start}}\;or\;\dot{Q}^{t_{start}}) \cdot \frac{\delta_{t_h}^{t_{end}}}{2}\]