Optimization with time-dependent constraint

I am trying to find the optimal operation of a battery to perform energy arbitrage, which consists in charging energy when prizes are low and discharging when prizes are high.

The information that I have is:

P_bat = 10 %Battery power in MW
E_bat = 40 %Battery capacity in MWh
price = rand (1,24) %1x24 array with hourly energy prices in USD/MWh

And the results I would expect from the optimization are:

d %1x24 array with energy discharged every hour in MWh
c %1x24 array with energy charged every hour in MWh
SOC %1x24 with the battery State of Charge in MWh (initial value 0)

I have identified f as the function that I have to minimize to obtain maximum profit

Which has these constraints:

But there is another important constraint to the problem:

I am basing my code in this answer in the forum, but since the state of charge in a certain hour depends on its value on the previous hour, I am unable to translate this into the constraints function. I hope you can help me.

Answer 1

[Alan Weiss]

For an example of a similar problem, see Optimal Dispatch of Power Generators: Problem-Based.

Alan Weiss

MATLAB mathematical toolbox documentation

Answer 2

[Torsten]

Use "linprog" and define the 24 equalities

SOC_i - (SOC_i-1 + c_i - d_i) = 0 (i=1,...24)

in the matrix Aeq with SOC_1,...,SOC_24 as additional unknowns.

[Iñigo Azpiri]

Thanks, Torsten. I will try this approach.