## Optimizer Definition for a constraint n-forecast Trading-Problem


We want to optimize the performance of an energy trader given the forecast for n steps. 
The battery:
 - holds 1MWh
 - charges/discharges at max. 1MW per hour (we can add/loose x*1MW, x \in R )
Prices are stable for the given hour (t) and we sell and buy for the same price.


### Considerations:
 - Single variable, P (=x), for each hour t from 0 to n-1.

 - If P > 0, it represents discharging (selling power) with a magnitude of P.
 - If P < 0, it represents charging (buying power) with a magnitude of -P.
 - If P = 0, it represents holding (doing nothing).

 - if we have forecasts for t_n, t_n+m we might have to **interpolate** between n .. m
 - or... we work with the gaps and dt as charge time .... no

#### Variables:

 - price_t = price per MWH at t (eq)
 - B (t=0..n) = State of Battery in MWH
 - P (t=0..n) = Charge/Discharge factor given the possible base rate of 1MW/h
 - max_p = 1 (charge/discharge limits) & and battery capacity limits (both=1)
 - SoB_initial = 0
 - h = horizon \in N^+


### Objective
 - We **Maximize**: Sum_{t=0}^{n-1} (price_t * P)

### Constraints
 - Fixed starting state: SoB_0 = SoB_initial
 - Charge/Discharge Limit: (-max_p <= P <= max_p) for all t = 0, ..., n-1
 - Storage Limit: (0 <= B+(1*P) <= max_p) for all t = 0, ..., n-1
 - Future B State: SoB_{t+1} = (B + P) for t = 0 to n-1