Ok, lets go through your conclusions, one by one:
Diana> In these results, the relation Var_Act (T1)+Var_FIn(T1)–Var_FOut(T1)= Var_Act(T2) does not hold.
AL> Yes, of course it doesn't hold exactly, because you have defined storage losses as well (and a storage efficiency, which also affects the reported output flows). But if you take into account the storage losses and efficiency, it holds exactly. You already asked earlier how the storage losses are calculated, and I think I gave a rather comprehensive answer to you? See again Part II, pages 254–255, to see how the loss term is added to this equation.
I also verified from the results of your model that the relation holds, when you take into account the storage losses (and the storage efficiency).
Diana> Per instance, In the first period it is used (2020), in the first season S1Q1-S1Q12 there is a VAR_ACT = 0.43732, the same in each TS of all S1.
AL> No, the VAR_ACT are definitely NOT exactly the same in each TS of all S1. For example, for timeslices S1Q1‒S1Q2 we have the following results:
VAR_ACT(S1Q1) = 0.437319519
VAR_ACT(S1Q2) = 0.437311737
VAR_Fin(S1Q1) = 0.000012238
VAR_Fout(S1Q1) = 0.000000036
And the loss can be calculated as 0.000019969. So, we have the relation
VAR_ACT(S1Q2) = VAR_ACT(S1Q1) + VAR_Fin(S1Q1) ‒ VAR_Fout(S1Q1) / 0.7 ‒ LOSS(S1Q2)
0.437311737 = 0.437319519 + 0.000012238 ‒ 0.000000036 / 0.7 ‒ 0.000019969
The relation holds well. I think everything works exactly as one may expect. Would you not agree?
Diana> Summing up all VAR_FIn does not gives VAR_ACT plus losses.
AL> Of course not, as it shouldn't give that amount. Neither in reality nor in the model. However, summing up all VAR_Fin, and subtracting the sum of all VAR_Fout divided by the storage efficiency, gives exactly the sum of the storage losses. For example, the sum of the losses in the season S1 is 0.000240, and the sum of those net inputs flows is exactly the same, 0.000240.
Diana> As I understand, to have a charge (VAR_ACT) the process has to be charged up to that level inside the same season (DAYNITE Level).
AL> No, certainly not. VAR_ACT represents the aggregate storage level. If there is no charging/discharging and no losses, the storage level can of course stay at any constant value without any charging up! And if there are small losses (like in your case), only the losses need to be compensated for by some charging up. That should be very obvious, and so it seems you do have a fundamental misunderstanding here.
Diana> In these results, the relation Var_Act (T1)+Var_FIn(T1)–Var_FOut(T1)= Var_Act(T2) does not hold.
AL> Yes, of course it doesn't hold exactly, because you have defined storage losses as well (and a storage efficiency, which also affects the reported output flows). But if you take into account the storage losses and efficiency, it holds exactly. You already asked earlier how the storage losses are calculated, and I think I gave a rather comprehensive answer to you? See again Part II, pages 254–255, to see how the loss term is added to this equation.
I also verified from the results of your model that the relation holds, when you take into account the storage losses (and the storage efficiency).
Diana> Per instance, In the first period it is used (2020), in the first season S1Q1-S1Q12 there is a VAR_ACT = 0.43732, the same in each TS of all S1.
AL> No, the VAR_ACT are definitely NOT exactly the same in each TS of all S1. For example, for timeslices S1Q1‒S1Q2 we have the following results:
VAR_ACT(S1Q1) = 0.437319519
VAR_ACT(S1Q2) = 0.437311737
VAR_Fin(S1Q1) = 0.000012238
VAR_Fout(S1Q1) = 0.000000036
And the loss can be calculated as 0.000019969. So, we have the relation
VAR_ACT(S1Q2) = VAR_ACT(S1Q1) + VAR_Fin(S1Q1) ‒ VAR_Fout(S1Q1) / 0.7 ‒ LOSS(S1Q2)
0.437311737 = 0.437319519 + 0.000012238 ‒ 0.000000036 / 0.7 ‒ 0.000019969
The relation holds well. I think everything works exactly as one may expect. Would you not agree?
Diana> Summing up all VAR_FIn does not gives VAR_ACT plus losses.
AL> Of course not, as it shouldn't give that amount. Neither in reality nor in the model. However, summing up all VAR_Fin, and subtracting the sum of all VAR_Fout divided by the storage efficiency, gives exactly the sum of the storage losses. For example, the sum of the losses in the season S1 is 0.000240, and the sum of those net inputs flows is exactly the same, 0.000240.
Diana> As I understand, to have a charge (VAR_ACT) the process has to be charged up to that level inside the same season (DAYNITE Level).
AL> No, certainly not. VAR_ACT represents the aggregate storage level. If there is no charging/discharging and no losses, the storage level can of course stay at any constant value without any charging up! And if there are small losses (like in your case), only the losses need to be compensated for by some charging up. That should be very obvious, and so it seems you do have a fundamental misunderstanding here.