...
This solution is applied in divisions where Muskingum weighting x ≠ 1 to owners that have Storage(o,t) ≤ StoragedsMAX(o) + StorageHFTRatioHFT(o). To solve mass balance, the outflow volume (O(o)) is recast in terms of the division’s live storages. Recalling the index flow rate q‾(o) from equation (17) and rearranging for O(o):
| Equation 35 |
|---|
Equation (35) can be rewritten in terms of live storage using the proportional routing equation, equation (19), as follows:
| Equation 36 |
|---|
| Info | ||
|---|---|---|
| ||
| Note: Equation (36) does not work for the case where x = 1 as it would lead to an attempt to divide by zero. This reflects the fact that in this case there is no relationship between the division’s outflow and live storage. |
Combining the mass balance and proportional loss equations (i.e. equations (15) and (19)) with equation (36), and rearranging yields:
| Equation 37 |
|---|
Case of Muskingum weighting x ≠ 1, owner with a high flow loss
This solution is applied in divisions where Muskingum weighting x ≠ 1 to owners that have Storage(o,t) > StoragedsMAX(o) + StorageHFTRatioHFT(o). The approach is based on defining a modified proportional loss which is the remaining proportional loss not accounted for after considering those owners not contributing to the high flow loss:
| Equation 38 |
|---|
Combining the mass balance and proportional loss equations (i.e. equations (15) and (19)), the loss equation (31), equation (35), and equation (38) above, and rearranging yields:
| Equation 39 |
|---|
Potential issue with high flow loss calculation (Case of x ≠ 1): Outflow can be negative
A problem may occur with equation (39) as it is possible for the modeller to configure a perverse case where an owner would be required to borrow from other owners to pay for their share of the high flow loss. In the case where x = 1 this does not cause a problem as borrowing between owners does not affect the share of the division’s storage. For other cases, borrowing between owners will change the share of the division’s storage (Storage(o,t)) as changes in outflow will change q‾. In theory this would indicate that high flow loss should be solved iteratively. However, as iterative solutions tend to impact performance, and the situation will only occur where outgoing lateral fluxes are so large as to reduce an owner’s outflow to less than zero, a solution that uses borrow and payback on division outflow is proposed. This means accepting in these cases a mismatch between q(o) and .