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Live storage is shared according to the owner’s share of the index flow rate (q‾(o)) (see the Proportional Routing section, below, for an explanation of this approach). The live storage calculation is done after each owner’s outflow has been determined.
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The owner’s index flow rate (q‾(o)) is determined from the inflow volume I(o), outflow volume O(0) and the Muskingum parameter x (see the Link Storage Routing SRGfor details):
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Proportional routing is used to share the division’s active (live) storage between owners. This is based on the idea that ownership travels at the rate that each owners’ flow influences the flow in the division. If we consider a division and divide each owners’ inflow into a very large number of small pieces, that each time one more of these slices is passed through, the increment in division storage (Storage) can be approximated as a linear function g of the increment in the index flow rate q‾ :
| Equation 18 |
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It can also be assumed that go ≈ go-1 ; i.e. the ratio g is the same for each owner’s slice of water as it passes through the division. After summing up all of the slices the following relationship is obtained:
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In the Murray, losses caused by flows in excess of the regulated flow range are shared to owners in proportion to how far each of them is above their fixed share of this range. Hence, if q‾(o) is is owner o’s current flow rate, RatioHFT(o) is their share of the high flow threshold, q‾HFT is
is the high flow threshold, the
the total flow rate, and LossHF the total high flow loss, then owner o’s share of the high flow loss is:
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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 from equation (17) and rearranging for O(o):
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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. 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
and Storage(o,t).
Dead division model, assumptions and equations
Fully mixed (continuously stirred reactor) model
Proportional routing cannot be used to determine owner shares where there is no active storage in a division. In this situation, the fully mixed (continuous stirred reactor) model is used. This approach is based on the concept that ownership will travel as if it were a substance mixed uniformly throughout the routing storages. If a substance, i, is completely mixed throughout a volume and a sample from that volume is taken, the following relationship applies:
| Equation 40 |
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where:
M is the total mass in the volume;
Mi is the mass of the substance in the volume;
m is the mass of the sample taken; and
mi is the mass of the substance in the sample.
Owner’s storage formula
To calculate an owner’s storage volume (Storage(o,t)) in a dead division, the fully mixed principle expressed in equation (40) is applied to a stored volume of water (Storage), with outflow (due to fluxes) O as the sample, and ownership o as the substance of interest. The resultant relationship is:
| Equation 41 |
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Rearranging this:
| Equation 42 |
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Proportional losses in a dead division are shared in the proportions of the stored water, therefore:
| Equation 43 |
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Reiterating equation (15), the mass balance of a routing division over a time step is:
| Equation 44 |
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Substituting equations (42) and (43) into (44), and rearranging the unknowns to the left hand side yields:
| Equation 45 |
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This can be rearranged to solve for an owner’s storage volume, Storage(o,t), as follows:
| Equation 46 |
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Ownership adjustments
In some situations, adjustments need to be made to owners’ shares of a division’s mass balance equation so that the shares all add up to the correct total. An imbalance can occur when:
- The division is transitioning from dead to live, i.e. between the continuous stirred reactor and proportional routing models. In this case, owner fixed loss and the previous time step storage are adjusted.
- The fixed losses specified by the modeller exceed the owner’s ability to meet a lateral outflow flux requirement. When this occurs, owner shares of fixed losses are modified, and the changes tracked in the appropriate borrow-and-payback account balances. (Refer to the Borrow and PaybackSRG entry for a description of these balances).
- High flow losses cause outflow to be negative. When this occurs, owner shares of outflow are modified using the borrow and payback mechanism.
More details on the first two points are discussed in the following sections.
Division transitioning from dead to live
If the division has started flowing again in the current model time step - that is, it has gone from being dead to being live - a correction is required if there was airspace in the dead storage last time step (i.e. if dead storage was not full: Storage(t-1) < StoragedsMAX). Firstly the airspace volume to be filled is calculated:
| Equation 47 |
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Secondly, each owner’s fixed loss for the current model time step is increased to represent their contribution to filling the airspace storage volume, and is equivalent to reducing the volume available to contribute to filling live storage by the requisite amount. The relevant equation is:
| Equation 48 |
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This is also consistent with the fully mixed principle discussed in the section on Owner’s storage formula, above. The final step is to adjust the value of the storage for last time step, so it is the appropriate value to use in calculations for a live division in the current time step (note, this adjustment occurs after outputs for the last time step are recorded). That is:
| Equation 49 |
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| Equation 50 |
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Borrow and payback
If an owner’s share of fixed losses is greater than their available capacity to meet a lateral outflow flux requirement, then an adjustment is made which entails borrowing from other owners that have surplus capacity available, with later payback. Different methods are used to determine whether each owner has a capacity deficit or has surplus capacity, depending on whether the division is live or dead (see the following sections). The borrow and payback options available are:
- Fixed loss borrow and payback: Owners with a surplus that lend to others have their fixed loss Lossfixed(o) increased by the amount loaned, and those with a deficit that borrow have their Lossfixed(o) decreased by the amount borrowed.
- Outflow borrow and payback: Owners with a surplus that lend to others have their outflow volume O(o) decreased by the amount loaned, and those with a deficit that borrow have their O(o) increased by the amount borrowed.
Live Division – Fixed Loss Borrow and Payback
In a live division, the maximum fixed loss an owner could meet is that which would occur when their outflow is equal to zero (i.e. when O(o) = 0). The value of Lossfixed(o) that will result in O(o) = 0 can be found from equation (23), re-expressed as follows:
| Equation 51 |
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For each owner, the values of surplus and deficit for borrow and payback are therefore:
| Equation 52 |
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| Equation 53 |
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Live Division – Outflow Borrow and Payback
For each owner, the limiting surplus and deficit for potential borrow and payback is their share of the outflow volume.
Dead Division
In a dead division, determining whether each owner has a surplus or a deficit, and the magnitude, is based on the mass balance equation (equation (15)), re-expressed as follows:
| Equation 54 |
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A positive net volume is a surplus, and a negative net volume is a deficit. Hence the values for borrow and payback are:
| Equation 55 |
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| Equation 56 |
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Methodology
Model Phase: Configuration
Link ownership features are specified as input data by the modeller at the level of the ownership system and the individual link.
- At the ownership system level the modeller specifies for all links in the ownership system:
- Whether ownership of the time series flux is shared according to a fixed ratio or proportional to owner flow/storage in the link. (This setting can be overridden at the link level where the modeller can input a time series flux per owner if required).
- Whether ownership of other link lateral fluxes is shared according to a fixed ratio or proportional to owner flow/storage in the link.
- Whether the Murray style high-flow loss method is to be used for the flow based flux.
| Info | ||
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| Note: When the Murray style method of sharing high flow losses is used, the configured method of sharing the flow based flux is overridden. In this case the flow based flux is shared according to fixed ratio when the total division flow exceeds the high flow threshold. |