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Description and rationale

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Therefore, for an owner, i, the continuity equation gives for a time step:

Equation 1

where:

Vⁱ₂ - Owner ’s volume of water in the reservoir this time step.

Vⁱ₁ - Owner ’s volume of water in the reservoir last time step.

I_i - Volume of inflow this time step belonging to owner i.

C_i - Net volume of water ceded by owner i this time step to all other owners, negative if this owner has ceded less water to other owners than received.

P_i - Proportional loss attributed to owner i, negative if a gain.

F_i - Fixed loss attributed to owner i, negative if a gain.

IS_i - Net internal spill of owner i, negative if spill into owner’s share.

ES_i - External spill for owner i.

B_i - Total borrowed from other owners by owner i, negative if the owner is lending to other owners.

R_i - Volume of regulated water released this time step for owner i.

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If the reservoir is empty, this equation is modified to account for proportional gains (such as rainfall) that could be released, or that have dried up the reservoir over the period considered:

Equation 3

where:

r_i - Owner i’s ratio share of the reservoir’s storage capacity.

P - Total proportional loss, negative if a gain.

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Using the borrow method described in Borrow and Payback - SRG  B_i for each owner can be calculated. Any owner that had to borrow will have Vⁱ₂ = 0. 

B_i = Borrowed_i - Lent_i

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1. Accumulate amount the owner owes other owners at the priority level:

   Equation 7

2. If the owner  has borrowed from any other at the priority level ( (CanPayback(pl, i) > 0):

   1. Calculate the ratio to limit the owner’s payback to their current ability to repay:

      Equation 8

   2. For every other owner OtherOwner that shares with owner i at the priority level:

      1. Calculate the payback to the other owner using the ratio above

         Equation 9

      2. Update the borrow record for the payback to the other owner

         Equation 10

         Equation 11

      3. Adjust current time step borrow totals for the payback:

         Equation 12

         Equation 13

      4. Update the surplus remaining to be shared at the next priority level down:

         Equation 14

Forfeiture of credit 

If a reservoir is a payback storage, a check is made to ensure that no owner’s credit owing to them exceeds their capacity to store it. If any owner has more water owed to them than they have remaining airspace, the excess is forfeited back to the debtors in order of priority. This methodology is similar to that used for calculating payback, but in this case the forfeits run in the opposite direction to the repayments.

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Each owner is entitled to use a fixed proportion of the proportion** of the storage capacity of a reservoir (r_i) which means that at any time its storage cannot exceed:

Equation 23

where:

Vⁱ_max - Owner i’s maximum allowed volume of water in the reservoir this time step.

r_i - Owner i’s ratio share of reservoir storage capacity.

V_max - Storage capacity of reservoir.

V₂ - Storage volume of reservoir this time step.

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Proportional gain P is distributed in an iterative process. Initially the proportional gain remaining to be distributed (P_rem) equals the total P, and every owner’s spill Spill_i is zero. P_rem is distributed until there are no more owners that will spill. This process is outlined below:

1. A total of the estimated unassisted storage volumes is calculated:
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   Note: This is using the notation suggested by Knuth (1992 [http://arxiv.org/abs/math/9205211v1]). Essentially it’s saying that the summation is for all of the terms where the conditions in the square brackets are true. So [Spillj ≤ 0] indicates that we want the summation of Vjest for all j where Spilli is less than or equal to zero.

2. For each owner not already identified as spilling (Spill_j ≤ 0) we calculate an initial estimate of their spill volume totals:

   Equation 27

3. If an owner is discovered to be spilling (Spill_j > 0), the remaining proportional gain is updated:

   Equation 28

4. And a final estimate of total spill calculated:

   Equation 29

5. If during this pass any owners are discovered to be spilling then it is necessary to return to equation (27) and keep repeating the process until no more spilling owners are discovered.

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The current share of water in storage for each owner is saved when the result of the override function transitions from “false” to “true”, which is when the suspension is activated (If the reservoir is empty then V = 0 and Share_i = r_i.):

Equation 51

where:

r_i - Owner i’s ratio share of the reservoir’s storage capacity.

Vⁱ - Owner i’s volume of water in the reservoir.

V - Total volume stored in the reservoir.

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When the result of the override function transitions from “true” to “false”, that is the suspension is deactivated, each owner’s share of water in storage is reset to represent what it was before the suspension started:

Vⁱ = Share_i × V

The owners’ shares of the storage capacity (r_i) are restored to the values they were before the suspension started.

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As ownership is calculated after the physical reservoir model has been run the volume of water that was released from the reservoir in the current model time step is known:

Equation 54

where:

R - The total volume released from the reservoir this time step.

R_p - The total volume released from the reservoir this time step down outlet path p.

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