Ownership in Links - SRG

Description and rationale

Modelling of ownership on links is an essential component of modelling water ownership in Source, as it enables ownership to be tracked through links in Source models.  The rationale for modelling water ownership, and the overall principles, are discussed in the Ownership Systems SRG entry.

Scale

The concept of spatial scale in the context of ownership on links relates to the fact that it applies to link divisions and these have length, width and depth dimensions (even if they are represented as points for modelling).  Ownership status can be updated as often as at every model time step, or less often if required.

Principal developer

This version of modelling ownership on links has been developed by eWater CRC for Source.

Scientific Provenance

Ownership on links has been modelled in predecessors to Source, such as IQQM and MSM, for many years.  The concepts in these models have been updated and enhanced to suit the needs of Source.

Version

Source v3.8.8.

Dependencies

The requirement is that there should be at least two water users (as well as an ownership system) in the river system being modelled, in addition to at least one link.

Definitions

Other definitions can be found in the eWater River Systems glossary.

Assumptions and Limitations

Table 1. Assumptions and constraints applicable to modelling ownership on links

Background

In Source, components that are physically or logically connected are joined using a link.  If the connection is significant enough to have an effect on the time that water would take to pass along it then the link is modelled as a routing link.  Each routing link is subdivided into one or more divisions.

Figure 1 below shows a single routing division, its storage compartments and fluxes.  Each owner’s share of these storage compartments and fluxes must be determined for every division.  The principles on which these calculations are based are discussed in the Theory section and the sequencing of the calculations is described in the Methodology section.

Figure 1. Division ownership conceptual model

Definition of terms used in equations 

Table 2. Symbols used in equations

States of a division: live or dead

At any time-step a routing link division can be classed as being either:

• Live: A division is described as being live if a solution for the time step can be found that satisfies both the continuity and storage equations. The general form of the continuity equation is discussed in the Ownership Mass Balance section, below (see equations 15 and 16).

 OR

• Dead: If the only solution that can be found satisfies the continuity equation but not the storage equation, then the division is described as being dead (i.e. it has ceased to flow).

A division is dead if x = 1 and  or if x ≠ 1 and the following is true:

Equation 1

Ownership of division storage

At the start of division ownership processing, the last time-step’s storage (Storage(t-1)) and each owner’s share of it (Storage(o,t-1)) are known from the flow phase routing calculations. The volume of water in a routing link division can be divided into:

• Live Storage: If the division is live then the live storage is the result of evaluating the storage equation  for the current inflow and outflow. 
• Dead Storage: If the division is dead then the dead storage is the total volume of water in the division. It is possible for the dead storage to be larger than the maximum specified by the modeller as the division may be, for example, in the process of transitioning from dead to live but there is not enough water in the division to satisfy both the continuity and storage equations. If the division is live then the dead storage is equal to the maximum volume (Storage_dsMAX) specified by the modeller.

The live storage in a division is the storage above dead storage, as shown in Figure 1, and is obtained from:

Equation 2

The total volume of water in a division is then:

Equation 3

Sharing dead storage

Ownership of the dead storage is shared by fixed ratio (Ratio_ds(o)) to all of the owners. Hence:

Equation 4

Where the sum of all the ratios is equal to one. That is:

Equation 5

Also if the maximum total dead storage specified by the modeller is Storage_dsMAX then each owner’s share of this is:

Equation 6

Water is borrowed and lent between all of the owners so that the fixed ownership share given by equation (4) is always maintained.

Sharing live storage

Live storage is shared according to the owner’s share of the index flow rate () (see Proportional Routing, below, for an explanation of this approach). The live storage calculation is done after each owner’s outflow has been determined.

Equation 7

Equation 8

Ownership of fluxes

Inflow flux

In the flow phase, the outflow flux from the division upstream becomes the inflow flux to the current division (or the outflow flux from the upstream node becomes the inflow to the division if the most upstream division in a link is being considered). The same is true for the inflow flux per owner (the ownership processing for the upstream division or node has already determined its outflow per owner).

Equation 9

Equation 10

Outflow flux

The total outflow volume  is also known after the reach’s flow routing is complete.  Ownership processing determines each owner’s share of this outflow flux.  The owner’s share of this flux is calculated based on mass balance and proportional routing.  The resultant equation (equation (23)) and its derivation are explained in the section on Owner’s outflow formula (excludes high flow losses) below.

Lateral loss fluxes

The volume of lateral loss fluxes from every link division is known from the routing calculations for each time step after the flow phase.  These fluxes may be divided into two categories:

1. Fixed loss fluxes whose ownership is known a priori (because they are shared by fixed ratio or by some other means such as time-series or expression), and
2. Proportional loss fluxes that are shared in proportion to the ownership of the water in the division.

Proportional fluxes may be further categorised into those based on total water in the division, and those above a specified threshold, as in the Murray-style losses. These Murray-style losses are also referred to as high flow losses.

Fixed loss sharing

The total fixed loss, Loss_fixed, is distributed to owners based on the specified ratios, expression, or time series such that:

Equation 11

An owner’s share of fixed losses is adjusted if that owner does not have sufficient water in the division to cater for the loss.  In a live division, an owner’s highest possible fixed loss occurs when their outflow is zero.  In a dead division, the owner’s fixed loss cannot be larger than their share of dead storage.  The Borrow and Payback mechanism is used to adjust owner shares of fixed loss for these situations.  See Ownership adjustments for more information.

Proportional loss sharing

If the division is live, owner shares of proportional losses are based on each owner’s index flow rate (see Proportional Routing for an explanation of this approach).

Equation 12

Equation 13

If the division is dead, proportional losses are shared according to the owner’s fixed share of dead storage:

Equation 14

Murray-style High Flow Loss Sharing

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 the regulated flow range. When the modeller specifies Murray-style losses for an ownership system in Source, this method of sharing is applied only to the general purpose flux, which is a function of flow, when specified on links that fall within the ownership system.  The calculation of each owner’s share of the high flow loss is explained in the section on High flow loss formula (Murray-style), below.

Fill dead storage flux

The division’s dead storage is shared according to fixed ratios.  As this rule dictates each owner’s share of part of the total storage, it necessitates changing other parameters of the ownership conservation (mass balance) equation when a dead division is transitioning to the live state.  The volume required to fill the dead storage compartment is treated as an additional loss that is shared according to fixed ratios.

Ownership mass balance

It is assumed that ownership is conserved in every division of a reach. This is reflected in the ownership mass balance equation below.  This shows that the difference between an owner’s share of storage at the beginning and end of a time step should be the sum of their share of all the division’s fluxes.

Equation 15

When Murray-style high flow losses are specified, these are separated from other proportional losses, and the mass balance equation takes the form:

Equation 16

Live division model, assumptions and equations

Index flow rate

The owner’s index flow rate () is determined from the inflow volume I(o), outflow volume O(0) and the Muskingum parameter x (see Link storage routing - SRG for details):

Equation 17

Proportional routing of ownership

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 :

Equation 18

It can also be assumed that g_o ≈ g_o-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:

Equation 19

From this, lateral loss fluxes that are proportionally shared can also be shared in proportion to the ownership ratios in live storage, as this is the same as sharing in proportion to each owner’s index flow rate.  Hence, substituting from equation (19) into equation (13) and rearranging yields:

Equation 20

Owner's outflow formula (excludes high flow losses)

As ownership is conserved, each owner’s outflow can be determined using the proportional routing, mass balance and division flow equations (note, this formula does not cater for ‘high flow losses’).

To simplify later steps, r is defined as:

Equation 21

Combining mass balance equation (15) with equations (19) and (21) gives:

Equation 22

Rewriting (22) by substituting equation (17) for , and rearranging in terms of owner outflow volume , gives:

Equation 23

If high flow losses are not specified, the result of this outflow volume equation is used in the mass balance equation to determine division storage at the end of a time-step.

Note: If the routing division is using an attenuation factor x=0 (i.e. ), equation (23) can be rearranged to the continuous stirred reactor equation (equation (46)). This means that for fully forward weighted routing schemes, proportional routing is the same as fully-mixed.

High flow loss formula (Murray-style)

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 is owner o’s current flow rate, Ratio_HFT(o) is their share of the high flow threshold,is the high flow threshold,the total flow rate, and Loss_HF the total high flow loss, then owner o’s share of the high flow loss is:

Equation 24

Substituting equation (19) into equation (24) enables it to be re-written in terms of the division’s storages, as follows:

Equation 25

High flow losses are worn only by owners where Storage(o,t)-Storage_dsMAX(o) is greater than their share of the high flow threshold (Storage_HFTRatio_HFT(o)).  Owners that will not be required to contribute the high flow loss are identified by calculating Storage(o,t) for each owner assuming that Loss_HF=0 and finding those that fall short of their share of the high flow threshold.

From equation (25), for those owners that exceed their share of the threshold:

Equation 26

The denominator of equation (26) is the total volume above the high flow threshold of owners contributing to the high flow loss.  The equations that follow are used to determine values required to find this.

Firstly, a modified high flow threshold is defined that applies to the owners, j (where j is defined in Table 2, and the relevant owners are those that have, with Loss_HF=0, a trial value of Storage(o,t) such that Storage(o,t) > Storage_dsMAX(o) + Storage_HFTRatio_HFT(o)):

Equation 27

where:

 is set-builder notation to indicate that the modified high flow threshold is the sum of all the owners’ shares of the threshold where the high flow loss is not going to be zero.

Next, it is necessary to determine the total volume of water this time step belonging to owners that are not contributing to the high flow loss, i.e. Storage_exclude:

Equation 28

The denominator of equation (26) (current live volume contributing to high flow loss) is then:

Equation 29

Equation (26) can now be recast in terms of the total volume in the division:

Equation 30

Rearranged:

Equation 31

High flow loss solutions over a time-step

Case of Muskingum weighting x = 1 (Index flow rate = Inflow)

In this case, the index flow rate is the same as the inflow rate, and an owner’s share of inflow will determine their share of active storage.  The proportional routing formula (equation (19)) can be rewritten and applied to inflow as shown in equation (32), below and then rearranged to determine the owner’s share of storage.  

Equation 32

Equation 33

The owner’s share of proportional lateral flux is then found by substituting into equation (20) and rearranging.

Equation 34

Each owner’s storage Storage(o,t) - Storage_dsMAX(o) is compared to Storage_HFTRatio_HFT(o) to determine whether a high flow loss applies.  If it does, equation (24) is used to determine this value.  Mass balance is applied to determine each owner’s share of outflow.

Determining whether an owner has a high flow loss when x ≠ 1

If the division’s Muskingum weighting x ≠ 1 both inflow and outflow impact storage, equation (23) is used to determine each owner’s share of outflow O(o).  From this a trial estimate of each owner’s storage (Storage(o,t)) is calculated by assuming Loss_HF = 0.  It is then possible to determine if owners have a share of the high flow loss or not, i.e. whether Storage(o,t) - Storage_dsMAX(o) is less or more than their share of the high flow threshold: Storage_HFTRatio_HFT(o).

Case of Muskingum weighting x ≠ 1, owner without a high flow loss

This solution is applied in divisions where Muskingum weighting x ≠ 1 to owners that have Storage(o,t) ≤ Storage_dsMAX(o) + Storage_HFTRatio_HFT(o). To solve mass balance, the outflow volume (O(o)) is recast in terms of the division’s live storages. Recalling the index flow ratefrom 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

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) > Storage_dsMAX(o) + Storage_HFTRatio_HFT(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. 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

where:

M is the total mass in the volume;

M_i is the mass of the substance in the volume;

m is the mass of the sample taken; and

m_i 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

Rearranging this:

Equation 42

Proportional losses in a dead division are shared in the proportions of the stored water, therefore:

Equation 43

Reiterating equation (15), the mass balance of a routing division over a time step is:

Equation 44

Substituting equations (42) and (43) into (44), and rearranging the unknowns to the left hand side yields:

Equation 45

This can be rearranged to solve for an owner’s storage volume, Storage(o,t), as follows:

Equation 46

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 Payback SRG 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) < Storage_dsMAX).  Firstly the airspace volume to be filled is calculated:

Equation 47

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

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

Equation 50

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 sub-sections below).  The borrow and payback options available are:

• Fixed loss borrow and payback: Owners with a surplus that lend to others have their fixed loss Loss_fixed(o) increased by the amount loaned, and those with a deficit that borrow have their Loss_fixed(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 Loss_fixed(o) that will result in O(o) = 0 can be found from equation (23), re-expressed as follows:

Equation 51

For each owner, the values of surplus and deficit for borrow and payback are therefore:

Equation 52

Equation 53

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

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

Equation 56

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.

• • Owner percentages of dead storage: Ratio_ds(o)
  • Owner percentages of time series fluxes (for the fixed ratio option): Ratio_TS(o)
  • Owner percentages of other lateral fluxes (for the fixed ratio option): Ratio_loss(o)
  • Owner percentages of the high flow threshold (if applicable): Ratio_HFT(o)
• For each individual link, the following are input:
  • Initial owner shares of flow or active storage
  • Whether the time series flux is to be input per owner (if so, this overrides the ownership system method of sharing time series flux), or not.
  • If the time series flux is to be modelled by owner, Flux_TS(o), otherwise Flux_TS
  • If Murray style losses are to be modelled, the high flow threshold  above which high flow conditions apply.

Model Phase: Initialisation

When the model is initialised at the start of the scenario run, for a given link the ownership processing is initialised using the same values for all divisions in the link.  Each owner’s share of the initial storage volume is calculated.  In addition, if Murray-style losses are being modelled, the threshold storage volume and each owner’s initial share of this are determined, and the initial value of the volume of flow based flux caused by flow at the high flow threshold is also determined.

Model Phase: Ordering

No ownership calculations are involved in the ordering phase.

Model Phase: Flow distribution

In this phase, ownership processing for a link is performed after other link processing is completed.  Computational steps are summarised in Figure 2.

Figure 2. Flowchart- of main steps in ownership processing for a division in a time step

For every division, to determine owner volumes of inflow, lateral flux, outflow, storage and mass balance, the steps for each model time step are as follows:

1. Initialise temporary parameters.

2. Set each owner’s upstream inflow I(o) to equal that owner’s share of the upstream component’s outflow using equation (10). (The upstream component is the next division upstream, or if it is the first division in the link, the upstream node).

3. Determine the volume of the total flow based flux, fFlow_LG(), for this time step to be shared according to fixed ratio, expression or time series specified by the modeller.  Where relevant, also determine the volume, Loss_HF, that is to be shared using high flow rules (i.e. Murray-style losses), which is the volume that is in excess of the value of fFlow_LG().
   • If the ownership system uses Murray-style losses:

     1. The flux based on flow up to the high flow threshold is shared according to fixed ratio. This volume is:

        Equation 57

     2. The flux based on flow above the high flow threshold is shared according to high-flow rules. This volume is:

        Equation 58

   • Otherwise, all the flux is shared using fixed shares, expression or time series specified by the modeller:

     Equation 59

4. Determine every owner’s fixed flux total for this time step (Loss_fixed(o)), where the modeller has specified which of the link’s lateral fluxes (time series, general purpose flow based, groundwater and net evaporation) are fixed at the ownership system level (noting these must satisfy equation (11)).
   1. 1. Time series flux:
5. Determine the owner’s share of input time series:
   1. 1. 1. • If the time series flux for the current link is not input per owner, and for the ownership system time series flux sharing is by fixed ratio, apply the specified owner’s ratio to the total time series flux, 〖Flux〗_TS, that was calculated prior to ownership processing (〖Flux〗_TS has been adjusted to ensure total loss does not exceed the amount of water in the division).

   2. Time series flux:

6. Determine the total proportional flux, Loss_prop.
7. Determine the division’s state (‘dead’ or ‘live’).
8. Find the division’s storage using a method appropriate to the state of the division (i.e. whether ‘dead’ or ‘live’).  These methods are summarised in the two following sections.
9. Calculate each owner’s outflow, , using equations (15) or (16), as appropriate, rearranged so the term  is on the left hand side of the equation. 
10. Adjust owner shares of outflow as necessary to ensure that none are negative.  The principles are discussed in Ownership Adjustments, above.  In summary, if , the owner concerned borrows from other owners to ensure O(o)=0.
11.  Calculate owner mass balances to report, based on equation (15) or (16), as appropriate, using adjusted values of O(o)=0.

Data

Input data

Details on data requirements are provided in the Source User Guide.

Parameters or settings

Input parameters and settings for ownership specified at the level of the link (as distinct from input parameters and settings at the ownership system level which apply to all links, and nodes, in the ownership system) are summarised in Table 3.

Table 3. Link ownership parameters

Output data

Output potentially available is summarised in Table 4.

Table 4. Link Ownership Output Variables

References

Border Rivers Commission (xxxx) Border Rivers Bulk Water Sharing Plan

Commonwealth of Australia (2007) Water Act 2007 (Act No 137 of 2007 as amended, including amendments up to Act No. 46 of 2011 and SLI 2008 No. 106 (as amended by SLI 2011 No. 117)). Part 1A - The Murray‑Darling Basin Agreement.  Available at www.comlaw.gov.au/Details/C2011C00621/Download

NSW and Queensland Governments (2008) New South Wales – Queensland Border Rivers Intergovernmental Agreement 2008.  Available at www.derm.qld.gov.au/wrp/pdf/border/intergovt_agreement_2008.pdf and at www.water.nsw.gov.au/Water-management/Law-and-policy/Intergovernmental-agreements/Intergovernmental-agreements/default.aspx

                              i.        The flux based on flow above the high flow threshold is shared according to high-flow rules. This volume is: