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This function is not available in Source (public version). |
Introduction
Linear programming techniques assume that the data used is perfect. In particular, when activating these features, you are assuming that the objective function and constraint coefficients are correct, rather than the best estimates available. If this assumption does not hold, the solutions found may be sub-optimal.
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Solvers use arc-node networks as their inputs. Source creates arc-node networks automatically for each time-step from the node-link network in the Schematic Editorthe Schematic Editor. There may be differences between the solution found by the solver ("as predicted") and flows as modelled during the flow phase ("as released"). The flow distribution phase resolves these differences by considering the ordering phase as having provided a minimum target to aim for. The optimum flow for each arc is determined by the netLP so that water will flow along the least cost pathway.
About costs
The solvers use a scheme of costs to determine optimal flows. Costs To choose network linear programming as the ordering algorithm, choose Edit » Scenario Options Ordering Algorithm. This opens the Ordering Algorithm section of Scenario Options (Figure 1). To configure netLP, choose Iterative solution to network linear program, this will make the rest of the display active and allow you to configure options for netLP.
Figure 1. Enabling ordering
About costs
The solvers use a scheme of costs to determine optimal flows. Costs can range from -1.1E13 through to +1.1E13. The larger the cost, the greater the disincentive for water to flow along an arc. Conversely, the smaller the cost, the greater the incentive for water to flow along an arc. For example, in the above list of solver priorities, satisfying evaporative and transmission losses have the lowest cost, and therefore the highest incentive, in any model. The question of relative costs becomes relevant when defining cost functions for storages that are being operated in harmony.
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- RELAX-IV is generally faster than PPRN, particularly when PPRN is being run without side constraints. However, under certain circumstances, both PPRN without side constraints and RELAX-IV may become bogged down in excessive iteration;
- PPRN supports side constraints. RELAX-IV does not*. If side constraints are crucial to the correct outcome, or to avoid sub-optimal solutions, or to control excess iteration, RELAX-IV can not be used; and
- PPRN uses real numbers (at the precision of the underlying hardware) whereas RELAX-IV works with integers. Selecting the RELAX-IV algorithm implies integer conversion of the real numbers used internally by Source, during the optimisation process. Conversion implies rounding. Source scales values automatically before sending them to RELAX-IV to minimise loss of precision, and reconverts results returned by the solver to the proper range.
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| Note: You should not change the solver algorithm once your model has been calibrated. Changing the solver algorithm will invalidate your calibration. |
About side constraints*
Side constraints are implemented by translating between requirements set in various nodes and links in the Schematic Editor to the arcs and nodes used by the solver. In other words, you do not need to do anything to configure side constraints. Simply choosing PPRN with side constraints activates the necessary translations. For example, a Loss node creates an arc with high incentive that forces the solver to accept a particular loss.
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- Head versus outlet capacity relationships for storages, providing that the change in head across a single time-step does not also cause a change in the linear relationship (cross a control point in a piecewise linear function);
- Flow dependencies that cause excess iteration; and
- Circular constraints that prevent the solver from converging on an optimal solution.
Configuring optimisation
*side constraints have not yet been fully implemented and tested in Source
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To configure optimised ordering, begin by choosing Edit » Ordering Network Costs... or click Configure Ordering on the Ordering toolbar and choose Network Costs.... This opens the Network costs dialog (Figure 31).
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1. Network costs (summary)
Creating cost functions
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The graph on the right hand side of Figure 31 provides a graphical representation of the selected storage cost. You can also simultaneously display several functions based on your selection of storage cost functions from the list on the left hand side. The lower the cost (Y-axis) the greater the incentive to retain, or carry-over the associated storage volume (X-axis) to the next time-step. The same information is presented in the table, and can be exported to a .CSV file if required.
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Use the Storage Break Points tab (Figure 42) to design cost functions. Begin by using the Storage Cost Function popthe Storage Cost Function pop-up menu to select a cost function that you defined in the Cost Functions list in thethe Summary tab. Next, select the storage that should be associated with this cost function.
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42. Network Costs (Storage Break Points)
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| Although you can define storage breakpoints for storage A in terms of storage B, you should avoid doing so because it can lead to infeasible solutions. |
By default, Source provides four rows in the Break points table. Each row is associated with a percentage of the full supply storage volume, which is visible, and a cost, which is not visible. In the Break pointsBreak points table shown in Figure 42:
- The bottom-most 10% of the capacity of the storage is associated with a base cost of -11000.00;
- The next 40% (50%-10%) of the capacity of the storage has a cost of the base cost plus one unit of increment. Here, the increment is 10, so the cost will be -10990;
- The next 30% (80%-50%) of the capacity of the storage has a cost of -10980 (base plus two increments); and
- The remaining capacity of the storage has a cost of the base plus three increments: -10970.
A more formal specification of the cost calculation is:
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Recall that negative costs are incentives. Accordingly, in this table, the greatest incentive is to retain any water in the bottom-most 10% of the capacity of the storage (for carry-over to the next time-step), followed by the water in the next 40% of the capacity of the storage. By interleaving base costs and increment values, releases from multiple storages can be controlled quite precisely to maintain a desired balance.
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Table 1. Storage breakpoints (data file format)
| Row | Column (comma-separated) | |
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| 1 | 2 | |
| 1 | Carry over number | Active storage (%) |
| 2..n | n | point |
where:
n is an integer in the range 1..n representing the carry-over number.
point is the storage percentage (eg. “10” for 10%) when carry-over arc n takes effect.
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Table 2. Storage cost functions (example)
| Storage Carryover Arc | Storage 1 | Storage 2 | ||
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| Base Cost | -11000 | Base Cost | -11001 | |
| Increment | 2 | Increment | 2 | |
| Storage % | Arc Cost | Storage % | Arc Cost | |
| 1 | 5.6 | -11000 | 7.4 | –11001 |
| 2 | 11.1 | -10998 | 14.8 | –10999 |
| 3 | 16.7 | –10996 | 22.2 | –10997 |
| 4 | 22.2 | –10994 | 29.6 | –10995 |
| 5 | 27.8 | –10992 | 37.0 | –10993 |
| 6 | 33.3 | –10990 | 44.4 | –10991 |
| 7 | 38.9 | –10988 | 51.9 | –10989 |
| 8 | 44.4 | –10986 | 59.3 | –10987 |
| 9 | 50.0 | –10984 | 66.7 | –10985 |
| 10 | 100.0 | –10982 | 100.0 | –10983 |
The result of the design intention is shown in Figure 53. Note that, in the absence of any inflows that replenish the storages, the X-axis can also be interpreted as expressing time.
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53. Carryover Arc Costs vs Storage Volume
Applying cost functions
You apply cost functions in the Storage Targets tab (Figure 64). Each storage in your model is represented by a row in this table. The remaining columns contain pop-up menus where you can connect an appropriate cost function to a selected storage for a particular month.
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64. Network costs (Storage targets)
You can also import storage targets from a .CSV file. The format of the file is shown in Table 3. Note that the column ordering in the .CSV file does not match the display in Figure 64.
Table 3. Storage targets (data file format)
Row | Column (comma-separated) | |
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| 1 | 2..13 | |
| 1 | Storage | month |
| 2..n | sname | cname |
Where
month is the first three characters of the month of the year (eg “Feb”)
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You can specify the number of shortfall arcs that are generated for each of the demand nodes using the Number of Shortfall Arcs drop down menu (Figure 75). The default number is 4.
Figure
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5. Network
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costs (Demand
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priorities)
You can control the order in which shortfalls are satisfied. The Priority column in Figure 75 shows which demand component of the model has will receive water via its shortfall arcs.
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Table 4. Demand priorities (data file format)
| Row | Column (comma-separated) | ||
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| 1 | 2 | 3 | |
| 1 | Demand | Priority | Computation Sequence |
| 2..n | sname | pri | seq |
Wherewhere:
sname is the name of the supply point
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Table 5. Shortfall arc costs (example)
| Scenario | Supply point | Priority | Computation sequence | Shortfall arc | Satisfaction order | |
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| Number | Cost | |||||
| A | 1 | 1 | 1 | 1 | 11000002 | 2 |
| 2 | 11000004 | 4 | ||||
| 3 | 11000006 | 6 | ||||
| 4 | 11000008 | 8 | ||||
| 2 | 1 | 2 | 1 | 11000001 | 1 | |
| 2 | 11000003 | 3 | ||||
| 3 | 11000005 | 5 | ||||
| 4 | 11000007 | 7 | ||||
| B | 1 | 1 | 2 | 1 | 11000001 | 1 |
| 2 | 11000003 | 3 | ||||
| 3 | 11000005 | 5 | ||||
| 4 | 11000007 | 7 | ||||
| 2 | 1 | 1 | 1 | 11000002 | 2 | |
| 2 | 11000004 | 4 | ||||
| 3 | 11000006 | 6 | ||||
| 4 | 11000008 | 8 | ||||
| C | 1 | 2 | 1 | 1 | 11000001 | 1 |
| 2 | 11000002 | 2 | ||||
| 3 | 11000003 | 3 | ||||
| 4 | 11000004 | 4 | ||||
| 2 | 1 | 1 | 1 | 11000005 | 5 | |
| 2 | 11000006 | 6 | ||||
| 3 | 11000007 | 7 | ||||
| 4 | 11000008 | 8 | ||||
NetLP ordering at nodes
In Source, networks are composed of arcs which have maximum capacities and costs associated with them. These costs can either be positive (disincentive) or negative (incentive). The optimum flow for each arc is determined by the netLP so that water will flow along the least cost pathway.
Storage node
While the minimum and maximum order times have to be calculated upstream of each relevant node to the relevant storages, the system also has to calculate the downstream order time which is the timeframe time frame over which the node must forecast the downstream orders for in advance.
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| Note: In a netLP system, refer to Operating targets when setting up a storage node. |
Controlled splitter node
For the splitter node, in the netLP order phase, the main stream has two parallel arcs - a forced flow arc and the natural flow arc. The capacity of the forced flow arc in the mainstream is the opposing flow volume of the maximum effluent flow. The maximum effluent flow is dependant on the inflow to the controlled splitter node. The effluent side of the controlled splitter also has a forced and a natural flow arc, with the capacity of the forced flow arc the minimum volume of flow down the effluent side.
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To ensure that the above are met, you can configure each of the branches with a cost incentive. In a maximum flow constraint node that is immediately below the controlled splitter on one side of the stream, this will ensure that the flow rules are met. The maximum flow constraint should be configured with parallel arcs with either a cost for flowing water down the effluent or a cost incentive for flowing water down the main stream. The cost should be small, to ensure that the overall cost structure of the network is not impacted too much. Figure 12 shows 9 shows an example of this.
Inspecting an arc-node network (simple)
It is possible to inspect textual representations of the arc-node networks that are generated by Source for processing by the solver. The steps are:
- Select the active scenario in the Project Hierarchy (Project Explorer);
- Locate the Arc-Node Network entry in the Model Parameters list (also Project Explorer) and enable it for recording;
- Run the scenario; and
- In the Values column of the Recording Manager (Complex time series, summary), locate and open View Multiple Supply Path Setup. The window will be similar to Figure 9.
Figure 9 is a work in process so most of the labels are inaccurate. The graph provides a measure of computational complexity (iterations to solve) at each time-step. Note that the peaks are clipped to the value of the Maximum Iterations parameter in Figure 1.
Figure 9. Inspecting the generated arc-node model
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| Note: Ensure that the forecast supply is greater than the total order size. If this is not the case, the following error message will occur at the first node that this occurs at: Forecast supply greater than total order size in interpretation of network LP. Try checking inflow forecasts at inflow nodes, and that all confluence inputs are defined to be regulated. |
Troubleshooting It is possible to visualise the arc-node networks that are generated by Source using third party software. For more information please see: Troubleshooting and Debugging