Univercity of Canterbury Visually representing river chain capabilities for market purposes

MANAGEMENT SCIENCE HONOURS PROJECT 2001

Nicola McSporran

1. Introduction

New Zealandís electricity market has become increasingly competitive over the last few years. Consequently, some generating companies have been rethinking their approach to offer formation, focusing more on market issues. For one such generating company, this has meant that it is no longer desirable to expend a large amount of computational time and effort on details of their river chain during the offer process.

 

However, ignoring the physical details of the river chain altogether could result in the need to retract or amend an offer in later periods. Alternatively, it may mean that offers are conservative, since traders are unsure of the riverís capabilities. Therefore, some indication of the river chainís capabilities may still be desirable, provided it does not demand excessive time from traders or have high computational requirements.

 

This website outlines the results of a study that investigated ways that the river chainís physical capabilities could be visually represented for market purposes. It is intended for these results to provide ideas for further research and is therefore by no means comprehensive. In particular, ways of representing reserve capabilities have not been explored.

2.  Checking feasibility of a generation scenario

At some stage during the offer process, it will be necessary to check whether a generation scenario is going to be feasible. This feasibility test may be desired prior to submitting offers to the centrally coordinated market, by testing various forecasted demand scenarios against a proposed set of offers. Alternatively, it may be used once a pre-dispatch schedule is received from the market. This section outlines some possible ways that the feasibility of a particular generation scenario can be determined.

2.1 Physical constraints of the river

For the purposes of this discussion, the following physical constraints have been recognised:

 

In reality, there may also be:

         Upper and lower bounds on spill at each station,

         Time delays between when water is released and when it is received in a downstream reservoir or head pond,

         Tributary inflows into reservoirs or head ponds, and

         Limits on the rate at which storage levels, at reservoirs or head ponds, may change.

 

However, at this conceptual stage of development it was not considered necessary to include these constraints. They should be able to be included at a later stage with only minor adjustments to the models discussed.

2.2 Using a linear program

One way of checking the feasibility of a generation scenario is to run a linear program (LP), with the constraint that demand must be met. If there is no solution to the linear program then the generation scenario is not feasible. Linear programs have been used in the past to check the feasibility of offers prior to submission to the New Zealand Energy Market (NZEM). However, one of the criticisms has been that the resulting test was overly stringent and required too much computational time. Moreover, there was no consideration for when the infeasibility may occur. As the time before the infeasibility increases, the likelihood of it being a real problem decreases, due to increased demand uncertainty.

 

Ultimately, using a LP or mixed integer linear program (MILP) would still provide an accurate feasibility test, given a particular demand scenario Ė and the generating company that sponsored this project does have such a model. However, it is unlikely that a generating company will want to use such a model frequently during offer formation.

2.3 Using single storage trajectory diagrams

An appealing alternative to using technical tools, such as linear programs, is to have a visual representation of whether a generation scenario is feasible. This representation could be viewed along side visual representations of the market, letting traders know if they need to be mindful of the physical system when submitting offers.

 

In its simplest form, the visual representation could be a series of storage trajectories, one for each reservoir or head pond. By looking at these trajectories, traders could see whether any storage bounds will be approached, or even breached, during the day ahead. Moreover, colour, sound, or animation could be used on these diagrams to indicate if throughput at the station directly below the reservoir was nearing or exceeding maximum capacity, or its maximum rate of change in release. These single-reservoir storage trajectories would look similar to Figure 1 below, which shows that this reservoir is expected to reach its upper storage bound towards the end of the day, given a particular generation scenario.

 
Figure 1: A single storage trajectory plot, showing the levels of storage over time for a given generation scenario

 

One of the advantages of this visual representation is that traders can see when an infeasibility may occur, and therefore can use judgement to determine whether this is likely to be an imminent problem or not.

2.4 Using 2-reservoir storage trajectory diagrams

It would also be possible to display the storage levels at two reservoirs on the same diagram, as shown in Figure 2 below.

 
Figure 2: A two-reservoir storage trajectory diagram

 

In the above diagram, the pink and red sections indicate that throughput levels are nearing maximum capacity for at least one station. Therefore, traders know that they need to check their offers carefully. Over time, traders should learn to recognise the most useful angles from which to view this diagram, to gain the information they require.

 

In some situations, it may only be necessary to consider storage levels at two of the reservoirs in the river chain. Generally, this will be when it is highly unlikely that storage levels will be reached, during the day, at the other reservoirs in the river. When this is the case, Figure 2 could prove extremely useful as it provides all relevant information on one diagram.

 

When it is necessary to consider more reservoirs, the above diagram can still be used, by coupling reservoirs up the river. In other words, a separate diagram may be displayed for each adjacent couple of reservoirs in the chain.

 

A further advantage that this diagram may have over showing several single storage trajectories is that trade-offs between storage levels can be displayed. Moreover, it is possible to provide an indication of the flexibility that exists in that sub-section of the river. The following section discusses these trade-offs and flexibility issues in more detail.

3.  Providing information about the flexibility of the river chain

At any point in time, total generation can be produced by a number of different combinations of station output. Each different combination will result in different storage levels at the corresponding reservoirs. Therefore, at a given point in time, there is likely to be a range of different storage pairs that can be reached in Figure 2, while still maintaining the same level of total output.

 

For the generating company investigated, the following shape represented this range of storage pairs, at a given point in time:

 
Figure 3: Showing the range of storage pairs possible in a given period, while maintaining the same level of total generation

 

As can be seen in this above diagram, there is generally a trade-off between reducing the level of storage at one reservoir, and increasing the level of storage at the other.

 

By displaying this area around the existing storage pair, it is possible to indicate how much flexibility exists in that subsection of the river, at that point in time, given the current generation scenario.

 

Figure 3 shows alternate storage pairs that are possible in a single period. It is also possible to approximate the range of storage pairs that are possible in future periods, given the current storage levels and generation scenario. This range will get larger over time, as shown by Figure 4 below. 

 
Figure 4: Areas of possible storage pairs can be compounded over time, showing how much flexibility there is in the river, while maintaining the same level of output.

 

The red rectangular prism represents the upper and lower bounds on storage at the two reservoirs, and the blue-green shape within this prism shows possible future storage pairs that are possible, while still maintaining the same level of total output. The black line meandering through the diagram represents the storage trajectory for the existing generation scenario, as in Figure 2.

 

There may also be desired future storage positions to aim towards, especially when end-of-day (EOD) targets need to be reached. Therefore, the above diagram can be modified to show, not only, where you can get from a current position, but, where you need to be to reach the EOD targets; as demonstrated in Figure 5.

  
 
Figure 5: Showing traders both where they can go, and where they need to be to reach EOD targets

 

This time, the blue-green shape gets larger over time, but then narrows down again towards the end of the planning horizon, reflecting the need to reach the EOD target.

 

Clearly, the fatter this shape is, the more flexibility there exists in the river. If very narrow, this will indicate to the traders that it may be difficult to find a feasible solution, and therefore it will be necessary to pay careful attention to the details of the river chain during offer formation. If there is generally a lot of flexibility in the river, then it is foreseeable that this diagram may serve to reassure traders that they can continue to focus mainly on market issues during the offer process.

4.  Providing information about the remaining energy burst capabilities

In the above discussion, the river chain was deemed flexible if there existed a large number of alternate ways to produce the same level of total output, over the planning horizon. The flexibility of the river chain may also be measured in terms of the energy capabilities remaining in the river. Given a particular generation scenario, how much additional energy can be produced if demand turns out to be greater than expected, or if reserves need to be used to maintain frequency? Furthermore, how long can this energy Ďburstí be sustained?

 

Factors that influence the size and duration of an energy burst include:

         The current level of generation in each period of the burst

         The upper and lower storage bounds at each reservoir

         The throughput capacity at each station

         The rate of change of release constraints at relevant stations

 

It has been assumed, in this study, that the increase in energy released is constant throughout the duration of the burst. Moreover, the levels of activity before and after the burst are to be the same as in the generation scenario under investigation. Consequently, for stations with constraints on the rate of change in water released, the upper limit on throughput during the burst will look like Figure 6 below. 

 
Figure 6: The throughput levels before and after a burst may restrict the upper limit on throughput during the burst

 

The initial upward sloping line, in the above diagram, reflects the throughput level increasing as much as possible in each period of the burst. Once the stationís throughput capacity is reached, the upper limit levels for as long as possible. Then, towards the end of the burst, it is necessary to reduce the upper limit so that it is possible to reach the desired throughput level after the burst. If, during the burst, the station releases at these upper limits, then other stations will need to release more at the beginning of the burst and then reduce their release. This will ensure that the increase in generation remains the same for each period of the burst.

 

It is possible to determine the largest possible energy burst that can be sustained for a specified duration, by using a linear program (LP). By running a separate LP for each energy burst of a required duration, a graph can be produced, such as Figure 7 below. This graph shows the maximum energy increase possible, starting from a specific point in time, which can be sustained for the desired duration.

  
Figure 7: A graph can be used to display the size and duration of possible energy bursts

 

Interestingly, the size of the energy burst does not always decrease, as the duration of the burst increases. The trough at the beginning of the graph, followed by a rise, is the result of the upper limit on throughput at stations with rate of change in release constraints discussed earlier. For energy bursts of short duration, these stations are unable to increase throughput levels very much. However, as the duration increases, it becomes possible for these stations to reach higher throughput levels in the middle of the burst, and hence the size of the total increase in generation is larger. Once the station reaches its throughput capacity in the middle of the burst, throughput can no longer increase as the duration of the burst increases. Consequently, the size of the energy burst begins to decrease again.

 

There may be several of these graphs of interest representing energy bursts starting from different initial points. It is possible to display more than one graph on the same diagram by connecting the points on the graph to the initial period of the burst, as shown in Figure 8. The black series in this diagram is the same as in Figure 7, while the pink series represents the energy bursts that commence ten periods later.


Figure 8: By indicating when the energy burst begins, more than one graph can be included on the same diagram

 

If traders do not believe that there are sufficient energy capabilities remaining in the river, they may choose to adjust their set of offers accordingly. It also seems possible to extend these ideas to representing reserve capabilities, although this has not been investigated in this study.

5.  Displaying more than 2 reservoirs on a single diagram

It may be desirable to represent the state of all reservoirs and stations in a river chain on the one diagram. Figure 9 illustrates a way that information relating to spare throughput capacity, current throughput levels, rates of change in storage levels, and even time before a storage bound is reached, can all be included in one diagram. The discussion below should assist in interpretation of this diagram

 
Figure 9: Displaying all stations and reservoirs on one diagram

 

The tributary contribution at each station is the sum of all tributary inflows into reservoirs downstream of this station. The throughput capacity at the corresponding station is then added to this tributary contribution. These throughput capacities are shown as the black lines on the above diagram. The actual throughput levels at any point in time can be displayed over these black lines (the green lines), which will then also show the spare throughput capacity available at each station.

 

The purpose of the tributary contribution is to scale the axis at each station in such a way that the difference between the sum of the tributary contribution and the actual throughput level at two adjacent stations represents the difference between inflows and outflows at the reservoir between the two stations. When inflows exceed outflows, the reservoir is filling up, and when outflows exceed inflows, the reservoir is drying out.

If, on the other hand, the inflows and outflows are equal, then there will be no change in storage level at the reservoir, and the current level of activity could be continued indefinitely. If all reservoirs were balanced in this way, and hence the river chain was in balance, the green dashed lines displayed in Figure 9 would all be aligned.

 

The rate at which the storage levels are changing is represented by the size of the difference between inflow and outflows, Traders can get an indication of how long before a storage bound will be reached, if a particular rate of increase or decrease were to be maintained over several periods, by overlaying the spare storage capacities on Figure 9. These spare storage capacities (the red lines in Figure 9) are the difference between the current level of storage and either the upper or lower bound, depending on which bound is currently relevant.

 

The length of time before reaching a storage bound can be simply calculated as the spare storage capacity divided by the rate of change, and can even be estimated by visual inspection. For example, in the above scenario, the reservoir above station 5 will reach its upper bound in approximately three periods, whereas, the reservoir above station 4 will reach its lower bound in about ten periods; if the current rates of change were maintained.

 

The greatest advantage of this Figure 9 is that a large amount of information can be displayed and interpreted reasonably simply, and there is no restriction on the number of stations or reservoirs that can be represented.

6. Using multiple demand scenarios to account for uncertainty

The diagrams discussed above assume that there is a particular generation scenario to be investigated. This may be a pre-dispatch schedule returned from the New Zealand Energy Market (NZEM). Alternatively, it may be a scenario created by matching forecasted demand with a particular set of offers. Clearly there will be uncertainty in this forecasted demand; therefore, it is advisable to use more than one such demand scenario.

 

Multiple scenarios demand scenarios may be created by first creating confidence intervals around the forecasted demand schedule. There will possibly be more uncertainty regarding demand during peak periods and consequently, the confidence intervals may be wider in these periods. Different demand scenarios could then correspond to the maximum and minimum values of the confidence interval. For example, to simulate high demand, the upper limit of a 90% confidence interval could be used in each period. The associated generation scenario could then be determined by matching the demand scenario with the proposed set of offers.

 

The implications for Figure 2, for example, will be that there will be several storage trajectories meandering through the same diagram. Other diagrams may need to be rerun for different scenarios.

 

Conclusion

This study was conceptual by nature, and the diagrams discussed have not been tested to assess their usefulness or timeliness. However, several of the ideas seem worthy of further investigation, especially if it means that generating companies have the confidence to make timely offers, and seize more market opportunities that may present themselves.

 

About the author:

 

Nicola McSporran has a bachelor of Commerce degree majoring in Management Science / Operations Research, from the University of Canterbury, New Zealand. She is currently completing a bachelor of Commerce (honours) degree in Management Science, part-time, at the same university. This study was carried out as part of this degree. Dr. E. Grant Read was the supervisor for this project.

 

 

E-mail address:

Nlm17@student.canterbury.ac.nz