Water Values in the EMarket Model
Water Values Defined
The concept of a water value is useful to an owner of some hydro electric generation which has some storage because it tells the owner exactly how much the next MWh of generation is worth at any point in time. Knowing this, the owner could offer their hydro generation into a spot market, for example.
Alternatively, if the owner operated an entire utility which had both hydro and thermal plant, then they could establish a place for the hydro in their "merit order", more simply the order in which the plant should be dispatched given its marginal cost of generation. The assumption here is, of course, that cheaper plant has more merit than expensive plant and should therefore be dispatched first.
The marginal cost of thermal plant is made up of fuel and other variable costs of generation. Hydro electric plant has either small or negligible marginal costs, so the water value is effectively an opportunity cost of operation – this occurs because a MWh of hydro electricity can, in principle, displace a MWh of thermal generation.
Two key characteristics of hydro electric schemes tightly govern what the owner may do with the scheme. One is that storage is limited and the other is that inflows are finite. Hydro generators have strictly limited supplies of water with which to generate.
Consequently, at any particular point in time, the hydro owner must decide if they use water in storage to generate now, or store it longer to use later. Thus the water value can be thought of as the expected future value of water in storage. The hydro owner should take the opportunity to generate whenever the nodal price received for generation is equal to or exceeds the water value.
Although we talk of water value, more correctly we should refer to marginal water value, which is defined as the expected future value of the next cubic meter of water arriving as storage for generation. This implies that the water value should be expressed, for example, in dollars per cubic meter of water. In practice, however, it is more convenient to express it in $/MWh.
Water Value Contours
The useful concept of the "operating guideline" arose from the development of various models for ECNZ in New Zealand. An operating guideline is a curve on a chart of total storage for the reservoir in question versus time. The guideline establishes the hydro generator's place in the merit order at any given time of the year.
For example, the Huntly operating guideline shown, for any given time of year, the storage level at which Huntly should be operating, assuming a centrally planned system. At any time during the year Huntly should operate if storage is at or below the thick black operating guideline. In practice, Huntly actually would come on progressively over a range of storage around the guideline. In addition, ECNZ's models calculated guidelines for all major thermal plant based on storage in the two islands.
Another way to think of the operating guideline is a curve joining points of equal water value – in this case the value of generation from Huntly. Thus, by definition, an operating guideline is a (marginal) water value contour. In designing EMarket, Energy Link decided to continue with the concept of an operating guideline because it is a useful way of visualizing how water value changes with storage and time.
The chart to the left shows typical water value contours for water in the big storage lakes of the Waitaki hydro electric system, Lakes Pukaki and Tekapo, as produced by EMarket. Each contour relates to the offer band from a thermal generator modeled in EMarket. At any given time in the year, if storage is exactly on a contour then it is obvious what it's water value is. If storage is between two contours then the water value is linearly interpolated between the two contours. For example, if storage is one third of the way between two contours of $50/MWh contour and $75/MWh, respectively, then the water value is simply
2/3 x 50 + 1/3 x 75 = $58.33 / MWh
Water Value Theory
Water value contours arise from the process of optimising the use of water in storage lakes, typically upstream of one or more hydro power stations. In fact, the water value contours in EMarket allow the (simulated) hydro owner to calculate water values and simulated offers that are optimal, in a convenient way.
The optimisation of a hydro generation scheme with seasonal storage is different to the optimisation of a thermal generator for three main reasons:
- the "fuel supply," ie total inflows, is finite in any given year;
- inflows are highly volatile;
- storage is finite.
By contrast, thermal generators usually (but not always) have access to unlimited amounts of fuel on call, albeit at perhaps widely varying prices, which can be stored by suppliers in many locations. The three extra restrictions on hydro generators make the optimisation of a hydro system with seasonal storage more difficult and more interesting than operating a thermal generator.
There is one further constraint on hydro generation which is particularly important in New Zealand, where storage is relatively small compared to total inflows. The constraint is known as the "1-in-N" dry year security criteria, where N is in years. For example, the government's policy statement on electricity includes a 1-in-60 security criteria for dry year security of supply. In principle, this means that a shortage situation, in which demand curtailment becomes necessary or desirable, should only occur 1 year in 60 on average. The 1-in-N constraint is included in EMarket's water value calculations for each of the four major hydro systems modeled in EMarket. The four major hydro systems listed below are actually modeled in some detail.
- Waitaki
- Major storage lakes Pukaki and Tekapo; inflows into lakes Pukaki, Tekapo, Ohau, and the Ahuriri River; eight power stations some with smaller storage lakes immediately upstream.
- Clutha
- Major storage Lake Hawea; uncontrolled storage and inflows into lakes Wanaka and Wakatipu; other inflows from the Shotover River; Clyde and Roxburgh power stations each with small storage upstream.
- Waikato
- Lake Taupo with major inflows; eight power stations, each with small storage upstream, modeled downstream, of Taupo; five tributaries inflows modeled below Taupo.
- Manapouri
- Partially controlled storage Lakes Manapouri and Te Anau; power station at West Arm of Manapouri; outflows also down the Mararoa River.
At this point we resort to some simple maths, but only for a while, and ask the question - if a hydro station has operated for one year from time T1, at which point it started out with storage of S1, until time T2 and storage of S2, then what would have been its optimal set of releases at each point in the year?
We use the term release to refer only to water released from storage to generate electricity, any other water leaving storage being classed as spill past one or more stations. In the following we make the assumption that generation is a constant function of release. In fact, generation in MW is related to release in cumecs by a number that is approximately constant for most power stations, at least over a wide range of output.
The time step used to calculate the water values is typically one week, so at some week t between T1 and T2 the revenue from release is given by:
Rt = Pt x rt
where Pt is the generator's average weekly nodal spot price at time t. Total revenue for the year is given by:
for all weeks between T1 and T2
Marginal costs for hydro electric stations are very low, so if a hydro owner wishes to optimise gross profit over the year then they can just as easily optimise the revenue function Rtot.
A potentially important consideration is the effect of the release on the nodal price, ie whether the hydro generator has any market power. In practice a higher release will often have a negative effect on the price, although for small amounts of generation this effect will usually be negligible. But to account for this possibility the nodal prices can be expressed as a function of release, written mathematically as P = P(r). The hydro revenue equation becomes:
for all t weeks between T1 and T2
Before we can maximise revenue for the year we have to account for constraints on what we can achieve with our hydro electric station:
- maximum and minimum releases at each station;
- maximum and minimum storage;
- conservation of mass - at each step within the year the storage increases by inflows less releases and spill;
- minimum flow below stations, eg for resource consent;
- 1-in-N dry year security criteria.
At this point we could get very mathematical – but we won’t. Suffice it to say that we can apply a technique known to mathematicians as the method of “Lagrange multipliers” to come up with a simple result. If we assume that the generator either has no market power or chooses not to use it if they have it, then the result simply states that the hydro owner will maximise revenue over the next week when he releases at a constant marginal water value equal to the expected future value of water.
Calculating Water Value Contours
The marginal value of water in each reservoir is potentially dependent on a number of factors. Principally among these are the time of year and the amount of storage in the reservoir concerned. Another major factor is the amount of storage in each of the other major reservoirs. Given this information there are a number of ways of calculating the value added by extra water in a given reservoir.
The optimal hydro behaviour can be expanded upon now:
- when minimum and maximum storage constraints are not in effect optimal hydro releases are based on a constant marginal water value;
- at any given week, as long as the effect of release on price is small the optimal release will be achieved by offering in at the current marginal water value.
In this way, disregarding the use of market power at this stage, the behaviour of a hydro system can be determined by its current marginal water value. In order to determine the marginal water value for all values of storage, it can first be observed that the marginal water value for an over-full reservoir is zero. This is not to say that a generator can not spill at a price above zero, eg to keep the price up and maximise revenue by reducing output, which might typically occur when a reservoir is nearing full. The point here is that when the reservoir is already totally full then the marginal value of another cubic meter of water flowing into the reservoir is zero.
The marginal water value for an empty reservoir is harder to define, but when a reservoir is empty the value of an extra MWh of storage should be at least the nodal price that would occur if the system's generation was removed from the market. The final marginal water value for an empty reservoir can be expressed as a shortage cost, CS. The shortage cost for a reservoir is calculated from the average nodal price for the year for the reservoir and its security factor.
Given a marginal water value at any time of year, a projection of future dispatches and inflows can be made using forecast demand, offers from other generators and historical inflow data. A different projection will be made for each inflow sequence. For a given starting storage a certain number of these projections will eventually reach maximum storage and the others will eventually hit the bottom of the reservoir - this typically happens after a few months. Since optimal releases are based on a constant marginal water value in between the two storage limits, the average resulting value of the marginal water should be equal to the starting marginal water value. This is the criterion used by EMarket for determining marginal water value's.
The figure at right illustrates how this works, though it must be noted that EMarket does not use this method of projecting storage forward to actually calculate the water value contours. The figure shows storage versus time of year starting from time T1 and projecting forward for a number of historical inflow sequences. The dashed black lines are water value contours - storage at T1 is equal to S1 which is on a water value contour. The thin traces are the result of dispatching forward with the different inflow sequences. Most inflow sequences meet one or other storage constraints within a few months.
Every inflow sequence that hits the top of the reservoir, resulting in spill, must have had a marginal water value of zero, given our result that constant marginal water value is optimal. Every inflow sequence that hits the bottom of the reservoir, resulting in shortage, must have had a marginal water value of the shortage cost, CS.
The figure actually shows a storage buffer zone, an input into EMarket, which allows for the fact that extra measures will be taken by hydro companies to avoid shortage once shortage becomes likely. A buffer size in GWh is entered as the 'Low Buffer' input for the Waitaki hydro system only. Once storage is in this region then Waitaki takes action to reduce releases, reducing the maximum output by a set MW per GWh ratio entered as the 'Reduction Ratio'. The effect of this is to scale down releases as storage falls within this buffer zone which. For calculation of Waitaki's water values in EMarket, the top of the Low Buffer region is taken as zero storage.
Thus, the marginal water value at S1 is equal to the average of the marginal water values of all inflow sequences at their respective end points at either maximum or minimum storage. The position of the contour at time T1 is obtained by adjusting the contour position until the average water value over all inflows is equal to the price of the contour.
While this section has illustrated how EMarket calculates water values, the calculations are not actually done in this way, and a shortage cost does not need to be calculated, though the end result is the same. An advantage of using projections that hit either the top or the bottom of the reservoir is the ability to include the 1-in-N security criteria directly in the calculation of water values. This method also captures the value of time correlations within inflows.
Calculating the Marginal Water Values
EMarket actually contains four major hydro systems for which contours must be calculated. Because of the time it takes to make a full multi-dimensional estimate of marginal water values as a function of the storage at all four major systems, EMarket first calculates the marginal water values for the major hydro reservoirs given that there is no knowledge of the storage in other reservoirs.
The aim of the calculation of marginal water values in EMarket is to determine the position of marginal water value contours that correspond to the prices of offers from the non-hydro offers included in EMarket simulation runs. This leads to a method that is fast and produces results that converge quickly and accurately.
The algorithm used has two stages, which are applied in successive iterations for each major hydro system: simulation and contour calculation. For the purposes of both stages, a simplified dispatch of generation offers is used which stacks offers in order of price but also constrains maximum generation on each island according to the constraints on the HVDC link. The simplified dispatch includes a fixed allowance for losses on the Grid.
Simulation Stage
The first stage is a simulation of the entire system over all inflow years. Each of the four major hydro generators offers in as much generation as it can at its current marginal water value, which is determined by the initial marginal water value contours. If a system reaches maximum storage it offers over-full storage in at a price of zero. When storage gets low, maximum generation is cut accordingly. All other offers are constant over the year but are calculated based on an assessment of how the offers vary over the year. The offer quantities, prices and spill are all recorded for the entire set of historical inflow sequences available to EMarket. These results are used solely to provide the contour calculation phase with a realistic set of offers to work with for all weeks and inflow years - in a wet year more must-run offers may occur and in a dry year generation may be limited from some systems.
Contour Calculation Stage
In principle, each contour's position could be calculated for each week using the method of forward projection described above. However this would be time consuming, so a faster method is used which also does not require the shortage cost to be calculated.
The contour prices are chosen to be those of the thermal offers in the simulation phase. In the contour calculation phase, for any given contour for any given hydro, a constant offer is made for the hydro at the appropriate thermal offer price. This offer is dispatched against the thermal offers and the offers calculated for the other hydros in the simulation phase, over all inflows sequences, and together with the inflow records is used to calculate a relative storage trajectory for all years of inflow data, starting with the last week of inflow data and working backwards. The process is repeated over all inflows to get a converged, "back projected" trajectory spanning all I inflow years. This process is a form of optimisation known as dynamic programming since it starts at an optimum storage position and works back from that assuming the optimum offer strategy is to offer at constant marginal water value.
Using this trajectory it is then possible to quickly choose the storage value at each week such that the average outcome over all inflows, either spill or shortage, is equal to the contour price, our condition for optimality. Above the contour, more inflow sequences will result in spill and less in shortage, and vice versa below the contour.
A corollary of this is that, starting from any given contour of price Pc, the number of sequences resulting in shortage is equal to 
where 
is the average nodal price over all inflow sequences modeled in the simulation phase of contour calculation.
Implied Shortage Cost for a Reservoir
Let's write the security setting for the reservoir as 1-in-N, as it appears in EMarket's inputs, and assume that there are I years of historical inflow data available for EMarket to use in calculating water values. Then the security factor is defined simply as N.
As an example, if N = I then there is 1 inflow from I that reaches bottom, which is the top of the Low Buffer region for Waitaki, zero storage for all other reservoirs. In this case, starting from a contour priced at the average nodal price, , 1 inflow sequence reaches bottom which means that the average nodal price is given by
assuming of course that all inflows are equally likely from that point. This equation can be easily rearranged to show that the shortage cost is 
. For example, if the security setting is 1-in-60 then the implied shortage cost is 60 times the average nodal price.
Hydro Offers in EMarket
We can now see that the owner of a hydro generator with storage should calculate the marginal water value, which turns out to be a function of time of year and storage, and in principle offer all releases at that price. EMarket calculates the water value contours in one go for an entire year, after which the water value can be looked up week-by-week during any simulation run. By virtue of the method used in EMarket, the water values include the cost of providing the target security setting in each of the four major hydro systems.
It might seem contradictory, but the water value is not actually constant throughout the year, although this would be the case if inflows and all thermal offers were known in advance. But inflows are highly volatile and EMarket has to work with historical inflow data going back at least to the 1930's. At the start of a week the water value is read from the water value contours, but by the end of the week, due to uncertainty in inflows, storage may be higher or lower than what was expected at the week's start. Thus the water value will most likely be different again at the start of the following week.
In practice, hydro stations have a more complex offer structure than indicated simply by examining water value contours, and this is mirrored in EMarket. For example, a minimum flow requirement at a station will be offered as must run, which means the generator output that achieves the minimum flow will be offered at a very low price to ensure dispatch.
Offers above must run in EMarket reflect water values, but modified to account for changes in the value of water downstream from the main storage lakes. These changes may or may not be large, but reflect the additional value, or loss of value, due to constraints and storage down the river chain.
Offers for hydro systems in EMarket reflect more than just the marginal water value. For example, a set of offers for the Waitaki system are shown at right during a week when the water value is $99.29/MWh. There are 5 offer bands priced from $1/MWh up to $215/MWh, but only the third band of 1,180 MW is priced at water value.
Band 1 reflects the need to satisfy the must run requirement of 120 cumecs at the Waitaki power station. However, this requirement only amounts to about 20 MW at the Waitaki dam, so the rest of the band is a reflection, in this particular instance, of all the water moving through the complex Waitaki river chain to satisfy the minimum flow constraint.
The second band in this example is 180 MW at $87/MWh and in this case is most likely a reflection of the water value of Lake Tekapo relative to Lake Pukaki.
The fourth and fifth bands are priced at $199/MWh and $215/MWh, respectively, and are most likely a reflection of the need to spill past some stations, reducing the overall efficiency of the Waitaki system, when output goes above 1,600 MW.
The number, size and price of these offer bands varies during each simulation run with EMarket as the water value changes and as the distribution of water within the Waiatki river chain changes. When Waitaki is dispatched, EMarket ensures that the movement and storage of water along the river chain is correctly updated. In this way, the behaviour of each of the major hydro systems can be modeled with some considerable detail in EMarket. An refinement on EMarket's basic hydro offers is provided by the optional "STRCO" feature, standing for Short Term River Chain Optimisation. The STRCO option performs a dynamic programming optimisation on offers from stations along the entire river chain, with the water value as a key input.
EMarket does not take account of market power when calculating water values, i.e it assumes all hydro generators offer at their marginal water value without using market power. EMarket provides other, more flexible facilities for the user to model and experiment with the use of market power, for example the optional Company Optimisation feature which optimizes gross profit across all generators, hedges and retail load modeled for a generating company. Effectively, EMarket calculates water values in the same way as a centrally planned utility would if they assumed the modeled thermal offers used in the calculations were actually marginal costs for those generators.
Multi-Reservoir Effects
Tests indicate that the marginal water value at Waitaki tends to be a function of total storage in the country and that marginal water values at the other reservoirs are more or less independent of this except for when total storage is very low. It is conceivable that this would result in significant cheap generation occurring when all systems are running low even though they all face a greatly enhanced probability of shortage. However, assuming the burden of dry year security lies with the Waitaki system, it appears from our work that this system can behave almost independently and will carry all the vital storage for the other systems. This is what occurs in all simulations of the algorithm described above so it appears that enhanced shortage costs can be isolated to the Waitaki system without loss of generality.
For example, if Manapouri were nearly full when Waitaki was empty, then the marginal water values for Manapouri calculated in EMarket could cause it to offer too cheaply. However, typically what happens is that other lakes empty out before the main Waitaki lakes, matching what is implemented in EMarket. Observations of the market over a number of years confirm that Waitaki is operated in such a way that storage never gets into the very low region.
It must also be noted that the optimization of water values does not explicitly account for unexpected outages of major plant, surges in demand, or line outages, so prudent operation of Waitaki in particular, necessitates allowance for a storage buffer in case of such an eventuality. This further reduces the likelihood of operation of Waitaki near empty. EMarket includes adjustments for Waitaki, centered on the concept of the low storage buffer, where generation is removed gradually in order to protect supply. This raises marginal water values near the bottom of Waitaki as would a full multi-dimensional water value calculation.
Similarly, when a reservoir is full EMarket includes an input parameter for Waitaki which allows water values to reduce earlier in order to avoid spill.
Effect of Hedges and Risk Aversion
If the hydro system owner also has hedges or retail load then the calculation of marginal water value is changed only to the extent that the incentive to use market power is reduced, which means that EMarket’s method ignoring market power is still valid.
A point to note here is that the water values do not take any account of risk aversion on the part of the hydro generator, e.g. the generator may value greater certainty over revenues than achieving the maximum possible revenue. Some allowance is made for the user to adjust for the impact of risk aversion using other features within EMarket’s hydro modeling input set.
The Mathematics of Water Values
Using the method of Lagrange multipliers, assuming we are clear of constraint conditions such as being at minimum or maximum storage, we arrive at a result involving only a relatively simple differential equation:
for all t between T1 and T2, where L is a constant – the marginal water value.
Note that we can use the derivative of Pt rather than the partial derivative, since for our purposes Pt is a function only of rt at each t.
It is worth noting here that dP/dr is always zero or negative and indicates the market sensitivity at time t for a release of rt. If the term is zero for all rt then the hydro generator has no market power and is a price taker.
When a generator has market power then the curve relating nodal price to release might look like the complex curve shown to the right. The flat portion of the curve at medium and high values of rt might represent a portion where the generator is on the margin, setting the price.
At the point shown where the tangent line intersects the curve, rt has reduced to the point, effectively by reducing the amount offered into the market, where the generator is no longer on the margin and the price is being set by a higher priced generator. At this point, of course, dP/dr is the slope of the curve.
In the case of a generator exercising market power, it can then be concluded from the above equation that when the release is not constrained to a maximum or minimum the nodal price will be greater than the marginal water value. The rt multiplier on the first term of our differential equation indicates that when the nodal price rises, release will be increased. If dP/dr is very small compared to Pt, that is the nodal price is minimally dependent on the release, then the nodal price will be more or less equal to the marginal water value and as such, constant over storage trajectories where release is not constrained.
This shows that optimal behaviour can be achieved by offering at a price that consists of a constant base price representing the marginal water value and an extra price that is added when nodal prices can be significantly raised by reducing release, i.e. by exercising market power.
In EMarket, the market power component of the water value equation is ignored when water value contours are calculated to give our result that hydro output should be offered each week at constant marginal water value:
Pt = L ≡ Marginal Water Value