EMK: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

Figure 1 - Huntly Operating Guideline

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.

Figure 2 - Waitaki Water Value Example

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:

1. the "fuel supply," ie total inflows, is finite in any given year;
2. inflows are highly volatile;
3. 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:

R_t = P_t x r_t

where P_t 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 R_tot.

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, C_S. 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.

Figure 3

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, C_S.

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 P_c, 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.