EMK:Water Values in the EMarket Model - Revision history

Sisep: /* The Mathematics of Water Values */

2012-11-28T22:33:14Z

The Mathematics of Water Values

Sisep: /* The Mathematics of Water Values */

2012-11-28T22:32:58Z

The Mathematics of Water Values

Sisep: /* The Mathematics of Water Values */

2012-11-28T22:32:05Z

The Mathematics of Water Values

Sisep: /* The Mathematics of Water Values */

2012-11-28T22:22:10Z

The Mathematics of Water Values

Sisep: /* The Mathematics of Water Values */

2012-11-28T22:17:45Z

The Mathematics of Water Values

Sisep: /* Multi-Reservoir Effects */

2012-11-28T22:14:16Z

Multi-Reservoir Effects

Sisep: /* Hydro Offers in EMarket */

2012-11-28T21:29:25Z

Hydro Offers in EMarket

Sisep: /* Implied Shortage Cost for a Reservoir */

2012-11-28T21:22:18Z

Implied Shortage Cost for a Reservoir

Sisep: /* Implied Shortage Cost for a Reservoir */

2012-11-28T21:20:01Z

Implied Shortage Cost for a Reservoir

Sisep: /* Contour Calculation Stage */

2012-11-28T21:07:35Z

Contour Calculation Stage

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 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:
-''P<sub>t</sub> = L &equiv; Marginal Water Value''
+'''''P<sub>t</sub> = L &equiv; Marginal Water Value'''''
 [[EMK:Water Values Bulletins | Back to Water Values Technical Bulletins]]

← Older revision		Revision as of 22:32, 28 November 2012
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−	[[EMK:Water Values ~~Technical~~ Bulletins \| Back to Water Values Technical Bulletins]]	+	[[EMK:Water Values Bulletins \| Back to Water Values Technical Bulletins]]

← Older revision		Revision as of 22:32, 28 November 2012
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	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 r<sub>t</sub>. If the term is zero for all r<sub>t</sub> then the hydro generator has no market power and is a price taker.		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 r<sub>t</sub>. If the term is zero for all r<sub>t</sub> then the hydro generator has no market power and is a price taker.
	[[File:WVEM Figure 5.jpg\|400px\|thumb\|right\|Figure 5]]		[[File:WVEM Figure 5.jpg\|400px\|thumb\|right\|Figure 5]]
		+	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 r<sub>t</sub> 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, r<sub>t</sub> 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 r<sub>t</sub> 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 P<sub>t</sub>, 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:
		+
		+	''P<sub>t</sub> = L &equiv; Marginal Water Value''
		+
		+
		+	[[EMK:Water Values Technical Bulletins \| Back to Water Values Technical Bulletins]]

← Older revision		Revision as of 22:22, 28 November 2012
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	Note that we can use the derivative of P<sub>t</sub> rather than the partial derivative, since for our purposes P<sub>t</sub> is a function only of r<sub>t</sub> at each t.		Note that we can use the derivative of P<sub>t</sub> rather than the partial derivative, since for our purposes P<sub>t</sub> is a function only of r<sub>t</sub> 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 r<sub>t</sub>. If the term is zero for all r<sub>t</sub> then the hydro generator has no market power and is a price taker.
		+	[[File:WVEM Figure 5.jpg\|400px\|thumb\|right\|Figure 5]]

← Older revision		Revision as of 22:17, 28 November 2012
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	== The Mathematics of Water Values ==		== 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:
		+
		+	[[File:EWV 5.jpg\|170px]]
		+
		+	for all t between T<sub>1</sub> and T<sub>2</sub>, where ''L'' is a constant – the marginal water value.
		+
		+	Note that we can use the derivative of P<sub>t</sub> rather than the partial derivative, since for our purposes P<sub>t</sub> is a function only of r<sub>t</sub> at each t.

← Older revision		Revision as of 22:14, 28 November 2012
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	== Multi-Reservoir Effects ==		== 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 ==

← Older revision		Revision as of 21:29, 28 November 2012
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	[[File:EWV 3.jpg\|140px]] 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 [[File:EWV 4.jpg\|60px]]. For example, if the security setting is 1-in-60 then the implied shortage cost is 60 times the average nodal price.		[[File:EWV 3.jpg\|140px]] 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 [[File:EWV 4.jpg\|60px]]. 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 ==	+	== 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.
		+	[[File:WVEM Figure 4.jpg\|400px\|thumb\|right\|Figure 4 - Waitaki Offers Example]]
		+	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 ==

← Older revision		Revision as of 21:22, 28 November 2012
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	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, [[File:P.jpg\|17px]], 1 inflow sequence reaches bottom which means that the average nodal price is given by		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, [[File:P.jpg\|17px]], 1 inflow sequence reaches bottom which means that the average nodal price is given by

−	[[File:EWV 3.jpg\|140px]] 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	+	[[File:EWV 3.jpg\|140px]] 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 [[File:EWV 4.jpg\|60px]]. 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 ==

← Older revision		Revision as of 21:20, 28 November 2012
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	=== Implied Shortage Cost for a Reservoir ===		=== 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, [[File:P.jpg\|17px]], 1 inflow sequence reaches bottom which means that the average nodal price is given by
		+
		+	[[File:EWV 3.jpg\|140px]] 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

@@ Line 112: / Line 112: @@
 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<sub>c</sub>'', the number of sequences resulting in shortage is equal to [[File:PIPN.jpg|40px]] where [[File:P.jpg|17px]] is the average nodal price over all inflow sequences modeled in the simulation phase of contour calculation.
+A corollary of this is that, starting from any given contour of price ''P<sub>c</sub>'', the number of sequences resulting in shortage is equal to [[File:PIPN.jpg|34px]] where [[File:P.jpg|17px]] is the average nodal price over all inflow sequences modeled in the simulation phase of contour calculation.
 === Implied Shortage Cost for a Reservoir ===