View source for EMK:Company Optimisation in EMarket

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== Introduction ==
This technical bulletin is intended to provide users and interested parties with a detailed explanation of how ''EMarket’s'' company optimisation feature works.

This paper includes a brief overview of other features in ''EMarket'' and a note on enhancements planned in the short to medium term.

=== Other Documents ===
This bulletin is one of a series of technical bulletins relating to Energy Link’s ''EMarket'' model.  Taken together, the bulletins replace the old ''EMarket'' User Guide.  The full series of bulletins covers an overview of the ''EMarket'' model,  the details of the four major New Zealand hydro systems modelled in ''EMarket'', water values and hydro offers, power flows, dispatch and nodal pricing, short term river chain optimisation and company optimisation.

== Company Optimisation ==
''EMarket'' is designed to produce stable offering schemes for generation that are, in principle, based on static short or long run marginal costs for each generator.  However, when a company owning a generation, retail and hedge portfolio is considered then the optimal offering behaviour for that company as a whole will affect the value of each generation offer.  ''EMarket'' can now adjust offers to simulate this behaviour, which is called Company Optimisation (CO).

=== Market Sensitivity ===
Market sensitivity is loosely defined as the change in price that can be affected by removing or adding generation to the available supply.  On an incremental basis, given a supply curve P(s), where P(s) is the price at which a supply of generation s is available, then market sensitivity can be defined as

[[File:dpds.jpg]] or the negative derivative of the supply curve.

In practice the supply curve is a lumpy, non-differentiable curve and is not all that useful.  In the following analysis we look for more general and approximate relationships between supply and price.

=== The Supply Curve ===
[[File:CO Figure 1.jpg|400px|thumb|right|Figure 1 - Average Market Sensitivity VS Price 26/9/00 - 03/10/00]]
Figure 1 is taken from supply curve data over the period of a week from the New Zealand market.  For a given price market sensitivity is estimated by first finding the generation available at that price and then finding the price at which 100 MW less generation is available.  This shows the average market sensitivity for a range of prices. The average standard deviation is shown by the dotted lines.

This particular week's data indicates that it is possible to estimate market sensitivity (MS) as a linear function of price:

MS = - dP(s) / ds &asymp;&alpha; p

In this case &alpha; = 0.005.

The analytical solution to [[File:dpdsp.jpg]] is P = e<sup>-&atilde;s</sup> so an exponential fit to the supply curve is indicated.

=== Company Optimisation Calculators ===
In the following analysis, location factors are ignored.  The problem of optimisation taking location factors into account is considerably more complicated and most of the benefits can be achieved without taking them into consideration, at least for longer term modelling (Optimisation on a short term basis, eg period-by period, is recommended using our EMarketOffer model with its optional Offer Optimiser).  The effect of reserve offers is also ignored.

==== Gross Profit ====
The gross profit including generation costs for a company in one period (reserve revenue aside) is given by:

&pi; = P<sub>E</sub>g - &mu;g - P<sub>E</sub>L + P<sub>S</sub>L + (P<sub>R</sub> - P<sub>E</sub>)*Q

where &pi; is gross profit, g is total generation, P<sub>E</sub> is the energy price, &mu; is the costs for generation, L is the retail load and P<sub>S</sub> is the retail price,  Q and P<sub>R</sub> are the hedge quantity and reference price respectively.

==== Optimum Gross Profit ====
With respect to generation, and treating P<sub>E</sub> as a function of g to account for market sensitivity, the optimum gross profit is given by:

&part;&pi;/&part;g = 0 =>

dP<sub>E</sub> / dg * (g - (L + Q)) + P<sub>E</sub> - &mu; = 0

==== Optimum Price/Generation Relationships ====
[[File:CO Figure 2.jpg|400px|thumb|right]]
Using the '''''Optimum Gross Profit''''' equation, where s<sub>0</sub> is the supply without any generation from the company in question.

P<sub>E</sub> = Exp(&alpha;(s<sub>0</sub> + g))

dP<sub>E</sub>/dg = &alpha; Exp(s<sub>0</sub> + g) = -&alpha;P<sub>E</sub>

which becomes

-&alpha;P<sub>E</sub> * (g - (L + Q)) + P<sub>E</sub> - &mu; = 0 or

P<sub>E</sub> = &mu; / (1 + &alpha;((L + Q) - g))

This gives a relationship between price and total generation for optimal solutions, given the marginal cost of generation.  Figure 2 shows an example of the optimum price generation relationships for a company with 500 MWh load plus hedges, with a marginal cost of generation at $20/ MWh, $40/ MWh and $45/ MWh.  Note that where generation equals load plus hedges, the optimum price is equal to the marginal cost.
[[File:CO Figure 3.jpg]]
Figure 3 illustrates how the price-marginal cost relationship determines optimal offering.  The figure show the optimal offering profile if the company has 300 MWh of generation with a marginal cost of $20/ MWh,  200 MWh at a marginal cost of $40/ MWh and 200 MWh at a marginal cost of $45/ MWh.

This shows that each block of generation will have an offer or offers determined by its marginal cost and the amount of generation with a lower marginal cost.  The optimal offer curve for each generation has to be approximated by a number of discreet offers and some compromise between optimality and complexity must be made.  For example, to make the simplest possible offers for the generators in figure 3, Gen 1 might be offered at $8/ MWh, Gen 2 at $35/ MWh and Gen 3 at $55/ MWh.