4. Process for the calculation of price and volume signals
4.3. Optimisation formulations for SRP
The optimisation of the SRP procurement for some player can be stated as follows:
[
( , )]
min ) , ( minf u x ≡ π u−B u x (1)Where π, u and B (u,x) are the cost and benefit of using the volume of SRP u, respectively. Specifically:
• π Price of SRP, €/MW • u Volume of SRP, MW
• B(u,x) Benefit of procuring the SRP, €. This benefit function includes the option of doing nothing u = 0 and should reflect the potential benefits of using other sources of flexibility.
Equation (1) is subject to a “generic” set of constraints, which states how the need or needs of the market player must be satisfied by the SRP.
b
x
u
F(
,
)≤
(2)We note at that stage that (2), is general enough to encompass requirements, which are modelled by equalities as well. This can be done using appropriate combined “greater than or equal to” and “less than or equal to” inequalities.
A first analysis of the optimisation problem by Lagrange method leads to the following:
λ
π
Tu
F
u
B
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
∂
∂
=
(3)Equation (3) states that at the optimum the price the player is willing to pay for the SRP product is
equal to the marginal value of player’s benefit from using the SRP
u
B
∂
∂
adjusted by marginal penalties
imposed by its active technical and commercial constraints
⎟⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
−
λ
Tu
F
. It is worth noting that in
theory, the marginal benefit of consuming the SRP(i.e.
u
B
∂
∂
) is not necessarily higher than π.
ADDRESS Technical and Commercial Conceptual Architectures - Core document ADD-WP1-T1.5-DEL-EDF-D1.1-Technical_and_Commercial_Architectures
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SRP product. Obviously, that amount depends on the value of u, the desired SRP volume. Any price below for the given volume would be acceptable to the player.
The SRP product is standardised so it can be provided by any other player, and not just by aggregators. Therefore, a market for SRP with a specific delivery time is made up of many suppliers with different offering prices for the same product.
The problem in (1) and (2) is generally a non-linear constrained optimisation problem. From a mathematical viewpoint, it can turn out to be hard or easy to solve it depending on the specific shape and properties of functions
B(u,x)
andF(u,x)
. In any case, in the literature general approaches to such problems have been largely studied. These include penalty methods and barrier methods. Penalty and barrier methods have complementary advantages and disadvantages. The best choice also depends on the specific functions.In particular, an example of an iterative approach is presented in Appendix G to solve the problem and find a pair [u, π ], i.e. a pair (volume, price), that optimises the objective function of the player while fulfilling the constraints.
It is not unreasonable to see also that a given player may wish to “draw” an explicit relationship between its willingness to pay for an active demand product and its corresponding optimal volume. This process boils down essentially to computing the value of the following parametric optimisation problem [equivalent to (1) and (2)] over a specified range of prices
π
∈[ ]π,π
.( )
[
u
B
u
x
F
u
x
b]
u
*π
=
argmin
π
−
(
,
):
(
,
)≤
(4)Where the function
u
*( )π
corresponds to a map of the optimal product volume for a given priceπ
. The pairs(π,
u
*( )π
)
can then be plotted to give a request curve. A example of such curve is given in Figure 21.We note that this may be the preferred way players may wish to attack AD product procurement because of the visual aspect of formulating a request curve. This kind of exercise may also be useful to reveal radical changes in optimal AD procurement decisions as the willingness to pay is varied. “Radical changes” refer here to the appearance of discontinuities in the optimal volume (i.e. jumps in the volume for a very small change in the price parameter). Furthermore, future markets for AD or flexibility products in general may call for the submission of request curves by potential buyers rather than single price-volume pairs. Thus, this is giving further credence to this approach.
* ( )
u
π
π
Figure 21. Example SRP demand curve obtained by computing the optimal SRP volume (crosses) at regular intervals
ADDRESS Technical and Commercial Conceptual Architectures - Core document ADD-WP1-T1.5-DEL-EDF-D1.1-Technical_and_Commercial_Architectures
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Another issue to consider is the formulation for time-coupled SRP. Players may be facing highly complex issues, which inevitably evolve over time. Therefore, players may have needs to procure AD products with different requirements over successive time intervals. Moreover, as technical and commercial constraints may also be dynamically coupled over time, AD product request behaviour in one time period also becomes coupled to the request behaviour in all other periods. The player then faces the problem of having to specify sequences of price-volume pairs or request curves, which reflect those couplings and would therefore result in AD market procurement outcomes, which are also consistent with all of its constraints.
The optimisation principles necessary here are identical to those introduced above for the single time- period case. Fundamentally, the only difference lies in the multi-dimensionality of the search for the optimal price and volume signals. Difficulties arise, however, because the process becomes a combinatorial search. For instance, the request for an AD product in one period becomes a function of not just the price in the given period as it now depends on the prices in the other periods as well. This topic is further discussed in Appendix G.