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EQUILIBRIUM MODELLING OF
BENEFICIARY-PAYS
TRANSMISSION CHARGES
P ROF. A NDY P HILPOTT, D R . A NTHONY D OWNWARD
EPOC W INTER W ORKSHOP 2013
BACKGROUND / MOTIVATION
The EA has proposed that, in order to fund current/future
investments in the transmission grid, a beneficiary-pays scheme
ought to be introduced.
The fundamental aim of this scheme is that those who benefit from
the investment will be required to pay for the investment (in
proportion to their benefit).
While the aim of this pricing mechanism may be fair, problems arise
in actually being able to compute the benefits created by adding an
asset to the grid.
The key proposal was that SPD would be run, with and without an
asset (e.g. a transmission line), and the difference in an offers’ inframarginal rent would be treated as the benefit.
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OVERVIEW
• Beneficiary-pays Pricing
• Computation of Rentals / Benefits
• Incentives to Reduce Rentals / Benefits
• Supply Function Modelling
• Uniform Pricing
• Pay-as-bid Pricing
• Supply Function Equilibrium Examples
• Single-node – tax on rentals
• Two-node symmetric quadropoly – tax on benefits
• Conclusions
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Transmission Pricing Methodology Consultation Paper (2012)
BENEFICIARY-PAYS PRICING
1.
Run dispatch software with transmission asset and record
dispatch and price.
2.
Compute infra-marginal rental 𝝆𝟎 (𝒊) of agent 𝒊, for each 𝒊.
3.
Re-run software with transmission asset derated to represent
previous (counterfactual) state, recording new dispatch and
price.
4.
Compute counterfactual rent 𝝆𝑪 (𝒊) of agent 𝒊, for each 𝒊.
5.
Charge a proportion of benefits 𝑩(𝒊) = 𝝆𝟎 𝒊 − 𝝆𝑪 𝒊 + to
agent 𝒊. Suppose the cost of the asset is 𝑲. Then the proportion
of benefits paid by agent 𝒊 is:
𝑲
𝜶 = 𝐦𝐢𝐧 𝟏,
𝒊𝑩 𝒊
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Transmission Pricing Methodology Consultation Paper (2012)
BENEFICIARY-PAYS PRICING
Solve 1
Solve 2
Change
Demand
A + B + C +D
A
B+C+D
Supply
E+F+G
B+E
F+G–B
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INCENTIVES TO REDUCE CHARGES
After the transmission pricing proposal was announced, there were
concerns over the way the benefits would be computed.
Particularly, the ‘profit’ of a firm would be assumed to inframarginal rental.
Price
P
D
Quantity (MW)
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INCENTIVES TO REDUCE CHARGES
Thus, in a context where firms are charged based on infra-marginal
rentals, there may be incentive to mark-up infra-marginal offers so
as to reduce these rents.
We will explore these incentives through a supply function
equilibrium duopoly.
Price
Price
P
P
D
Quantity (MW)
D
Quantity (MW)
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INCENTIVES TO REDUCE CHARGES
EA TPM Consultation Presentation November 2012:
“Parties may be able to alter their offers to avoid the charge e.g. South
Island generators could reduce their beneficiaries-pay charge for Pole
3 by offering as if only Pole 2 was available.
“To the extent parties can do this it would reveal the asset is not
economically justified unless the SPD charge recovered costs from
other beneficiaries e.g. costs of Pole 3 may be able to be recovered
through the SPD charge from consumers.”
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SUPPLY FUNCTION AUCTION
Agents offer supply functions 𝑺𝒊 (𝒑) indicating how much they will
supply at price 𝒑. Let 𝒑𝒊 (𝒒) be the corresponding offer curve.
There is a demand curve 𝑫(𝒑) and a random demand shock 𝒉.
Demand realization occurs and all agents are paid:
a uniform price 𝒑 defined by the relation
𝑺𝒊 (𝒑) = 𝑫 𝒑 + 𝒉,
𝒊
for their respective dispatch quantity 𝑺𝒊 (𝒑).
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SUPPLY FUNCTION AUCTION
Price
D(p)
D(p)+h
∑S(p)
p
Quantity (MW)
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Anderson and Philpott (2002), Wilson (1979)
MARKET DISTRIBUTION FUNCTION
The market distribution function 𝝍(𝒒, 𝒑) defines the probability
that a supplier is not fully dispatched if they offer the quantity 𝒒 at
price 𝒑.
It can be interpreted as the measure of residual demand curves that
pass below and to the left of the point (𝒒, 𝒑).
Random residual demand curves faced by a supplier. Here 𝝍(𝒒, 𝒑) is
the probability of a curve being red.
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Anderson and Philpott (2002)
UNIFORM PRICE DISPATCH
The optimal offer curve 𝒑(𝒒) for a supplier with profit 𝑹(𝒒, 𝒑)
facing a market distribution function 𝝍(𝒒, 𝒑) maximizes:
𝐄𝑹
𝒒𝑴
=
𝑷
𝑹 𝒒, 𝒑 𝒒 𝒅𝝍 𝒒, 𝒑 𝒒
𝟎
+
𝑹 𝒒𝑴 , 𝒑 𝒅𝝍 𝒒𝑴 , 𝒑
𝒑(𝒒𝑴 )
+ 𝑹 𝒒𝑴 , 𝑷 𝟏 − 𝝍 𝒒𝑴 , 𝑷 .
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Anderson, Holmberg and Philpott (2013)
PAY-AS-BID DISPATCH
When the market clears at quantity 𝒒 for a supplier under a
particular demand realization, the supplier receives payoff:
𝒒
𝚷=
𝒑 𝒕 𝒅𝒕 − 𝑪 𝒒 .
𝟎
The expected payout earned from a offer curve 𝒑(𝒒) is then
𝒒𝑴
𝐄𝚷 =
𝒑 𝒒 − 𝑪′ 𝒒
𝟏 − 𝝍 𝒒, 𝒑 𝒒
𝒅𝒒 .
𝟎
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INFRAMARGINAL RENTALS
When the market clears at quantity 𝒒 for a supplier at price 𝒑 𝒒 ,
then the regulator observes a rent of:
𝒒
𝑷 𝒒, 𝒑 = 𝒒𝒑 𝒒 −
𝒑 𝒕 𝒅𝒕 .
𝟎
Given 𝝍, the total expected rent earned by the curve 𝒑(𝒒) is:
𝒒𝑴
𝐄 𝑷(𝒒, 𝒑) = 𝐄 𝒒𝒑 −
𝒑 𝒒
𝟏 − 𝝍 𝒒, 𝒑 𝒒
𝒅𝒒 .
𝟎
This is expected revenue from uniform pricing minus expected
revenue from pay-as-bid pricing.
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MODELLING A TAX ON RENTALS
Suppose that some fraction 𝒂 ∈ 𝟎, 𝟏 of these rentals is paid to the
regulator as a tax. The after-tax total expected payoff 𝚷 will be:
𝚷 = 𝐄 𝑹 𝒒, 𝒑 − 𝜶𝐄 𝑷 𝒒, 𝒑
𝒒𝑴
= 𝟏 − 𝜶 𝐄 𝑹 𝒒, 𝒑 + 𝜶
𝒑 𝒒 − 𝑪′ 𝒒
𝟏
𝟎
𝚷 is a convex combination of uniform price and pay-as-bid payoffs.
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SYMMETRIC DUOPOLY
Suppose we examine a symmetric duopoly:
• each firm has no costs, and capacity 𝒒𝑴 ;
• there is no demand elasticity (𝑫 𝒑 = 𝟎) and the demand shock
𝒉~𝑼 𝟎, 𝟐𝒒𝑴 + 𝝐 .
At the limit as 𝝐 → 𝟎, the equilibrium can be found to be:
𝑺 𝒑 = 𝒒𝑴
𝟐𝜶
𝟏−𝜶 𝒑
+
𝟑𝜶 − 𝟏 𝟏 − 𝟑𝜶 𝑷
𝟏−𝟑𝜶
.
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SYMMETRIC DUOPOLY
Symmetric equilibrium with no tax (black) and with 25% tax on observed profit
(blue). Generators mark-up low priced offers.
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SYMMETRIC DUOPOLY
No Tax
With Tax
Increase
Suppliers’ rentals
3,240
2,560
–680
Suppliers’ payoffs
6,480
6,344
–136
810
640
–170
Suppliers’ payoffs after tax
5,670
5,704
34
Consumer welfare
3,240
3,376
136
Tax
Welfare with 25% tax on observed rentals (times 4860).
Taking into account the tax on rentals, suppliers offer to improve
their actual after-tax payoffs. A side-effect of this is a transfer of
some wealth to consumers.
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TAX ON BENEFITS
However, in the proposed transmission pricing methodology, the
benefits would be computed based on the difference between the
rentals in the current market, and those computed for a
counterfactual without a transmission asset.
In this context, the incentive to increase one’s offer curve is reduced.
Price
P
P’
D’
D
Quantity (MW)
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INCENTIVES TO REDUCE CHARGES
However, in the proposed transmission pricing methodology, the
benefits would be computed based on the difference between the
rentals in the current market, and those computed for a
counterfactual without a transmission asset.
In this context, the incentive to increase one’s offer curve is reduced.
Price
Price
P
P
P’
P’
D’
D
Quantity (MW)
D’
D
Quantity (MW)
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INCENTIVES TO REDUCE CHARGES
However, in the proposed transmission pricing methodology, the
benefits would be computed based on the difference between the
rentals in the current market, and those computed for a
counterfactual without a transmission asset.
In this context, the incentive to increase one’s offer curve is reduced.
Price
Price
P
P
P’
P’
D’
D
Quantity (MW)
D’
D
Quantity (MW)
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INCENTIVES TO REDUCE CHARGES
However, firms may be incentivised to mark-up supra-marginal offers.
Consider the situation where a transmission investment has allowed
additional supply into a node.
• In the current market, a generator at that node is unlikely to be
dispatched for their final tranche;
• whereas in the counter-factual they could be.
Price
D(p)
counterfactual
pc
pc
p
Quantity (MW)
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TRANSMISSION EXAMPLE
• Four (identical) suppliers.
• Independent uniform demand shocks, 𝒉𝟏 and 𝒉𝟐 .
• With a line capacity of 𝑲 the market distribution function is 𝝍𝒄 .
• A grid investment increases the capacity to ∞, changing the
market distribution function to 𝝍.
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PREDICTED BEHAVIOUR?
The supplier benefit in a demand outcome giving 𝒒, shown shaded
for two candidate offer curves. The counterfactual (e.g. a lower
capacity line) reduces dispatch to 𝒒𝒄 .
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TRANSMISSION EXAMPLE
We model line expansion as a change in market distribution function from
𝝍𝒄 to 𝝍.
Recall that the benefits are the difference in rentals.
𝑩 𝒒, 𝒑 = 𝝆 𝒒, 𝒑 − 𝝆𝒄 𝒒, 𝒑 .
Thus, the expected benefit is:
𝑬 𝑩 𝒒, 𝒑
= 𝐄𝝍 𝝆 𝒒, 𝒑 − 𝑬𝝍𝒄 𝝆𝒄 𝒒, 𝒑
𝒒𝑴
= 𝐄𝝍 𝒒𝒑 −
𝟏 − 𝝍 𝒒, 𝒑 𝒒
𝒅𝒒
𝟎
𝒒𝑴
− 𝐄𝝍𝒄 𝒒𝒑 −
𝒑 𝒒
𝒑 𝒒
𝟏 − 𝝍𝒄 𝒒, 𝒑 𝒒
𝒅𝒒
𝟎
So the expected payoff will be
𝚷 𝒑 𝒒
= 𝐄 𝑹 𝒒, 𝒑 − 𝜶𝐄 𝑩 𝒒, 𝒑
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TRANSMISSION EQUILIBRIUM
Now let us consider the symmetric equilibrium:
• each firm has no costs, and capacity 𝒒𝑴 ;
• there is no demand elasticity (𝑫 𝒑 = 𝟎) and the demand shock
at each node is 𝒉𝒊 ~𝑼 𝟎, 𝟒𝒒𝑴 + 𝝐 .
At the limit as 𝝐 → 𝟎, the equilibrium can be found to be:
𝑺 𝒑 = 𝒒𝑴
𝒑
𝑷
𝒕 𝜶
,
𝒕 𝜶 =
𝜶 𝟏𝟎𝑲 + 𝟔𝒒𝑴 − 𝟓 + 𝟏
𝟑
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TRANSMISSION EQUILIBRIUM
𝟑
𝟏
Supply function equilibrium when 𝑲 = 𝟖 , 𝐪𝐌 = 𝟒:
• with no benefits tax (black)
• 25% benefits tax (blue)
• 100% benefits tax (green)
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CONCLUSIONS
On the surface, if you consider a single dispatch point, clear
incentives to avoid transmission charges exist.
However, since the charge is based on benefits, the incentive to
increase infra-marginal offers is limited.
Ability of firms to mark-up in a supply function, with many demand
realisations is restricted.
Furthermore, competition restricts ability of firms to mark-up to
reduce transmission charges.
FUTURE EXTENSIONS
Asymmetric players
Demand response
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