1. Submodularity in capacity auctions
Eddie Anderson
University of Sydney Business School
Joint work with Bo Chen (Warwick)
Lusheng Shao (Melbourne)

2. Outline
How does an energy only market ensure that generators recover their capacity costs?
Why are capacity markets introduced and how do they operate?
A new form of capacity mechanism that combines capacity and energy auctions
How will participants bid in this auction?
What does the equilibrium behaviour look like?
Why is submodularity necessary and what conditions will ensure that it holds?

3. The theory for energy-only markets
Total cost for different technologies e2 e3
Slope r the cost of not meeting demand e1
Slope c1 the marginal cost of supply
Hours of supply
This is called the screening curve

4. The theory for energy-only markets
Total cost for different technologies e1 e2 e3
Hours of supply

5. The theory for energy-only markets
Optimal mix of capacity to install
It is not worthwhile to add more of the high cost technology 3 to meet the demand in these few peak demand hours q1 q2 q3
Turns out these are optimal choices
The demand during a year laid out in decreasing sequence of demand: “load duration curve”
Hours of supply

6. The theory for energy-only markets
How much money is made
Money made by technology 2 in the hours when it is paid more than its costs exactly meets its capacity cost. Magic!
Hours of supply
technology 3 sets price of c3
technology 1 sets price of c1
days where there is insufficient supply: price is r
technology 2 sets price of c2

7. What is needed to ensure generators get full cost recovery?
The technology mix must be system optimal. When there is a non- optimal mix there no longer full cost recovery, i.e. there is an incentive for more of some technologies and less of others. So the analysis of full cost recovery is the same analysis that makes the market move towards an equilibrium with the right technology mix.
There needs to be a scarcity price (a line through the origin at slope VOLL that defines some high demands where there is rationing).
Generators must offer power at short run marginal prices.
Demand has to be distributed continuously (no flat sections in the load duration curve). We cannot have capacity built that exactly matches the demand for a non-zero number of hours. (A Lagrangian argument has dual variables as prices, and this would make prices undetermined).
Capacity is available in continuous amounts. (The optimal mix solves a continuous not a combinatorial problem).
Generators are risk neutral – they recover their costs but only in expectation.

8. What is NOT needed to ensure generators get full cost recovery?
We do not need non-zero short term prices. Everything works the same with wind having zero marginal cost.
We do not need a single future scenario for the load duration curve. Uncertainty in demand scenarios is handled by reinterpreting the load duration curve on as (which is just the inverse of where F is the distribution of demand.
We do not need a single future scenario for marginal costs. Everything works fine with uncertainty over future fuel prices (even if this changes the merit order) provided all the players agree on what the probabilities are.
We do not need guaranteed power availability. Everything is fine provided everyone agrees about the probabilities of an intermittent power generator being available under different scenarios.

9. Why might we introduce a capacity market to get cost recovery?
VOLL is very high and system scarcity events are rare. In an energy only market we see occasional hours with very high prices (given to everyone who is generating during those hours).
With an optimal mix of technologies the total amount paid out during these short periods is very large (it is enough to support the fixed cost components of the highest cost generation used).
PROBLEM 1 This translates to uncertainty in return which reduces investment
PROBLEM 2 Periods of very high prices are politically difficult if reflected in consumer charges.
PROBLEM 3 Intentionally planning for rationing and scarcity pricing is politically difficult, and measures that increase total capacity above optimal levels mean not enough time at high prices and generators do not recover their costs.

10. How do capacity markets operate?
Every country has their own version of a capacity market.
Simplest to think of the VOLL component being replaced by a fixed payment to all generators q3
Capacity payment made to everyone matches the cheapest capacity used. q2 q1
The energy market is left unchanged. So in a capacity auction the generators bid their capacity cost minus what they expect to make in the energy market

11. A new form of capacity mechanism
Rather than separating capacity and energy auctions into two stages, combined capacity and energy bids are submitted at the same time.
The system operator can determine the portfolio of bids to accept and does this in a way that minimizes expected total costs
Accepted bids are paid their cost for capacity at the start of the year. Generators must make available the capacity that has been purchased.
At dispatch generators are dispatched in merit order determined by their energy bids and are paid these energy prices.
Both capacity and energy operate on a pay-as-bid basis rather than having a uniform price.
The total cost for capacity is passed on to consumers (in the same way as for existing capacity auctions).
An average price is calculated for energy in any given 15 minute period and this is the amount paid for energy used.

12. An example of something similar: Short term operating reserve tenders to National Grid
Tenders in 2018 for Season 13.2 (which is 29 April 2019 to 19 August 2019) for power available within 20 seconds.
Rejected bids
Accepted bids
Utilisation price
£/MWh
Availability price £/Mw paid each hour

13. An equivalence to a supply chain problem
System operator aims to maximize welfare:
Buyer aims to maximise profit:
Consumers with varying and uncertain demand and VOLL = V
Generator 1
Transmission network
Generator N
Consumers with varying and uncertain demand pay V per unit
Supplier 1
Buyer
Supplier N

14. Setting up the model
For each generator there is a cost of capacity as a function of amount, and a cost of energy as a function of amount. We define marginal costs e(x) for capacity, and c(x) for energy.
e(x) = Marginal cost of capacity in $/MW
r(x) = Bid price for capacity in $/MW
p(x) = Bid price for energy in $/MW
c(x) = Marginal cost of energy in $/MWh
Generators bid into the market higher amounts (r(x) and p(x)) shown as dashed lines.
The system operator decides how much capacity to purchase. When demand occurs the system operator meets this using the cheapest energy available subject to the capacity purchased.

15. What does this look like with a total cost basis?
Ei(t) is the cost for generator i to provide an amount t of capacity, and Ci(x) is the cost for generator i to provide an amount x of energy. The generator bids prices Ri (t) for capacity and Pi (x) for energy.
Ri (t) = Bid price for capacity in $/MW
Generator profit is sum of these two amounts
Ei(t) = Cost of capacity in $/MW
Pi(x) = Bid price for energy in $/MWh
Ci(x) = Cost of energy in $/MWh
Energy amount dispatched from generator i
Capacity amount purchased from generator i

16. System operator choices
Let N be the set of generators. The system operator receives a set of bids
Given a choice for the capacity amounts reserved, demand D occurs and the ISO chooses amounts with the constraints
The system operator chooses the dispatch amounts x to minimize costs given the available capacities.
When there are fixed marginal energy prices pi then the system operator choice is to dispatch in merit order (cheapest first, up to capacity)
But when marginal energy prices are not constant (e.g. allowing generators to charge extra when they bring a new unit online) then the best choice is more complex.
We allow arbitrary choices for the dispatch amounts and simply say they depend on the demand.

17. Notation:
Given the generator bids and a given policy (t, x(D)) for the system operator we can calculate the expected social welfare as
The system operator maximizes this over policy choices (t, x(D)) given an available set of generators

18. Theorem 1: Best response by a generator
Given bids by the other generators, what is the response from generator i that gives it the maximum profit?
Define
Then the highest achievable profit is
(This is the additional benefit achieved when generator i becomes available and bids at cost. We can think of it as the extra value created by generator i given the bids of the other generators)
If generator i has preferred status (in the sense of its capacity being chosen when there is a tie) then an optimal offer is
If generator i does not have preferred status it takes epsilon off its capacity offer.
Offer at cost in the spot market
Make profits with capacity bids

19. Submodularity for the social welfare (supply chain profits)
Consider the overall social welfare (or supply chain profit) where the ISO makes choices based on the underlying cost structure.
If generators in S are available we have
where
The social welfare is submodular (supply chain profits are submodular) if for any we have
i.e. the value of adding generator i into the available set is reduced when the set is larger (left hand side has j available, right hand side does not)

20. Theorem 2: A Nash equilibrium solutionIf
(A) the set function is submodular; and
(B) there is a unique system optimal solution (t*, x*(D)) with
Then the following bids are a Nash equilibrium
In this equilibrium generator i makes a profit and the system operator makes the efficient policy choice (t*, x*(D))

21. Theorem 3: Conditions to ensure submodularityIf
(A) there are just two generators; or
(B) there are more than two generators and each generator has a constant marginal energy cost (so ci(x) = ci ) and a marginal capacity cost ei(t) that is non-decreasing
Then the function is submodular (so that the Nash equilibrium of Theorem 2 exists.)

22. Conclusions
This is a new form of capacity mechanism that is a long way from existing practice, but we investigate what it would imply.
By combining capacity and energy auctions there is an incentive for generators to bid their true short run marginal costs for the energy component and just make money on their capacity bids
This is true very generally: no assumption is required on marginal cost behaviour or on the market being competitive. This approach has potential to avoid the need for uplift payments arising from non-convexities
In the equilibrium each generator makes a profit that matches its contribution to the system. This is a VCG type result but arises not from mechanism design, but from a pay as bid auction with capacity and energy simultaneously
In equilibrium the system operator choices of capacity amounts are efficient.
The equilibrium results only hold when there is submodularity, but this can be proved to hold when marginal costs are well-behaved