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American options for energy balancing

Jan Palczewski (Leeds) John Moriarty (Manchester) Milton Keynes, 1st June 2015

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 1 / 18

Real-time market: supply curve and demand process

Reproduced from Stephenson and Paun (2001) (highlighting added): Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 2 / 18

UK Electricity Market Reform

UK is introducing a capacity market to supplement the real-time market. Reproduced from

www.gov.uk: Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 3 / 18

Questions

1

Can battery storage for grid balancing be a viable business?

2

How can such a scheme be organised?

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 4 / 18

Two actors

We model a two-agent problem of electricity balancing: The system (network) operator seeks to control deviations in the AC frequency away from 50Hz (i.e., imbalance between generation and demand). The provider of energy balancing services (storage operator) delivers energy to the grid under a ‘call off’ contract.

+ has battery capable of storing one arbitrary unit of energy + may buy electricity at any time at the prevailing market imbalance price, and without any price impact + the battery charge-discharge cycle incurs round trip costs + the battery charge-discharge cycle gradually decreases capacity

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Call-off contract

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The model

Ornstein-Uhlenbeck process (Xt)t≥0 models the imbalance price (time in days) dXt = θ(D − Xt)dt + σdWt Baseline level D = 60, θ = 0.77, σ = 20.81 (07/2011-03/2014, 8am) Exercise occurs when Xt ≥ x∗ = 70 Interest rate is r > 0 (3% per annum in examples) Aim: To value an American call option on real-time electricity with physical delivery. Approach: Real Options valuation for electricity storage (battery) operator (≡ expectation of discounted future cashflows)

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 7 / 18

OU process for APXUK data

20 40 60 80 100 20 60 100 Days OU Price 20 40 60 80 100 40 60 80 120 Days APXUK Price Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 8 / 18

Questions

Research questions:

1

The battery operator’s optimal strategy and expected payoff from a) one call option b) a sequence of call options (lifetime contract)

2

Is there mutual benefit? Difficulty:

1

Small price (=large excess supply) means cheaper purchase of power greater expected time until exercise when the strike price Kc is received Unclear whether it is optimal to buy energy when the price x is very small.

2

Payoff is non smooth (kink at x∗)

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 9 / 18

Mutual benefit

• S1. The storage operator has positive expected profit from the offer and exercise of

the option.

• S2. The option cannot lead to a certain financial loss for the system operator.

S2 is equivalent to: pc + Kc < x∗ Sufficient conditions: detailed analysis

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 10 / 18

The optimal stopping problem

(Xt)t≥0 the price process Time of exercise by the system operator: ˆ τe = inf{t ≥ 0 : Xt ≥ x∗} Expected net present value of the strike price: hc(x) = Ex{e−r ˆ

τeKc} =

   Kc, x ≥ x∗, Kc

ψr (x) ψr (x∗) ,

x < x∗. (1) So, optimally timing the purchase of electricity corresponds to solving the following optimal stopping problem with non-smooth payoff: Vc(x) = sup

τ

Ex{e−rτ − Xτ + pc + hc(Xτ)

• }.

(2)

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 11 / 18

Solution

We apply characterisation via concavity of excessive functions for one-dimensional regular diffusions due to Dayanik and Karatzas(2003).

Theorem

Under pc + Kc < x∗, an optimal stopping time is given by τ ∗ = inf{t ≥ 0 : Xt ≤ xL}, with xL < x∗.

−40 −20 20 40 60 −10 10 20 30 40 50 60 stopping boundary: £ per MWh present value £

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 12 / 18

Lifetime contract

Lifetime contract = an infinite sequence of call-off contracts back-to-back. Battery deterioration: A = 0.9999 remains after cycle

2 4 6 8 10 0.4 0.5 0.6 0.7 0.8 0.9 1.0 thousands of cycles Capacity

Value this contract using a fixed point argument. Provides a value of a storage unit on the balancing market.

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The optimal stopping problem

ξ(x) - the continuation value Time of exercise by the system operator: ˆ τe = inf{t ≥ 0 : Xt ≥ x∗} Expected net present value of the strike price: hξ(x) = Ex{e−r ˆ

τe(Kc + Aξ(Xˆ τe)} =

   Kc + Aξ(x), x ≥ x∗, [Kc + Aξ(x∗)] ψr (x)

ψr (x∗) ,

x < x∗. (3) Fixed point argument (non-linear equation for ξ) ξ(x) = sup

τ

Ex{e−rτ − Xτ + pc + hξ(Xτ)

• }.

(4)

Theorem

Under pc + Kc < x∗, an optimal stopping time is given by τ ∗ = inf{t ≥ 0 : Xt ≤ xL}, with xL < x∗.

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Lifetime value of storage

20 40 60 80 100 120 140 29240 29250 29260 29270 29280 current £ per MWh present value £

x∗ = 70, pc = 20, Kc = 40 = ⇒ xL = 39.4 expected time xL → x∗ is 2.63 days, expected time x∗ → xL is 4.72 days

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Is it worth optimising?

20 30 40 50 60 5000 10000 15000 20000 25000 30000 stopping boundary: £ per MWh present value £ Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 16 / 18

Summary and related work

Closed form solutions obtained for

(i) single, and (ii) lifetime American-type options with physical delivery to support real-time power system balancing.

Model calibration is possible from publicly available data in the UK John Moriarty, Dávid Szabó

(i) model imbalance process + price stack function, (ii) put options

Jhonny Gonzalez, John Moriarty: estimation (previous talk) Alessandro Balata, John Moriarty: numerical solution with periodic variations

Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 17 / 18

Carmona R. and M. Ludkovski (2010): Valuation of energy storage: an optimal switching approach, Quantitative Finance, 10:4, 359-374 Dayanik S. and I. Karatzas (2003). On the optimal stopping problem for

• ne-dimensional diffusions. Stochastic Processes and their Applications. 107:2,

173-212 Moriarty J. and J. Palczewski (2014). American Call Options for Power System

• Balancing. Available at SSRN: http://ssrn.com/abstract=2508258

Stephenson P . and M. Paun (2001), Electricity Market Trading. Power Engineering Journal, 277-288. Zervos M., T.C. Johnson and F. Alazemi (2013). Buy-Low and Sell-High Investment Strategies. Mathematical Finance, 23(3), 560-578

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