Competing stores in an arbitrage market Lisa Flatley Joint work - - PowerPoint PPT Presentation

Competing stores in an arbitrage market

Lisa Flatley

Joint work with Stan Zachary and James Cruise

1st June 2015

Lisa Flatley (University of Warwick) Competing stores in an arbitrage market 1st June 2015 1 / 15

Introduction

The main questions

Consider an electricity network which includes a number of energy stores. Each store earns its profits through arbitrage and the actions of each store will impact on the price of electricity. If the stores are competing against each other, how would they behave in order to maximise their profits? What is the resulting impact on the market?

◮ e.g. on prices and consumer surplus?

How does this compare with the case of co-operating stores?

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Introduction

Description of the competing stores model

Consider n stores, each with its own capacity (MWh), power ratings (MW) and round-trip efficiency (%). Assume prices p∗

1, . . . , p∗ T are known in advance, with no storage on

the system. Assume also that the impact of the stores’ actions on these prices is known, leading to a time-series of price functions: p1, . . . , pT. Here pt(x) is the price at time t if the total energy bought by all of the stores at that time is x MWh (where x < 0 if selling).

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Introduction

Formulation of the optimisation problem

For each time t, each store i chooses a quantity xit - the amount of energy it plans to put into its store at time t (or take out if xit < 0). The energy bought is h(xit) = xit if xit ≥ 0, or h(xit) = ǫxit if xit < 0 (taking round-trip efficiencies ǫ into account). Each store i chooses x∗

i = (x∗ i1, . . . , x∗ iT) to minimise its cost: T

• t=1

pt

 

n

• j=1

h(xjt)

  h(xit)

• ver all xi which satisfy the store’s capacity and power rating

constraints.

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Introduction

Discussion of the market model

Seeking a Cournot (or Nash) equilibrium: each store competes on

• quantities. An extreme example of supply function bidding (e.g.

Klemperer and Meyer (1989), Holmberg and Newbery (2010), Anderson (2013),...) The other extreme is Bertrand competition (competing on prices), but here each store would continue to undercut each other until they reach zero profits. A full game of supply function bidding would require a lot of information e.g. for each vector of prices (over time), each player declares a corresponding vector of quantities to trade.

◮ For two timesteps, this reduces to one-shot results in the literature. ◮ For a higher number of timesteps, this amount of information seems

unrealistic.

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Main results

Results: General price functions

Existence of a Nash equilibrium is guaranteed whenever the cost functions pt(x)x of each store are convex. Algorithm to identify optimal strategies. This algorithm also requires Lagrangian theory (extension of Cruise, Flatley, Gibbens, Zachary 2015)

◮ Sensitivity results are derived from the multipliers. Lisa Flatley (University of Warwick) Competing stores in an arbitrage market 1st June 2015 6 / 15

Main results

Results: Linear prices

We can say a lot more in the special case where prices are linear, of the form pt(x) = p∗

t + λtx.

◮ x is the total energy bought by all stores at time t ◮ p∗

t is the price before storage is introduced

◮ λt ≥ 0 is the market impact factor

Should be a good approximation when the stores are small in comparison to the entire network. Existence and uniqueness of a Nash equilibrium are now guaranteed. Can relate the actions of competing stores with the action of a larger single store...

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Main results

Linear prices - scaling of quantities trading

Theorem: Consider n identical competing stores, each with round-trip efficiency ǫ, and with given capacity, power ratings, and start and end levels of stored energy. Consider also a single store, also with efficiency ǫ, but with all other parameters increased by a factor (n + 1)/2. Then the energy traded by each competing store (at each time) is a fraction 2/(n + 1) of the energy that would be traded by the above single store.

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Main results

Linear prices - comparison between competing and cooperating stores

Assume n identical stores, each constrained to start and finish empty. Some immediate deductions in extreme cases: Large stores: competing stores trade more than cooperating stores by a factor 2n/(n + 1) ⇒ greater market impact, but less profit per store. Large n : again competing stores trade more than cooperating stores by a factor 2n/(n + 1).

◮ Increasing with n : extra competition makes stores work harder.

Large n : the price in the cooperating case remains unchanged by the addition of further stores but competing stores continue to affect prices.

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Main results

Linear prices - scaling of profits

Theorem: Consider n identical competing stores, each with efficiency ǫ and fixed start and end stored energy levels. If the capacity and power ratings never restrict the action of the stores, then the profit per store is proportional to 1/(n + 1)2. In the cooperating case, the profit per store is proportional to 1/n.

Lisa Flatley (University of Warwick) Competing stores in an arbitrage market 1st June 2015 10 / 15

Examples

Parameters

Huntorf parameters: 72.5MW charging, 290MW discharging, 580MWh capacity. Assume 80% round-trip efficiency (an adiabatic store). A linearised price function of the form pt(x) = p∗

t + λp∗ t x, with

λ = 0.01. Prices p∗

1, . . . , p∗ T are Elexon’s half-hourly Market Index Prices for

2014 (missing 1 day in March). Plots are for the first two weeks of June.

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Examples

Three identical competing and cooperating stores

7200 7300 7400 7500 7600 7700 7800 7900 100 200 300 400 500 600 Comparison of strategies for three stores Time (half hours) Level of stored energy (MWh) Competing stores Cooperating stores

Competing stores trade more than cooperating.

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Examples

Prices resulting from three identical stores

7200 7300 7400 7500 7600 7700 7800 7900 25 30 35 40 45 50 55 60 Comparison of prices for three stores Time (half hours) Price of electricity (GBP/MWh) Competing stores Cooperating stores

Competing stores have greater market impact than cooperating.

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Examples

Profits of identical stores

5 10 15 20 25 30 100 200 300 400 500 600 700 800 900 Comparison of annual profits per store Number of stores Annual profit per store GBP k Competing stores Cooperating stores

Competing stores earn less than cooperating.

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Conclusions and summary

Conclusions and summary

Algorithms derived from Lagrangian theory to identify optimal strategies for n competing stores with market impact. These constitute a Nash equilibrium under a reasonable market model. Existence when each cost function is convex. Uniqueness when prices are linear: a good approximation for stores which are not large compared to the whole system. Linear prices ⇒ behaviour of n identical competing stores can be related to a single larger store. In the above setting, competing stores tend to over-trade in comparison to cooperating stores, for a lesser profit.

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