Retail Electricity Markets with Risk Aversion and Asset Swaps

Retail Electricity Markets with Risk Aversion and Asset Swaps

A. Downward, D. Young, G. ZakeriUniversity of Auckland.

EPOC Workshop: Optimization under Uncertainty

• Motivation

• Risk Aversion Background

• Risk Aversion Example

• Retail Market Model– Differentiated Products Model– Single-node Example

• Two-Node Model Inspired by New Zealand

• Conclusions

Overview

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MotivationWolak Report

In early 2009, Frank Wolak’s report to the Commerce Commission was released.

It highlighted some shortcomings of the NZEM, including:– limited competition for thermals in dry years,– only one firm with generation in both islands.

It suggested that asset swapping may improve market outcomes.


MotivationDistribution of Generation

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Genesis

Meridian

Mighty River

Contact


MotivationMinisterial Review

The Electricity Technical Advisory Group produced a discussion paper that presented three asset swap proposals.

In December 2009, the government stated its intent to transfer:Tekapo A & B to Genesis, andWhirinaki to Meridian.

Virtual swaps were also proposed, where contracts for energy in either island are compulsorily traded.


Retail MarketOverview

• In this work, we investigate how risk-aversion affects entry and pricing of retailers in electricity markets.

• Consumers enter into contracts with retailers, reducing the risk that they would otherwise face buying from the spot market.

• Retailers compete with each other for the same consumers through mainly price competition.

• Retailers must pass on the risk of purchasing from the spot market to consumers.


Retail MarketMarket Risks

There is significant risk involved in participating an electricity retail market.

Retailers purchase electricity at the spot price and sell to consumers at predetermined fixed prices.

In New Zealand, vertical integration is common; this acts as an internal hedge against spot price fluctuations.

However, even with generation, transmission price risk may still be of concern.


Retail MarketTransmission Price Risk

For example, consider the following situation.

The profit for the firm is:

Profit = 100p1 – 50p1 – 50p2 + Retail Payments

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50 MW retail contracts

50 MW retail contracts

100 MW generation


Retail MarketModel Overview

The full model consists of three stages:

– Entry – here firms make 0/1 decisions regarding whether they have a retail base at each node.

– Retail competition – each firm sets a retail price at each node.

– Wholesale market – the uncertainty is resolved and wholesale prices and profits are computed.


Risk AversionDefinition

Often in stochastic optimization problems we wish to maximize expected profits, given various scenarios that may eventuate.

This may be a fair thing to do if the problem is repeated daily – within a week any positive outcomes will likely balance out the negative outcomes.

However, if the problem being solved involves a one-off decision or covers an extended time horizon with limited recourse decisions, it may be important that the firm protects itself from the worst-case outcomes.

In such a situation the firms may wish to behave in a risk-averse manner.



For example, consider a firm with a binary decision of whether to purchase an asset costing $50,000.

Depending on whether the firm chooses to purchase the asset, the density functions for its returns may look like:



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Purchase No Purchase

Expected Return $100,000 $80,000

Probability of Loss 2.3% 0.003%

VaR5% -$17,758 -$47,103

CVaR5% $3,135 -$38,746


Risk AversionRisk-adjusted Profit

ρ(Profit) = (1 – θ) E[Profit] – θ CVaR5%[Profit]


Risk AversionRisk Averse Newsboy Example

Suppose that a newsboy needs to decide how many papers to buy in the morning at $0.30 each. Unfortunately, his demand is uncertain and has the following three equally likely outcomes:

Low demand: dL = 80 papers (pL=1/3)

Medium demand: dM = 100 papers (pM=1/3)

High demand: dH = 120 papers(pH=1/3)

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Choi and Ruszczyński



If the newsboy were risk-neutral then his optimization problem would look like:

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13max 0.3

. . , , ,

, , ,

0, , ,

0

L M H

i i

i

i

y y y x

s t y d i L M H

y x i L M H

y i L M H

x



The optimal solution to this problem is to purchase 120 papers. The number sold in each of the scenarios are:

yL=80, yM=100, yH=120.


Shapiro et al. (2009) discuss the relationship of weighted mean deviation from quantile to conditional value at risk, in mean-risk optimization problems.

With some algebra, this can be reduced to a weighted sum, maximizing expected profits and minimizing conditional value at risk.

Risk AversionWeighted Mean Deviation from Quantile

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1 1

1 1

| |

min : E max 1 ,

=E max 1 ,

1 E EZ Z F Z Z F

r Z Z Z

F Z Z F

Z Z

1|

E 1 E E

1 E CVaR

Z Z FZ r Z Z Z

Z Z


This formulation of CVaR can be solved using convex optimization.

Using a substitution of the form:

We can formulate an optimization problem for a risk averse agent:

Risk AversionMean-risk Measures

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maxE 1

,

0

, , 0

Z v w

Z v w

Z f x

g x

v w x

v Z w Z



Back to the newsboy: if the newsboy were interested in taking into account risk, rather than maximizing expected revenue, then we solve the following problem:

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, , , ,

max 1

. . , , ,

, , ,

0, , ,

0.3 , , ,

, , ,

, 0, , ,

0

i i i i ii L M H i L M H

i i

i

i

i i

i i i

i i

p p w v

s t y d i L M H

y x i L M H

y i L M H

y x i L M H

v w i L M H

v w i L M H

x


Retail MarketMarket Share

Each firm competing in the retail market at a given node chooses a price at which it will offer power to consumers.

Using a differentiated products model (similar to that found in Vives (2001)), we compute the market share for each of the firms.

This model assumes that total demand is inelastic, and consumers merely switch between retailers.

The market share of retailer f is:

Where p is a vector of retail prices (one for each firm), and b is the cross-elasticity.

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1...

11f i f

i n

b p pn


Retail MarketWholesale Market

We allow retailers to also own generation (vertical integration).

We assume that the generation is bid into the market at cost. This is said to be a competitive equilibrium.

At the time that retail contracts are determined, the future wholesale prices are unknown, due to uncertainties around hydro inflows and outages.


Retail MarketRetail Market Competition

All firms optimize the following profit maximisation problem simultaneously.

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1...

max 1

1

0

i i

i i j i ij n

i

E CVaR

Xp c b p p P

n

p


Single Node ExampleSetup

First we consider a situation with two gentailers at a single node.

Firm A owns a thermal plant whereas B owns a hydro, each with capacity of 100MW.

The firms compete for customers in the retail market.

The total demand is 150MW.

Water value: h ~ U[0,100].


Single Node ExampleWholesale Prices and Profit

As a function of the water value, h, the wholesale prices and profits can be computed:

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50, 50,

, 50.

0, 50,

100 50 , 50.

100 50 , 50,

0, 50.

A

B

hc h

h h

hP h

h h

h hP h

h


Single Node ExampleRisk-neutral Equilibrium

If firms are risk-neutral, it can be shown that the profit from the wholesale market has no bearing on the retail pricing.

We can compute the equilibrium retail prices for both firms to be $137.50 in this case.

In this situation, both firms share the retail demand equally.


Single Node ExampleBest-response

Now let us examine the optimal retail prices for the firms as they increase their risk-aversion.

The hydro plant makes more profit when water values are low, whereas the thermal plant makes more profit when the water values are high.

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Best Response Prices

130

135

140

145

150

155

160

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Risk Aversion

Ret

ail

Pri

ce

Firm A

Firm B


Single Node ExampleBest-response

These plots show the distributions of profits for the firms, under risk-neutral and risk-averse pricing strategies.

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Firm A Profit

6000

6500

7000

7500

8000

0 20 40 60 80 100

h

Profi

t

Risk Neutral Risk Averse

Firm B Profit

4000

8000

12000

16000

20000

0 20 40 60 80 100

h

Profi

t

Risk Neutral Risk Averse


Two Node ExampleBackground

In this model we have 2 nodes and 3 firms:

– firm A owns 2 thermal plants in the North,

– firm B owns 2 hydro plants in the South,

– firm C owns 1 thermal in the North and 1 hydro in the South.

The North and South have separate retail markets.

Firm B is not in the North, whereas firm A is not in the South.

There is a fixed cost of entry C associated with entering a market.


Two Node ExampleAsset Swap

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Before Swap After Swap

H H H

T T T

H H H

T T T

Firm A

Firm B

Firm C

Firm A

Firm B

Firm C

Virtual Swap

H H H

T T T

Firm A

Firm B

Firm C


Two Node ExampleWholesale Market

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In the wholesale market, we have two plant-types: hydro and thermal.

The thermal cost function is:

The hydro cost function is:

where h (the measure of water scarcity) is normally distributed.

The nominal capacity of the line is 250MW, although there is a 5% chance of an outage in any period.

Demand in the North is 2000MW, and in the South in 1000MW.

250 0.1Tc q q q

210Hc q q hq


Two Node ExampleWholesale Nodal Price Differences


Two Node ExampleWholesale Profits for the Firms


Two Node ExampleRisk Averse Behaviour

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In order for each firm to determine its best pricing strategy it must now solve a similar optimization to what we encountered in the single-node example, adjusted for multiple nodes (and hence multiple retail markets).

It is interesting to note that when an agent is risk averse, its pricing decision at one node may affect the optimal price at another.


Two Node ExampleRisk Averse Behaviour

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If the asset swap does not incentivise an additional firm to enter into each retail market, then we find the following change in prices.

North Price South Price Cost

A B C A B C

Before Swap 283.35 – 288.90 – 344.84 344.84 916,775

After Swap 287.20 – 290.83 – 345.49 346.13 923,697

Virtual (800) 283.90 – 289.18 – 346.21 345.52 918,664


Two Node ExampleNew Retailer in Each Market

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On the other hand, if firm A enters the retail market in the South and firm B enters in the North, we find the following prices.

North Price South Price Cost

A B C A B C

Before Swap 225.12 226.90 226.90 271.34 274.05 268.53 691,235

After Swap 227.79 227.79 227.79 278.17 278.17 278.17 733,751

Virtual (800) 217.27 223.93 222.89 218.39 249.47 256.05 671,121


Two Node ExampleCost of Entry


Two Node ExampleConclusions

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Risk aversion for firms can affect whether or not they enter a market.

If they do enter, whether there exists a risk premium or discount for the consumers depends on the particular circumstances.

From our model, we find that physical asset swap will only have a beneficial effect on consumer prices if additional retailers enter each market.

The ‘virtual asset’ improves retail competition since the hydro risk is eliminated for the firm not owning hydro.


Retail Electricity Markets with Risk Aversion and Asset Swaps

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