An optimal trading problem in intraday electricity...

An optimal trading problemin intraday electricity markets

Workshop on Commodity Markets and their FinancializationIPAM, Los Angeles May, 2015

Rene Ad Pierre Gruet Huyen PhamEDF R&D - Finance for Energy Market Research Centre

LPMA, University Paris Diderot

FiME Lab Trading on intraday electricity models 1 / 50

Agenda

1 Trading in the intraday marketIntraday marketA problem of optimal trading

2 Optimal trading modelNo jumps, no delayJumps in the residual demand forecastDelay in generation

3 Numerical applications and simulations

4 Conclusion


Disclaimer

DisclaimerAny views or opinions presented in this presentation are solely those of the authorand do not necessarily represent those of the EDF group.


Trading in the intraday market

Trading in the intraday market


Trading in the intraday market Intraday market

A bit of context

Development of renewable energy in continental Europe

windfarm: Germany 31 GW of 177 GW total installed capacity, Spain 22 GWof 105 GW

solar power: Germany 32 GW, Italy 16 GW of 124 GW of installed capacity.

source: Department Of Energy, Energy Information Agency



Effects on generation management and trading

Increasing forecasting error on short time horizon

root mean square error (RMSE) of the error forecast for the production of awind farm in six hours can reach 20% of its installed capacity (Giebel et al.(2011))

Producers / retailers endure inbalance costs. Imbalance: difference betweengeneration (plus purchases) and consumption (plus sales).

They are penalised for their imbalances because the TSO has to buy energyfrom someone else to insure the equilibrium of the system.

Increasing need for producers to find ways to balance their short-termposition.

Development of intraday electricity market



Intraday market

Intraday market

Operated by a market operator

Market for next hours where firms

exchange power to balance their positionminimize the cost of their imbalances

Exchanged volume in Epex intraday market in Germany has grown from 2TWh (2008) to 25 TWh (2013).

EPEX Intraday market

Continuous trading

Opens at 15:00 the day before.

Possibility to buy/sell physical delivery contracts for the 24 periods 0:00 1:00, ..., 23:00 24:00.

Closes 45 minutes before beginning of delivery.



Example of the quotation of a given hour of delivery

15/2248 16/0000 16/0112 16/0224 16/0336 16/0448 16/060060

70

80

90

100

110

120

Figure: Epex intraday Germany market december 16th 2010, 7 a.m.




videoPrix2014115-a.mp4Media File (video/mp4)


Relation between intraday and dayahead prices

500 0 500 1000 1500 2000200

0

200

400

600

800

1000

1200

Epex dayahead 2012

Ep

ex in

trad

ay 2

012

Figure: Epex intraday and dayahead hourly prices during 2012.



Liquidity and players

EpexSpot - for a given hour (2014)

Total exchanged volumes can reach 3.000 MW

Average buy/sell volume 350 MWExchanges are cut in orders 20 MW

EpexSpot - members (2014)

100 actors on the French intraday market 240 actors on the all Epex intraday markets (France, Germany, Austria,Switzerland)

Power utilities (EDF, E.On, Enel...), oil companies (BP), agregators(Voltalis), banks and financial institutions (Morgan Stanley, JP Morgan,CitiGroup, Barklays, Merril Lynch Commodities Ltd...)


Trading in the intraday market A problem of optimal trading

Problem

Settings

Demand and renewable production forecasts are frequently updated (every 2hours for demand, every 4 hours for wind production)

Shocks in forecast happens.

Short-term generation management involves many complex constraints ofpower plants and many different technology costs.

Mainly, it takes time to mobilize generation.

QuestionsKnowing that the objectif of a power producer is to minimize the total cost ofproduction and trading, what can be the optimal trading strategy?

Knowing that shocks may happen, should traders take anticipatedprecautionnary positions?

Dilemna between waiting for a possible better price and closing immediatlyan imbalanced position.



Previous works

References

Henriot (2014): discrete time model with wind production error forecast asonly source of randomness

Garnier and Madlener (2014): continuous time model analysing the trade-offbetween entering into a deal right now and waiting for a better quote. Windfollows a arithmetic Brownian motion and intraday prices a geometricBrownian motion.



This talk

Analysis of the problem with a stylised model simplifying the generation side andyet preserving the main feature of the dilemna, the information structure

The producer minimises the expected total cost of production using

Thermal plants (oil, gas, nuclear): can be dispatched with anticipation (delay)

+ trading costs

+ Penalization of the imbalance with residual demand.

Model inspired by Almgren-Chriss (2000) optimal execution model withmarket impact

Here, random target, as we only have a forecast of final residual demand

Residual demand: total final demand minus renewable energy production.


Optimal trading model




Trading

Xt : net sale/buy position (inventory) on the intraday market with tradingrate control qt = Xt

Xt = X0 +

t0

qsds,

Transactions occur at price

Pt(q) = Pt +

t0

qsds Yt :=observable quoted price

+qt ,

Pt : unaffected price on (,F , (Ft)t ,P) permanent impact factor, : instantaneous impact factor.



Demand, generation and penalization

DT residual demand at time T . Continuously updated forecast (Dt)0tT .

Thermal production is chosen at time T h. Length h is the delay inproduction.

cost function c() = 2 2, > 0.



Objective function of the agent

Minimize over trading rates q A: F-adapted and generation L0+(FTh)

E( T

0

qsPs(q)ds trading cost

+

22

generation cost

+

2(DT XT )2

imbalance penalization

)

where

C (DT XT , ) := 2 2 + 2 (DT XT )

2 is the cost of holding XT whilethe final demand is DT and the production has been chosen.


Optimal trading model No jumps, no delay

Case with no jumps and no delay

We assume that thermal production can be decided instantaneously. Theproduction decision has to be made at final time T .

The unaffected price process is

Pt = P0 + 0Wt , thus dYt = qtdt + 0dWt ,

the residual demand forecast has dynamics

dDt = dt + ddBt ,

with R, 0 > 0, d > 0 and d t= dt, [1, 1].



Dynamical formulation

Value function:v(t, x , y , d) = inf

qAtL0+(FT )

J(t, x , y , d ;q, )

where

J(t, x , y , d ;q, ) = E( T

t

qs(Yt,ys + qs)ds + C (D

t,dT X

t,xT , )

),

At being the set of adapted processes q such that E( T

tq2s ds

)< +.

Note: The production quantity is chosen at time T , after the choice of thewhole trajectory q.



Solving the problem

v(t, x , y , d) = infqAt

E( T

t

qs(Yt,ys + qs)ds+ inf

L0+(FT )C (Dt,dT X

t,xT , )

)Optimal generation:

T =

+ (Dt,dT X

t,xT )1Dt,dT X

t,xT 0

=: +(Dt,dT Xt,xT )

Thus:


E( T

t

qs(Yt,ys + qs)ds +

2+(Dt,dT X

t,xT )

2

+

2(Dt,dT X

t,xT

+(Dt,dT Xt,xT ))

2)

We do not expect to get an explicit formula, due to the indicator function.FiME Lab Trading on intraday electricity models 21 / 50


Auxiliary relaxed problem

Relax the positivity constraint on the production :


L0(FT )

J(t, x , y , d ;q, ).

Optimal generation level:

(Dt,dT Xt,xT ) =

+ (Dt,dT X

t,xT )

New value function expression:


E( T

t

qs(Yt,ys + qs)ds +

1

2

+ (Dt,dT X

t,xT )

2).

Linear-quadratic problem. Value function is quadratic. Straightforward but tedious computations.



Optimal strategy in the auxiliary problem

Optimal trading rate

qt =r(, )(Dt + (T t) Xt) Yt

(r(, ) + )(T t) + 2

where r(, ) := + and Xt , Yt designate the inventory and price trajectory withqt as a control.



Interpretation

Define the forecast final production seen from time t s T :s =

+

(Ds + (T s) qs(T s) Xs

)The optimal trading rate satisfies:

Ys + qs(T s) + 2qs = c (s).

The optimal trading strategy consists in making the forecast marginal costsequal to the forecast price.

Close to the operational strategy.



Interpretation

Consequence

Suppose that at time 0, the intraday price is equal to the day-ahead spotprice and the producer is balanced.

Thus Y0 = c(D0 + T X0)

Thus the initial trading rate is null No anticipated precautionnary positionis needed.



A nice property of the optimal trading rate in the auxiliaryproblem

Property

The optimal trading rate is a martingale.

Proof

Itos lemma applied to qs = q(T s,Ds Xs , Ys

)gives:

dqs =(D2q

)ddBs +

(D3q

)0dWs .

Consequence

The expected inventory is thus a linear function of time: E(Xs)

= X0 + q0s.

Constant trading rate in Almgren and Chriss (2000).

Previous forecasts keeping the same control are thus pertinent.



Quality of the approximation of the original problem

The production has to be positive.

Suggested strategy:1 First follow strategy (qs)tsT of the auxiliary problem.2 Then choose production level + (= 10).

Denote

E1(t, x , y , d) = J(t, x , y , d ; q, +

) v(t, x , y , d),

E2(t, x , y , d) = v(t, x , y , d) v(t, x , y , d).

Bounds of Ei show very low error.


Optimal trading model Jumps in the residual demand forecast

Incorporation of jumps on demand

Add a compound Poisson process (N+t ,Nt )t with intensity , counting

positive and negative jumps, to the residual demand forecast.

At each jump time t,

with probability p+, (N+t )t has a jump.with probability p = 1 p+, (Nt )t has a jump.

New dynamics of demand:

dDt = dt + ddBt + +dN+t +

dNt

with + > 0 and < 0

Impact on intraday price:

dYt = qtdt + 0dWt + +dN+t +

dNt

with + > 0 and < 0.

Let := p++ + p and := p++ + p.



Value function and auxiliary problem

Value function:

v ()(t, x , y , d) = infq()AtL0+(FT )

J(t, x , y , d ;q, ).

Auxiliary problem, relaxing the constraint of positivity of the production:

v ()(t, x , y , d) = infq()AtL0(FT )

J(t, x , y , d ;q, ).

The solution to that problem is again explicit.

Yet straigthforward but (much more) tedious computations.

Tedious?



Pierre Gruets handmade computation



Optimal control in the auxiliary problem

Optimal trading rate

q()s = q(0)s +

r(, )(T s) + 4 (r(, ) + )(T s)2

(r(, ) + )(T s) + 2

Property

q()s :=(q()s +

2(s t)

)tsT

is a martingale:

if > 0, then (q()s )s is a supermartingale.

if < 0, then (q()s )s is a submartingale.



Interpretation

Expected demand at final time T seen at time t is

Dt := Dt + (T t) + (T t),

Expectation of XT seen at time t is

X t := Xt + q()t (T t)

4(T t)2

Expectation of the final price plus marginal cost of getting q()T is

Y t := Yt +(T t) +(q()t (T t)

4(T t)2

)+ 2

(q()t

2(T t)

)Thus,

Y t = c(Dt X t)

Optimal trading strategy is still to make the forecast marginal cost equal tothe forecast price.



Interpretation

Consequence

Suppose that at time 0, the intraday price is equal to the day-ahead spotprice and the producer is balanced.

Here, balanced means:

Y0 + T = c(D0 X0)

Thus, at time 0 shoud be taken the precautionnary position:

q()0 =

T + 4 (r(, ) + )T2

(r(, ) + )T + 2


Optimal trading model Delay in generation

Delay h in generation

Consider the auxilary relaxed problem without jumps.

It is always better to wait until T h to take the generation decision.Between T h and T , we face an optimal trading problem with nogeneration and with inventory XTh + .

Knowing the optimal trading rate as function of , it is possible to computethe optimal generation level.

Knowing the optimal generation level to be applied at time T h and theoptimal trading rate between T h and T , we are brougth back to aninstance of our problem between 0 and T h with a (slightly) more complexterminal cost function.


Optimal trading model Delay in generation

Consequences

After tedious computations, we found that:

Between 0 and T h, the control with and without delay is the same.Only the value functions differ.


Numerical applications and simulations

Numerical applications andsimulations



Parameter values for nice simulations

Time period T = 24 hInitial inventory level X0 = 0 MWhInitial demand forecast D0 = 50, 000 MWhInitial intraday price Y0 = 50 e(MWh)1Demand forecast trend = 0 MWhs1Intraday price volatility 0 = 1/60 e(MWh)1 s1/2Demand forecast volatility d = 1000/60 MWhs1/2Correlation = 0.8Cost function parameter = 0.002 e(MWh)2Inbalance penality = 200 e(MWh)2Permanent impact = 4.00 105 e(MWh)2Instantaneous impact = 2.22 es(MWh)2Probability of positive jumps p+ = 1Intensity of jump = 1.5/(3600 24) s1Size of the price jump + = 10 e(MWh)1Size of the demand forecast jump + = 1500 MWh



No jumps, no delay Optimal trading rate

Cont r ol

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Final oscillation to adjust production and demand.



No jumps, no delay Inventory

Int r aday inventor y and f inal pr oduct ion

For ecast r esidual demand

0

10000

20000

30000

40000

50000

60000

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Linear growth of the inventory with final generation to adjust to demand.



No jumps, no delay Prices

Pr ice w ithout impact

Pr ice w ith impact

49

50

51

52

53

54

55

56

57

58

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Small yet persistent impact on prices.



No jumps, no delay Marginal cost and price

The marginal cost decreases until it reaches the increasing price.



Jumps > 0, no delay Optimal trading rate

Cont r ol

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Decreasing trading rate. Starts with positive value (buy), at some point in time,becomes negative (sell).



Jumps > 0, no delay Inventory

Int r aday inventor y and f inal pr oduct ion


0

10000

20000

30000

40000

50000

60000

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

First, increasing inventory, then decreasing inventory, and final generation toadjust to demand.



Jumps, no delay Marginal cost and price

Pr ice w ith impactFor eseen mar ginal cost

0

20

40

60

80

100

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

First decreasing forecast marginal cost (positive large inventory means less to beproduced at maturity) then, increasing forecast marginal cost.



Delay h = 4 hours Optimal trading rate

Cont r ol

0.10

0.15

0.20

0.25

0.30

0.35

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

Trading rate after time T h is the only tool to fit the demand. Thus becomesmore volatile.



Delay h = 4 hours Inventory

Cont r acts bought on mar ket


0

10000

20000

30000

40000

50000

60000

0 10000 20000 30000 40000 50000 60000 70000 80000 90000

At T h, the rate at which the inventory increases suddenly changes.



Application Precautionnary position

QuestionWhat is the size of the anticipated precautionary position that should be taken inthe case of jumps in the residual demand forecast?

Answer

q()0 =

T + 4 (r(, ) + )T2

(r(, ) + )T + 2

with reasonnable parameter values: = 1.5 105 (1 jump per month), = 2 eh(MWh)2, = 0.1 e(MWh)2, = 500 e/MWh, = 200 e/MWh, = 2.0 103 e/MW2, T = 24 hours.

q()0 3.6 MWh/h

For one jump per week, the precautionnary position is 14 MWh/h.


Conclusion

Conclusion

ConclusionAnalysis of electricity intraday trading with a small and tractable stochasticcontrol model.

Extension of Almgren and Chriss (2000) optimal execution model with linearimpact to stochastic target.

Confort the operational strategy.

Jumps in the residual demand process lead to non-zero yet smallprecautionnary initial position.

Perspective

Intraday prices models.

Statistical arbitrage.

Risk management.


Conclusion

References

Talk based on paper An optimal trading problem in intraday electricitymarket available on arXiv and to appear in Mathematical and FinancialEconomics.

All presented data on intraday electricity markets are available on demand atEpexSpot website www.epexspot.com.

R. Almgren and N. Chriss. Optimal execution of portfolio transactions.Journal of Risk. 2000.

E. Garnier and R. Madlener. Balancing forecast errors in continuous-tradeintraday markets. FCN WP 2/2014, RWTH Aachen University School ofBusiness and Economics, 2014.

G. Giebel, G. Kariniotakis, R. Brownsword, M. Denhard and C. Draxl. Thestate-of-the-art in short-term prediction of wind power. A literature overview.2nd Edition. In Deliverable Report D1.2 of the Anemos Project(ENK5-CT-2002-00665), 2011.

A. Henriot. Market design with centralised wind power management:handling low-predictability in intraday markets. The Energy Journal. 2014.


www.epexspot.com

Trading in the intraday marketIntraday marketA problem of optimal trading

Optimal trading modelNo jumps, no delayJumps in the residual demand forecastDelay in generation

Numerical applications and simulationsConclusion

An optimal trading problem in intraday electricity...

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