Optimal switching problems for daily power system balancing

Transcription

1 Optimal switching problems for daily power system balancing Dávid Zoltán Szabó University of Manchester June 13, 2016 ávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

2 Overview 1 Introduction 2 Market modelling 3 Empirical analysis 4 Numerical work Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

3 Introduction National Grid is responsible for balancing generation and consumption National Grid acts as the system operator on the UK electricity market Balancing cost can be reduced with entering into contracts We introduce real financial options on physical delivery of the electricity Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

4 Introduction Two participants on the electricity market Storage operator offers options to the system (network) operator These options give a balancing opportunity for the system operator Moriarty and Palczewksi analysed the call option problem with the methods of Dayanik and Karatzas We then analysed the put option problem After empirical analysis we turned to an optimal switching problem Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

5 Market modelling Imbalance value Electricity price is the image of the imbalance process Imbalance process can be modelled as a linear transformation of a Brownian motion X t = µ + σb t or as an Ornstein Uhlenbeck process dx t = κ(µ X t )dt + σdw t Electricity price is handled as an exponential f (x) = D + de bx function or as an affine f (x) = c + bx function Figure 1 shows a loop between the market participants in the original put option problem Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

6 Market modelling Figure : Figure 1 Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

7 Empirical analysis We analysed empirical data from Elexon portal with covering 3 years We can see that the imbalance changes very quickly even on a daily basis A mean-reverting behaviour is also visible in this process Therefore, an Ornstein-Uhlenbeck is a natural idea to model the imbalance value Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

8 Empirical analysis Figure : Real life imbalance values in two different days Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

9 Empirical analysis We also analysed the electricity price We concluded that exponential pattern is not visible in the price value Fitting linear model for the whole data set results in a relatively low R-squared value We have 48 settlement periods a day which can be included into the model by splitting the whole data set R-squared values increased significantly after splitting data set into 48 parts Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

10 Empirical analysis Figure : Empirical data and fitted linear regression on the left for the whole data set, on the right for only one settlement period Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

11 Empirical analysis We fitted linear regression for each settlement period separately We receive a clear pattern on the slope and intercept point graph which contains 48 points It shows us that the settlement periods indeed must be included into the analysis We fitted Fourier series for these points with a very good accuracy Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

12 Empirical analysis Figure : Fitted Fourier series for the intercept points on the left hand side and for the slope points on the right hand side of the linear regressions Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

13 Optimal switching model We turn to Ornstein-Uhlenbeck imbalance values, where analytical results are no more available Due to the high volatility of the imbalance value process we either hit an imbalance value immediately or we basically never hit it The original problem seems to be trivial in this model A new model is introduced where American put and call options have a fixed (T=1 day) expiration date The storage operator can optimally switch between the possible states (under some constraints) Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

14 Optimal switching model Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

15 Optimal switching model The free boundary equations in the time dependent case are: t V + σ2 2 2 x 2 V κx V rv = 0 in C (1) x V (x, t) C = g(x, t) where g(x, t) is already known and it is related to the instantaneous state jump. C represents the continuation region. We are able to solve the perpetual problem numerically with using a finite difference scheme Time dependent price stack functions f (t, x) = c(t) + b(t)x have also been added to this model Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

16 Optimal switching model We split the day into two periods; in the first period options can be sold but not exercised, while in the second period options can be exercised but no more sold The exercise rate is quite small in this model ( 4%) Then we split the day into one point and two periods. Options can only be sold at the first time point of the day. No options related transactions can happen in the first period. Options can be exercised in the second period of the day The exercise rate significantly increased with this additional condition ( 40%) Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

17 Results Figure : The frequency of sold options during the day. In this case the day is split into two parts, and the second part lasts from 4 pm to 6 pm. Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

18 Results Figure : Convergence of perpetual value for both parties on the left and the change of contract values on the number of hours to exercise the options on the right. Number of hours to exercise the option is 2 on the left. Iterated number of days is 1000 on the right hand side. We use the time dependent price stack function with bid-ask spread 25. The other parameters of our model: r = 0.1, p p = 10, p c = 10, K p = 60, K c = 30, x p = 200, x c = 200. Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

19 Results Figure : Perpetual contract values on the left for both parties for different premium prices under a model when the day is split into one point and two parts with the second part lasting from 4pm to 6pm. On the right we can see the sum of these values for the different premium prices. Iterated number of days is 200 on both graphs. We use the time dependent price stack function with bid-ask spread 25. The other parameters of our model: r = 0.1, K p = 60, K c = 30, x p = 200, x c = 200. Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

20 Results Figure : The ratio between two results created with different number of grid points. The plots are received by dividing the jmax = 80, imax = 500 results with the jmax = 200, imax = 5000 results. We compared the results for 200 days with 40 bid-ask spread, 10 premium prices and K p = 70, K c = 40 strike prices. On the left we can see the values for the storage operator and on the right for the system operator. Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

21 Thank you for you attention Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22

22 Further Reading I J. Moriarty and J. Palczewski. American Call Options for Power System Balancing. Preprint Available. S. Dayanik and I. Karatzas. On the optimal stopping problem for one-dimensional diffusions. Stochastic Processes and their Applications, 107:2, A. Shiryaev and G. Peskir. Optimal stopping and free-boundary problems. Birkhuser Basel. Dávid Zoltán Szabó (University of Manchester) Short title June 13, / 22