1. Power Station Control and Optimisation Anna Aslanyan Quantitative Finance Centre BP

2. Background Tolling (spark/dark spread) agreements widespread in power industry Both physical and paper trades, usually over-the-counter Based on the profit margin of a power plant Reflect the cost of converting fuel into electricity Physical deals facility-specific Pricing often involves optimisation

3. Definitions Optimisation problem referred to as scheduling (commitment allocation, economic dispatch) Profit is the difference between two prices (power and fuel), less emissions and other variable costs The latter include operation and maintenance costs, transmission losses, etc. Objective function similar to a spread option pay-off

4. Definitions (contd) Examine power, fuel and CO2 price forecasts and choose top N MWh to generate, subject to various constraints, including volume (load factor) restrictions operational constraints –minimum on and off times –ramp-up rates –outages Apart from fuel and emissions costs, need to consider start-up costs operation and maintenance costs

5. Motivation Trading of carbon-neutral spark spreads of interest to anyone with exposure to all three markets Attractive as –speculation –basis risk mitigation –asset optimisation tools Modelling required to –price contract/value power plant –determine optimal operating regime and/or hedging strategy

6. Commodities to be modelled Electricity –demand varies significantly –sudden fluctuations not uncommon –hardest to model Fuel (gas, coal, oil) –sufficient historical data available –stylised facts extensively studied Emissions –new market, just entered phase two –participants’ behaviour often unpredictable –prices expected to rise

7. Methodology outline Given forward prices for K half-hours and a set of operational constraints, allocate M generation half- hours, maximising profit or, equivalently, minimising production costs C A. J. Wood, B. F. Wollenberg Power Generation, Operation, and Control, 1996 S Takriti, J Birge, Lagrangian solution techniques and bounds for loosely coupled mixed-integer stochastic programs, Operations Research, 2000 –combination of two techniques, dynamic programming and Lagrangian relaxation

8. Dynamic programming Forward recursive DP formalism implemented to solve Bellman equation Given an initial state, consider an array of possible states evolving from it States characterised by –cost –history –status –availability

9. Dynamic programming (contd) Ensure that only feasible transitions are permitted –if the plant is on, it can stay on if allowed by availability switch off if reached minimum on time –otherwise, it can stay off switch on if allowed by availability and reached minimum off time Update the cost for each of these transitions Maximise the profit over all possible states at every stage

10. Lagrangian relaxation Define combining –cost function C –penalty (Lagrangian multiplier) –actual number of half-hours, m and maximum to be allocated, M Solve primal problem for a fixed Update to solve dual problem Iterate until duality gap vanishes

11. Lagrangian relaxation (contd) Initialise and its range Update to move towards along a subgradient Anything more suitable for mixed-integer (non-smooth) problems?

12. Lagrangian relaxation (contd) Solution sub-optimal (optimal if using DP alone) Can be partly improved by redefining the ‘natural undergeneration’ termination condition Further optimisation may be required, for example over outage periods

13. Summary Understanding of tolling deals provides market players with –alternatives to supply and/or purchase power –risk-management instruments –power plants valuation tools –ability to optimise power plants –competence necessary to participate in virtual power plant (VPP) auctions Large dimensionality requires fast-converging algorithms