Power Station Control and Optimisation

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