1. Supported by Offer Construction for Generators with Inter-temporal Constraints via Markovian DP and Decision Analysis Grant Read, Paul Stewart Ross James and Deb Chattopadhyay

2. Presentation Outline Deregulated Electricity Market Offering Process Intertemporal Constraints Literature Two-Phase DP Algorithm Concept DA/DP Concept Conclusions 2

3. Electricity Markets 3 Quantity (MW) Price ($/MWh) Q* Q*’ P* P*’ Demand Curve Rest-of- Market Supply Curve Total Supply Curve offer

4. Player perspective 4 Quantity (MW) Price ($/MWh) Q* Q*’ P* Generator offer Residual Demand Curve P* ’ Generator marginal cost Generator profit Note, we are only optimising a single- player reaction to the assumed residual demand curve, not finding a multi-player equilibrium

5. Intertemporal Considerations 5 Market outcome expectations and correlations Reactions of rival market participants Linkages within a Company Water/Fuel storage limits Reservoir Balancing Inflow correlations Unit operational rules (ramping, startup/shutdown etc) We will focus on the impact of: Market price correlations For an energy limited (hydro or thermal) plant

6. Intuition Inter-temporal interactions make a difference Some may be managed by adjusting the shape of the (monotone) offer curve –Provided an increase in this period’s price always implies an increase in desired output Others may have to be managed by dynamically moving the (vertical) offer curve –Eg if an increase in this period’s price may imply a decrease in desired output, so as to save fuel/water for even higher prices implied for later 6

7. Optimal offer curves with correlation Optimal offer curves for various hours of the day –Derived by plotting price/quantity pairs –Assuming higher price now implies a higher price path for the rest of the day quantity Price

8. Literature 8 Philpott, Zakeri, Pritchard, etc Optimal offers for a Market Distribution Function = (roughly) a set of possible residual demand curves Rajaraman & Alvarado (2003). Optimal Bidding Strategy in Electricity Markets Under Uncertain Energy and Reserve Prices Two-Level Dynamic Program Lower Level: Determining the optimal offer for the given state Upper Level: Moving back through horizon, constructing a value curve (defined over reservoir level) for each previous market outcome. Two-Dimensional (Markov) Dynamic Program for upper level Reservoir Level, Previous Market Outcome

9. Upper Level Markov DP 9 T-2T-1 T 1 1 1 22 2 0.6 0.4 0.70.3 1 1 2 2 From To 0.9 0.1 0.5 1 1 2 2 From To Value Reservoir Level Market States

10. Lower Level DP 10 Quantity (MW) Price ($/MWh) (Similar to Philpott et al, noting that there may be many possible residual demand curves corresponding to each “market state”)

11. Our Goals Given that Alvarado et al have developed this basic approach, our goals are: –To improve its computational efficiency by efficiently separating the upper/lower DP levels in a two-phase approach –To extend it to cover more general uncertainty structures –(To test our intuition with respect to optimal offer patterns) 11

12. Two-Phase DP Approach 12 This offering process needs to be done once every half hour, and so it is important that the computational complexity of what needs to be done on the spot, in “real-time” is minimised. Two-Phase DP that we have developed recognises that the construction of the offers at a given market state is very repetitive and inefficient. Uses a concept we call marginal cost patching, to enable the production of all the offers to be brought out of real-time into a “Pre-Processing Phase” The “Real-Time Phase” that remains is highly efficient and can deal with highly detailed representations of a generator’s situation in a reasonable amount of time. Two-Level DP Pre-Processing Phase (Lower-Level DP) Real-Time Phase (Upper Level DP) +

13. Marginal Cost Patching 13 The optimal offer price at any particular quantity level depends only on the local marginal cost level. It is therefore independent of the marginal cost levels occurring at higher/lower quantities So segments of the optimal offers for constant marginal costs can be patched together to provide the optimal offer for a marginal cost curve that is stepped For example, the optimal offer for a firm with a marginal cost curve that steps from MC1 to MC2 at the quantity BP will be the combination of: The section of the offer from 0 to BP under the assumption that marginal cost is equal to MC1 over the entire range, and The section of the offer from BP to Qmax under the assumption that marginal cost is equal to MC2 over the entire range

14. Two-Phase DP Approach The actual marginal cost structure (eg water value curve) is unknown at the start of the analysis –It is determined internally by the DP, –So we can not pre-compute optimal offer curves But Marginal Cost Patching enables us to bring the lower level DP out of real time, by: –Producing a set of offers for a set of fixed MC levels in advance Then, in the real time DP, for each period and market state: –We know the end-of period expected marginal water value curve (by backwards recursion); so –We can quickly patching together segments of fixed-MC offers to produce an offer curve matching that marginal water value curve.

15. Pre-Processing Diagram 15 Quantity (MW) Price ($/MWh) Offer (MC=5) Offer (MC=10) Offer (MC=20) Optimal offer curves for various assumed MC levels Set of possible residual demand curves for this market state

16. Real-Time Algorithm 16 t-2t-1 T 1 1 1 22 2 0.6 0.4 0.70.3 1 1 2 2 From To 0.9 0.1 0.5 1 1 2 2 From To Value Reservoir Level Market States

17. Patching Diagram 17 Offer (MC=10) Offer (MC=20) Patched offer Quantity (MW) Price ($/MWh) Offer (MC=5) Q1 Q2 MC=5 MC=10 MC=20 (MC determined by backward recursion in real-time DP)

18. RT Phase – Finding Values 18 Quantity (MW) Price ($/MWh) (These price quantity pairs then provide a pdf for outcomes for the period in the real time DP recursion)

19. Results – Computation Time 19 Results over 320 Test Instances covering a wide range of the problem domain. Largest problem considered: 40 Periods 300 Reservoir Levels 40 Dispatch Levels 100 Possible RD Curves per Period 110 Fixed MC Levels AlgorithmMean Computational Time R&A Two-Level Our Pre-Processing Phase Our Real-Time Phase 80,324 seconds (22.3 hours) 4578 seconds (1.3 hours) 99 seconds

20. Results – Solution Quality 20 AlgorithmMean Percentage Error R&A Two-Level Our Two-Phase Approach 5.0% 1.7% These differences: Are defined in terms of deviation from (deterministic) optimality, as determined by ex post simulation on a dataset which contains some non-Markov correlations Arise from differences in the degree of approximation employed. The R&A algorithm could be made more accurate, but at even higher computational cost

21. Extension: DA/DP Not all uncertainty is well described by a Markov Chain Often it will become clear at some point in time that a new state of the world has arisen, and will remain for some time. For example: –A major breakdown –A change in weather –A different competitor strategy Our DA/DP approach models this structure using: –A DA “decision Tree” linking macro-states on a coarse time scale –A set of Markov DPs optimising behaviour within each macro- sate on a finer time scale

22. DA/DP Structure (example) 22

23. Results Despite added “complexity”, computational time actually reduces, for the same TOTAL number of micro-states –By about 25% in real time, and much more in pre-processing –because the number of possible state transitions etc reduces But the real issues are: –Does this structure exist in the real world? –How much do we gain by modelling it? And this obviously depends on the “strength” of the structure For our test problem set, the gain was actually quite marginal (1.6% error >> 1.3% on average, or 6.9%>>5.6% worst case) –Still, it is better… and significantly quicker

24. Conclusions MC patching can be used to create a TWO-PHASE DP This separation greatly improves computational efficiency, so more complex problems can be considered Solution quality is also superior to the original R&A algorithm Generalisation to the DA/DP structure can: –Further reduce computation time; and –Further improve solution quality …. Provided the real world actually exhibits this structure (Solutions exhibit the expected patterns with respect to offer curve dynamics) This methodology is quite workable for real-time application to a single reservoir/stockpile situation –Various extensions and variations are covered in the thesis (See http://www.mang.canterbury.ac.nz/people/stewart.shtml )http://www.mang.canterbury.ac.nz/people/stewart.shtml But a multi-reservoir model is the obvious next step