Offer Construction for Generators with Intertemporal Constraints via

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

• Deregulated Electricity Market Offering Process
• Intertemporal Constraints
• Literature
• Two-Phase DP Algorithm Concept
• DA/DP Concept
• Conclusions

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Electricity Markets
Rest-of-Market Supply Curve
Demand Curve
Total Supply Curve
P
P
offer
Q
Q
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Player perspective
Note, we are only optimising a single-player
reaction to the assumed residual demand curve,
not finding a multi-player equilibrium
Price (/MWh)
Generator offer
Residual Demand Curve
P
P
Generator marginal cost
Q
Quantity (MW)
Q
Generator profit
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Intertemporal Considerations

• 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

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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 periods 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 periods price may
  imply a decrease in desired output, so as to
  save fuel/water for even higher prices implied
  for later

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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

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Literature

• 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

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Upper Level Markov DP
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1
1
Market States
T
T-2
T-1
2
2
2
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Lower Level DP
Price (/MWh)
Quantity (MW)
(Similar to Philpott et al, noting that there may
be many possible residual demand curves
corresponding to each market state)
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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)

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Two-Phase DP Approach

• 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 generators situation in a
  reasonable amount of time.

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Marginal Cost Patching

• 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

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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.

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Pre-Processing Diagram
Optimal offer curves for various assumed MC levels
Set of possible residual demand curves for this
market state
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Real-Time Algorithm
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1
1
Market States
T
t-2
t-1
2
2
2
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Patching Diagram
Patched offer
(MC determined by backward recursion in real-time
DP)
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RT Phase Finding Values
(These price quantity pairs then provide a pdf
for outcomes for the period in the real time DP
recursion)
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Results Computation Time

• 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

80,324 seconds (22.3 hours)
4578 seconds (1.3 hours)
99 seconds
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Results Solution Quality
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 RA algorithm could
  be made more accurate, but at even higher
  computational cost

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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

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DA/DP Structure (example)

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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 gtgt 1.3 on average,
  or 6.9gtgt5.6 worst case)
• Still, it is better and significantly quicker

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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
  RA 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/ste
  wart.shtml )
• But a multi-reservoir model is the obvious next
  step