The Physical Market
description
Transcript of The Physical Market
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The Physical Market
AMSI Workshop, April 2007
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Overview
The NEM is the physical market for electricity in Australia.
This seminar aims to:
Describe the NEM with a view to its influence on the derivative markets
Expose some useful mathematical modelling techniques for certain aspects of the NEM
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Structure of the NEM
~
~
~
Pool
Retail supplier
Retail supplier
Generators
P,Q,T
$400
$350
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Market Rules
• Gross market• 5-minute intervals, 30 minute clearing• National market with 6 regions• Bid-based dispatch• System marginal price applies to all
participants• Demand always met with VoLL cap• Ancillary services also on-market
The need for the secondary derivative
marketRisk profile due to
deviation from a continuous marketInter-regional basis risk,
Accumulation of interregional funds
Analysis of ‘behaviour’ and gaming on physical
market
Models of volatile price processAnalysis 8
Optimisation algorithms for minimum cost dispatch under
constraints
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Regional Structure
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The Energy Market
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Spot market for Energy
5 minute Price ($/MWh) + dispatch (MW)
Minimum load
Ramp rates
HV link constraints
Network losses30 minute Price ($/MWh) + cleared MW
4-second target data for frequency control
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Losses
• Within a region:
• Payment to a generator
= MLF Pool Price MWh sent out
• System generation at the terminals
• Across regions, losses taken into account
• Equivalent volume “at the node”
= MLF MWh sent out
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Models of the NEM
• Structural models– Prophet, iPool, Prosym
• Parsimonious, hybrid models– Inhouse, academic research
• Risk, econometric models– Lacima, Accurisk
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Inputs to models
• Availability
• Demand
• Transmission
• Behaviour
• Regulatory
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Availability
• Reliability models of power stations
• “Complex systems” of elements – With/without redundancy
• Failure described with Poisson process
• Improved with time-varying hazard intensity
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Availability model implementationFailure time is exponential distribution: PDF
1. Model failure times directly: CDF
Let R ~ U(0,1) then
X = F-1(R) = -log(1-R)/2. Model failure times directly:
Pr(failure in [0, 0+dt])
xexf )(
x xedssfxF
01)()(
)(1 2dtOdte dt
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‘Behaviour’
• Generator bidding and commitment• More market rules:
– 10 bid price and volume bands– ‘Rebids’ permitted by shifting volume
• Portfolio optimisation subject to:– Market rules– Contract cash flows– Longer-term objectives– Response of competitors
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Demand
• A relatively exogenous variable
• Limited elasticity to price
• Driven by:– Industrial processes– Human-related influences– Random residuals
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Market data: modelling / visualising
• A regular 30-minute market
• Data matrices in daily resolution
Matlab
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Time Series Plots
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Intensity Plots (imagesc)
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Overlay plots (plot)
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PCA
• The overlay visualisation provides motivation for – Principal Component Analysis– Singular Value Decomposition
• “Daily Demand shape”
= “Average Shape” + “Perturbation”
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Modelling Demand
• Demand = f (i,d,s,T,H)– i = interval number (1-48)– d = day type (M,T,W,.., S,PH)– s = season– T = prevailing temperature– H = prevailing humidity
• Explanatory variables: – 48 8 4
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Principal Components Analysis
• Let v1, …, vn Rm
• Establish the correlation matrix of the vectors, M Rmm
• Determine the eigenvectors of the correlation matrix, e1, e2, .., em
• These are the principal components
• Determine the eigenvalues of the correlation matrix 1, 2, …, m
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Properties of the components
• Being a correlation matrix, j are real and positive
– {ei} are orthogonal
– That is principal components are independent– Eigenvalues of the correlation matrix sum to
the dimension m
• Explanation provided by component j (R2) is j / m
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Intuitive explanation
• Principal components are vector subspaces explaining most variation in the data
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Models using the PCs
• Model a process with say three components.• Fit each observation on the components:
Min { || vj - α1j e1 + α2
j v2 + α3j v3 || : [α3,α3 ,α3] }
• Then observation j is explained:vj = α1
j e1 + α2j v2 + α3
j v3 + rj
• Obtain sequence: A1, A2, A3, …, An
• Where Aj =
3
2
1
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Models using the PCs
• Then v1, v2, v3, …, vn (each in Rn) is simplified to A1, A2, …, An (each in R3).
• Now regress on the explanatory variables for a linear model:
A = f (d,s,T,H)
• “Katestone” demand modelling software
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Singular Value Decomposition
• Identical to the PCA, but from a different historical development
• Data in columns: D R48 365
• Find unitary matrices U and V and diagonal matrix S: D = U S VT
• U R48 48, S R48 365, V R365 365
• Elements of S are positive decreasing in magnitude
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SVD approximations
00000000
00000000
00000000
00000000
• D = U S VT
u1, u2, .., um
S1, S2, … Sm
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SVD approximations
000000000
000000000
000000000
000000001
4321
S
uuuu
0000000011uS
nVuSVuSVuS 11~
1121~
1111~
1 000000
An optimal weighting of a single shape u1.
Include two components: get an linear combination of u1 and u2.
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2004 demand
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Applications in demand models
• Step 1: Principal Components
• Step 2: Explanatory power:
• Step 3: Best fit to components:
• Step 4: Regressed on weather variables:
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Alternative application of PCA
• Say a demand model exists already
• Perform PCA decomposition of residuals:
• Describe perturbations with components– Dj = D*j + 1e1 + 2e2 + 3e3 + r
– Where j ~ N(0,σj2)
• Then future demand is simulated by perturbations of the modelled demand
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Weather-dependence of demand
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Models for Pool Price
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Pool Price and PCA
• Data structure is similar to demand
• Apply the same PCA methods
• Apply fits to price rather than log price– Note on Excel’s exponential fitting!
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Principal Components of Price
• Step 1: Principal Components
• Step 2: Explanatory power:
• Step 3: Best fit to components:
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Historical sampling
• Stochastic model
• Want a pool price process with behaviour representative of historical observations
• Perform historical sampling
• Extend by biasing subject to known variables
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Drawing a path
• Draw a half-hourly succession of U(0,1) random variables
• Lookup the resulting pool price
• Problem: more autocorrelation in the real world
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Inducing autocorrelation
• Require prices to be more autocorrelated than independent random draws
• Need successive U(0,1) variables to be correlated
• Need to generate very long sequences of autocorrelated random variables
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Tricks:• To generate U(0,1) variable:
=RAND()
• To generate N(0,1) variable =NORMSINV(RAND())
• To generate two -correlated N(0,1) variables:– X ~N(0,1), T~N(0,1)– Y = X + (1- 2)T– Then (X,Y) =
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For a long sequence• X1, X2, X3, … XN ~ N(0,1) iid• Y1 = X1
• Y2 = Y1 + (1- 2)X2
• Y3 = Y2 + (1- 2)X3
= [X1 + (1- 2)X2] + (1- 2)X3
• So Y = B*X• To speed the process:• Y=F-1F(B*X)=F-1(F(B)F(X))• Y1, Y2, Y3, …, YN ~ N(0,1) autocorrelated• P = NORMSDIST(X)
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Demand-driven price model
• Pool price = f(demand, bids, transmission)
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• We know process for weather,
• Know weather impact on demand,
• Know demand and price relationship
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Ito’s lemma
• Mechanism for describing a new random process which is derived from another
• Primary process:
dS = a(t,S) dt + b(t,S) dW• Secondary process: V(t,S)
dtS
VbdS
S
Vdtt
VdV
2
22
2
1
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Price from Demandy = 5.1894e0.0002x
0
5
10
15
20
25
30
35
40
3500 4000 4500 5000 5500 6000 6500 7000
0
10
20
30
40
50
60
70
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196
Actual PriceDemandPrice(demand) model
Valid with a time-varying bid stack: PCA again?
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Standard Price Model
• Derive a stochastic differential equation without reference to explanatory variables
• MRJD – to be introduced for option pricing
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Transmission and Prices
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Regional Structure
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Intraregional Constraints
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Negative prices?
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Interregional Constraints
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Price Separation
1000 MW flowing South (-10%)
QLD generators paid $18
NSW customers paying $1200
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Dispatch Process
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Black Hole Money
• NEMDE:– “What is the next cheapest dispatch to provide the
next MW of electricity?”
• When constrained:– QLD – isolated region with large generation supply– NSW – shortage of supply
• Inter Regional Settlement Residues• Electricity flows up the price gradient, so value
can only accumulate to NEMMCO.• Representative of a true discontinuous process
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Models of Flows and Price Spreads
• Require stochastic model of:– Flow level– Price in region 1– Price in region 2
• Under the physical models of:– Flow limits– Losses