1 1 the physical market amsi workshop, april 2007
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1 1
The Physical Market
AMSI Workshop, April 2007
2 2
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
3 3
Structure of the NEM
~
~
~
Pool
Retail supplier
Retail supplier
Generators
P,Q,T
$400
$350
4 4
Market Rules
• Gross market• 5-minute intervals, 30 minute clearing• National market with 6 regions• Bid-based dispatch• System marginal price applies to all
participants• Demand always met with VoLL cap• Ancillary services also on-market
The need for the secondary derivative
marketRisk profile due to
deviation from a continuous marketInter-regional basis risk,
Accumulation of interregional funds
Analysis of ‘behaviour’ and gaming on physical
market
Models of volatile price processAnalysis 8
Optimisation algorithms for minimum cost dispatch under
constraints
5 5
Regional Structure
6 6
The Energy Market
7 7
Spot market for Energy
5 minute Price ($/MWh) + dispatch (MW)
Minimum load
Ramp rates
HV link constraints
Network losses30 minute Price ($/MWh) + cleared MW
4-second target data for frequency control
8 8
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
9 9
Models of the NEM
• Structural models– Prophet, iPool, Prosym
• Parsimonious, hybrid models– Inhouse, academic research
• Risk, econometric models– Lacima, Accurisk
10 10
Inputs to models
• Availability
• Demand
• Transmission
• Behaviour
• Regulatory
11 11
Availability
• Reliability models of power stations
• “Complex systems” of elements – With/without redundancy
• Failure described with Poisson process
• Improved with time-varying hazard intensity
12 12
13 13
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
14 14
‘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
15 15
Demand
• A relatively exogenous variable
• Limited elasticity to price
• Driven by:– Industrial processes– Human-related influences– Random residuals
16 16
Market data: modelling / visualising
• A regular 30-minute market
• Data matrices in daily resolution
Matlab
17 17
Time Series Plots
18 18
Intensity Plots (imagesc)
19 19
Overlay plots (plot)
20 20
PCA
• The overlay visualisation provides motivation for – Principal Component Analysis– Singular Value Decomposition
• “Daily Demand shape”
= “Average Shape” + “Perturbation”
21 21
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
22 22
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
23 23
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
24 24
Intuitive explanation
• Principal components are vector subspaces explaining most variation in the data
25 25
26 26
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
27 27
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
28 28
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
29 29
SVD approximations
00000000
00000000
00000000
00000000
• D = U S VT
u1, u2, .., um
S1, S2, … Sm
30 30
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.
31 31
2004 demand
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34 34
35 35
Applications in demand models
• Step 1: Principal Components
• Step 2: Explanatory power:
• Step 3: Best fit to components:
• Step 4: Regressed on weather variables:
36 36
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
37 37
Weather-dependence of demand
38 38
Models for Pool Price
39 39
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!
40 40
41 41
Principal Components of Price
• Step 1: Principal Components
• Step 2: Explanatory power:
• Step 3: Best fit to components:
42 42
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
43 43
44 44
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
45 45
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
46 46
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) =
47 47
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)
48 48
Demand-driven price model
• Pool price = f(demand, bids, transmission)
49 49
• We know process for weather,
• Know weather impact on demand,
• Know demand and price relationship
50 50
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
51 51
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?
52 52
Standard Price Model
• Derive a stochastic differential equation without reference to explanatory variables
• MRJD – to be introduced for option pricing
53 53
Transmission and Prices
54 54
Regional Structure
55 55
Intraregional Constraints
56 56
Negative prices?
57 57
Interregional Constraints
58 58
Price Separation
1000 MW flowing South (-10%)
QLD generators paid $18
NSW customers paying $1200
59 59
Dispatch Process
60 60
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
61 61
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