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Pinpointing the security boundary in high-dimensional spaces using importance sampling
Simon Tindemans, Ioannis Konstantelos, Goran Strbac
Risk and Reliability Modelling of Energy Systems12th November 2014, Durham University
Off-line support for operational security
Day ahead and real time operation
operating conditions
forecasts
contingencies
Security analysis actions
severe computational constraints
1. Anticipate 2. Analyse 3. Classify
Offline analysis (e.g. week ahead)
Monte Carlosampling ofoperatingconditions
contingencies
dynamic simulation
impact analysis
machine learning
data driven heuristics
High-dimensional DT with training errors
Decision trees for security studies
insecure
insecure secure*
secure
parameter 1
para
met
er 2
Two-dimensional example
DT image courtesy of Pepite
The security boundary
parameter 1
para
met
er 2
Quality of predictive classification (secure / insecure)
Pinpointing the security boundary
Which one?
Scenarios ‘near’ the security boundary improve prediction quality
Importance sampling 1. Which states to sample2. How to sample those states
Importance sampling for classification
Previous applications have relied on three assumptions:
1. Meaningful definition of ‘distance’ from the security boundary.
2. ‘Easy’ sampling distributions.
3. ‘Nice’ properties of the security boundary.
We propose a data-driven iterative importance sampling method that does not rely on these assumptions.
Krishnan et al. (2011), IEEE Transactions on Power Systems
Lund et al. (2014), IEEE Transactions on Power Systems
What to sample?
insecure
insecure secure*
secure
parameter 1
para
met
er 2
High-dimensional DT with training errorsTwo-dimensional example
DT image courtesy of Pepite
Defining ‘interestingness’
How to sample?
Repurpose machine learning process to guide samplingDecision trees express security and interestingness in terms of pre-fault variables
Abort evaluation of uninteresting pointsPossible because of separation of time scales
contingencies
dynamic simulation
impact analysis
machine learning
sample random
‘external’ conditions
complete starting
point
~10ms* ~1 min N x ~1 min
Decision Trees
importance sampling
filter
reject manyuninteresting points
Considerations and challenges
How to control biasing?• Biasing parameter b which controls relative populations. • b=0.5 is a defensive choice (max 2x slowdown).• For very high rejection rates, initial stages are no longer negligible.
Weights
Weights should be used at every subsequent analysis step.
Two-stage filteringFurther gains can be made by exploiting gap in effort between sampling of ‘TSO-external’ variables (~ms) and completion of base state (~1 min)
Case study
• Dynamic simulation study based on French EHV grid• ~1500 nodes, ~2000 lines• 30,334 classifying variables• 1970 contingencies• 6 security indices (only overloads used)
• Computation on PRACE Curie HPC• 10,000 cores, 24 hours [ ~ 2.5 tCo2]• 2GB results file; 10GB decision trees• Unbiased sample of 10,044 valid initial conditions
PRACE Curie : http://www-hpc.cea.fr/en/complexe/tgcc-curie.htm
Case study [contd]
‘Offline’ simulation of importance sampling• Use 6,000 states x 1,970 contingencies as an unbiased sample ‘pool’• Process in batches
• generate trees after each batch
Importance sampling acceptance rate• average: 24% (1431 of 6000)• minimum: 16% (967) [least interesting]• maximum: 100% (6000) [most interesting]
Validation using 4,044 states x 1,970 contingencies to estimate errors• Importance sampling classifiers• Unbiased classifiers, using identical computational budget
(1431 states/contingency)
500 500 1000 1000 1000 1000
Results: error analysis
mean change in error : -0.0012
mean number (1431)
misclassification error
poin
ts a
naly
sed
increased attention on badly classified contingencies
computational budget for naïve implementation
decreased average error
without ISwith IS
per contingency:
Results: error analysis - continued
|dError| > 0.01 only : 101 of 1970 contingencies
misclassification error
poin
ts a
naly
sed
Focused analysis results in reduction of errors
Some trees are worse off
mean change in error : -0.016Most change is for the better
Summary and outlook
Summary• Offline analysis and machine learning can support power system
operation• Challenge to pinpoint security boundary with finite resources• Proposed data-driven importance sampling method that uses
‘interestingness trees’ and accept-reject sampling• Initial trials suggest increase in accuracy for given computational
budget
Outlook• Quantification of speedup• Two-stage importance sampling (extra early rejection step) • Implementation on HPC platform
Thank you
This research was supported by the iTesla project within the 7th European Community Framework Programme
Partners for this work
Two-stage importance sampling
contingencies
dynamic simulation and impact
machine learning
sample random
‘external’ conditions
complete starting
point
~10ms* ~1 min
Decision Trees
stage II importance sampling
reject manyuninteresting points
stage I importance sampling
reject manyuninteresting points
reduced classification
N x ~1 min
Example decision tree
Decision tree for classification 1 if MTAHUP6_S_VL6_PGEN<-0.153883 then node 2 else node 3 2 if ROMAIP6_S_VL6_QSHUNT<54.9982 then node 4 else node 5 3 class = false 4 if TAMAR6COND_11_SC_V<244.195 then node 6 else node 7 5 if BXLIEL61ZGRA6_ACLS__TO__ZGRA6P6_S_VL6_V<242.056 then node 8 else node 9 6 if ANSERL61PRRTT_ACLS__TO__ANSERP6_S_VL6_Q<-47.9415 then node 10 else node 11 7 class = false 8 class = true 9 class = false10 class = false11 if BOCTOL71N_SE1_ACLS__TO__N_SE1P7_S_VL7_V<408.882 then node 12 else node 1312 class = false13 class = true
Importance sampling
Importance sampling deliberately distorts the sampling of system states to focus on the “important” events (i.e. those that contribute to the risk metrics).
Simulation results are corrected for this bias by sample weights. If done correctly, this procedure leads to large speed-ups.
𝐸𝑓ሾ𝑞ሺ𝑥ሻሿ= න𝑞ሺ𝑥ሻ𝑓ሺ𝑥ሻ𝑑𝑥= න𝑞ሺ𝑥ሻ𝑓ሺ𝑥ሻ𝑔ሺ𝑥ሻ𝑔(𝑥)𝑑𝑥= 𝐸𝑔ቈ𝑞ሺ𝑥ሻ𝑓ሺ𝑥ሻ𝑔ሺ𝑥ሻ