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Applications of Network Partitioning and Clustering in Electrical
Networks
Applications for Reliability and Economics
Seth Blumsack
Dept. of Energy and Mineral Engineering
Penn State University
1Thanks to NSF and PJM for funding support
Blumsack, Hines, 5-April-2008
Outline
• Partitioning the PJM Network• Structural Metrics for Electrical Networks
• Topological vs. Electrical Distance Concepts• Three Clustering Techniques
• Hierarchical, K-Means, Spectral• Genetic Algorithm Refinements
• Clustering Results on the PJM Network• Further Applications: Economics of “Smart
Grids”
2
Blumsack, Hines, 5-April-2008 3
The PJM Footprint
Blumsack, Hines, 5-April-2008
Zonal analysis in PJM
4
PJM Zones are used for multiple planning and operations analyses:1. Reliability and resource adequacy
assessments2. Allocation of ancillary services costs3. Administration of load curtailment programs4. Identification and mitigation of market power
Blumsack, Hines, 5-April-2008
A Reliability Application in PJM
5
• Load delivery test for resource adequacy, RTEP, RPM
• For each zone within PJM:• CETO (transfer objective)• CETL (transfer capability)• Evaluated for seasonal peaks, transmission
contingencies (but not inter-zonal congestion), generator EFOR’d
Blumsack, Hines, 5-April-2008
Problems with Zonal Definition
6
• Zones defined based on TO territories• Prior to restructuring, TO territories were tightly
knit, few interconnections• Now, the PJM system is regional (multi-
regional?)• More long-distance transactions• “Big” transmission projects planned (NIEC, AEP
765 kV, APS projects, etc…)• Do the current LDA zones reflect the value of
these and future projects?
Blumsack, Hines, 5-April-2008
Criteria for a “Good” Zone
7
• “Closeness”• Nodes within a zone should behave similarly,
relative to the system outside the zone • “Connectivity”
• No significant bottlenecks• Robust to single-point contingencies
• Zones should be defined structurally, rather than by “rules”
Electrical vs. TopologicalNetwork Structure
Blumsack, Hines, 5-April-2008
Topological Distance Metrics
9
• Buses should be grouped in the same zone if:• They are close to one another – short distances• Highly connected – multiple paths
• Topological metric of closeness is the degree• Degree(k) = number of other nodes that are
directly connected to node k.• Distance (j,k) = number of hops required to get
from node j to node k
Blumsack, Hines, 5-April-2008
Degree distribution for PJM
10
Degree
Freq
uenc
y
Blumsack, Hines, 5-April-2008
Electrical distance and centrality
• We can get an estimate of the “distance” between two nodes using the inverse of the admittance matrix:
• Then we can get the total closeness (or centrality) of a node via:
11
Blumsack, Hines, 5-April-2008
Electrical Distances in PJM
12
Blumsack, Hines, 5-April-2008
Topological representation of PJM
13
Blumsack, Hines, 5-April-2008
Electrical representation of PJM
14
Clustering Techniques and Algorithms
Blumsack, Hines, 5-April-2008
• We have worked on defining zones within the PJM system by clustering based on electrical distances
• Three clustering algorithms:• Hierarchical• Spectral (k-means)• Genetic Algorithms
Electrical centrality clustering
16
Blumsack, Hines, 5-April-2008
Clustering algorithms
17
Define distance or similaritybetween points (nodes)
Arrange points based on good similarity scores (top-down or bottom-up)
Define similarity between clusters
Iterate until some convergence criteria is met
Blumsack, Hines, 5-April-2008
Four fitness metrics for cluster evaluation:1. Clustering Tightness Index (CTI):
2. Cluster Count Index (CCI): p* = desired # clusters
Fitness (I)
18
2 2
2*
exp( (ln ) / 2 )ln
ln
CCI pw n
p
μ σσ
μ σ
= − −=
= +
Blumsack, Hines, 5-April-2008
Clustering Count Index
• When p = p*, CSI = 1• When p = n, CSI = 0
19
2 2
2*
exp( (ln ) / 2 )ln
ln
CCI pw n
p
μ σσ
μ σ
= − −=
= +
Blumsack, Hines, 5-April-2008
3. Cluster Size Index (CSI): s = cluster size
4. Cluster Connectedness Test (CCT):CCT = 1 if the cluster is connectedCCT = 0 if the cluster is unconnected
Aggregate fitness = CTI×CCI×CSI×CCT
Fitness (II)
20
2 2
2*
* *
exp( (ln ) / 2 )ln
ln/
CCI sw n
ss n p
μ σσ
μ σ
= − −=
= +=
Blumsack, Hines, 5-April-2008
• Hierarchical Clustering• K-Means (Spectral) Clustering• Genetic Algorithms
Clustering Techniques
21
Blumsack, Hines, 5-April-2008
• Bottom-up clustering method
Hierarchical clustering
22
AVERAGE distance metric
CENTROID distance metric
MINIMUM distance metric (aka “single linkage”)
MAXIMUM distance metric can also be used
Blumsack, Hines, 5-April-2008 23
K-means clustering
© Andrew Moore, 2004
Blumsack, Hines, 5-April-2008 24© Andrew Moore, 2004
K-means clustering
Blumsack, Hines, 5-April-2008 25© Andrew Moore, 2004
K-means clustering
Blumsack, Hines, 5-April-2008 26© Andrew Moore, 2004
K-means clustering
Blumsack, Hines, 5-April-2008 27© Andrew Moore, 2004
K-means clustering
Blumsack, Hines, 5-April-2008
Performs k-means clustering on the spectral representation of the network
1. Calculate the graph Laplacian:2. Choose k, desired number of clusters3. Calculate k eigenvectors associated with k
largest eigenvalues4. Perform k-means clustering on the
eigenvectors.
Spectral clustering
28
L D W= −
Clustering on the PJM Network
Blumsack, Hines, 5-April-2008 30Overall Fitness = 0.25 (Not very good)
Current PJM Zones
Blumsack, Hines, 5-April-2008
Clustering Method #1
31
Fitness = 0.006
Blumsack, Hines, 5-April-2008
Clustering Method #2
32
Fitness < 10-4
Blumsack, Hines, 5-April-2008
Clustering Method #3
33
Fitness = 0.24
34
Hierarchical Clustering
K-Means Clustering
Spectral Clustering
Blumsack, Hines, 5-April-2008
Genetic algorithms
• Represent a solution (individual) as a string of bits• 01001011000110011010• Branch statuses
• Create an initial population• Random or from known solutions
• Evaluate fitness• Do crossover, mutation, gen. eng, cloning• Repeat until fitness reaches desired level
35
Blumsack, Hines, 5-April-2008
Crossover
• Choose two parents giving preference to the “most fit” individuals
• Choose a set of bits to select from each parent (adjacent strings or random)
• Swap bits:• 00111010101000010110• 01000000110111000110
becomes• 00111010110111000110
36
Blumsack, Hines, 5-April-2008
Mutation
• For each bit flip a (weighted) coin.
• If it lands “heads” flop the bit• 01001011000110011010• 01001011010110011011
37
Blumsack, Hines, 5-April-2008
Genetic engineering
• Select an individual for GE, giving preference to more “fit” individuals
• Flip random bits to see which bit flip improves the fitness most
• Select the best modification
• Tends to weed out “loners”
38
Blumsack, Hines, 5-April-2008
Cloning
• Select several (3) of the best individuals in the previous population and copy them to the new population
39
Blumsack, Hines, 5-April-2008
Demonstration on the 300 bus network
40
Blumsack, Hines, 5-April-2008
Clustering via Genetic Algorithms
41
Fitness = 0.89
Cloning
42
Flexible Transmission Topologies
“Smart Grid” is not simply “smart meters”
Blumsack, Hines, 5-April-2008
The Transmission Problem (I)
43
TLRs (Level 2 or higher)
Blumsack, Hines, 5-April-2008
The Transmission Problem (II)
44
Blumsack, Hines, 5-April-2008
Congestion and Reliability
45
Blumsack, Hines, 5-April-2008
Flexible Transmission Topologies
• New transmission is needed to achieve many different energy goals (renewables, reliability, competition)
• Another application of “smart grid” technology is to allow the system operator to adjust the grid topology in near-real time
• FACTS (power electronics), fast switching, are a couple of examples
46
Blumsack, Hines, 5-April-2008
Problem Formulation
• Formulate an optimization problem where the status of network branches is a decision variable for the system operator
47
( ), ,min ( )
. .
( , , , )
( )( )
Li Gi iji Gi i iP P u
i
transferij Gi Li
j
ij ij i j i j
C P VOLL UE
s t
F P P i
F u f V V θ θ
+ ×
= − ∀
=
=≤
∑
∑
g x 0h x 0
Blumsack, Hines, 5-April-2008
Placement Strategies?
• The mixed-integer programming problem is essentially impossible to solve globally.
• Do we really need all transmission to be controllable by the system operator?
• Are there diminishing returns to controllability? (Can we get 90% of the benefit by strategically selecting and controlling 15% of the lines?)
48
Blumsack, Hines, 5-April-2008
Graph-theoretic Strategies
Suppose you are given N controllers, where N
Blumsack, Hines, 5-April-2008
Finding Wheatstone Bridges
50
Applications of Network Thank You!
Questions?
Seth Blumsack
Dept. of Energy and Mineral Engineering
Penn State University
51Thanks to NSF and PJM for funding support
Applications of Network OutlineThe PJM FootprintZonal analysis in PJMA Reliability Application in PJMProblems with Zonal DefinitionCriteria for a “Good” ZoneSlide Number 8Topological Distance MetricsDegree distribution for PJMElectrical distance and centralityElectrical Distances in PJMTopological representation of PJMElectrical representation of PJMSlide Number 15Electrical centrality clusteringClustering algorithmsFitness (I)Clustering Count IndexFitness (II)Clustering TechniquesHierarchical clusteringSlide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Spectral clusteringSlide Number 29Slide Number 30Clustering Method #1Clustering Method #2Clustering Method #3Slide Number 34Genetic algorithmsCrossoverMutationGenetic engineeringCloningDemonstration on the 300 bus networkClustering via Genetic AlgorithmsCloningThe Transmission Problem (I)The Transmission Problem (II)Congestion and ReliabilityFlexible Transmission TopologiesProblem FormulationPlacement Strategies?Graph-theoretic StrategiesFinding Wheatstone BridgesApplications of Network