TREES General trees Binary trees Binary search trees AVL trees Balanced and Threaded trees.
Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem
description
Transcript of Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem
![Page 1: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/1.jpg)
Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem
David Tse
Dept. of EECS
U.C. Berkeley
Santa Fe
May 23, 2012
Joint work with Baosen Zhang (UCB) , Albert Lam (UCB), Javad Lavaei (Stanford), Alejandro Dominguez-Garcia (UIUC).
![Page 2: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/2.jpg)
Optimization problems in power networks
• Increasing complexity:– Optimal Power Flow (OPF)– Unit Commitment– Security Constraint Unit Commitment
• All are done at the level of transmission networks
• Smart grid: Optimization in distribution networks
![Page 3: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/3.jpg)
Power flow in traditional networks
Optimization is currently over transmission networks rather than distribution networks.
http://srren.ipcc-wg3.de/report/IPCC_SRREN_Ch08
![Page 4: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/4.jpg)
Power flow in the smart grid
Distributed Generation
Renewables
DemandResponse
DemandResponse
EVs
![Page 5: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/5.jpg)
Optimal power flow
• Focus on OPF, the basic problem
• Non-convex• DC power flow often used for transmission network• Not satisfactory for distribution network due to high
R/X ratios– Higher losses– Reactive power coupled with active power flow
![Page 6: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/6.jpg)
Transmission vs. distribution
• Transmission Network
• OPF is non-convex, hard
• Distribution Network
• OPF still non-convex• We show:
– convex relaxation tight– can be solved distributedly
tree topology
cycles
![Page 7: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/7.jpg)
Outline of talk
• Results on optimal power flow on trees.
• A geometric understanding.
• Application to the voltage regulation problem in distribution networks with renewables.
• An optimal distributed algorithm for solving this problem.
![Page 8: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/8.jpg)
Previous results for general networks
• Jabr 2006: SOCP relaxation for OPF
• Bai et.al 2008: rank relaxation SDP formulation of OPF:
Formulate in terms of VVH and replace by positive definite A.
• Lavaei & Low 2010: – key observation: rank relaxation for OPF is tight in many IEEE
networks– key theoretical result: rank relaxation is tight for purely
resistive networks
• What about for networks with general transmission lines?
![Page 9: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/9.jpg)
OPF on trees: take 1
Theorem 1 (Zhang & T., 2011):
Convex rank relaxation for OPF is tight if:
1) the network is a tree
2) no two connected buses have tight bus power lower bounds.
No assumption on transmission line characteristics.
(See also: Sojoudi & Lavaei 11, Bose & Low 11)
![Page 10: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/10.jpg)
Proof approach
• Focus on the underlying injection region and investigate its convexity.
• Used a matrix-fitting lemma from algebraic graph theory.
Drawbacks:
• Role of tree topology unclear.
• Restriction on bus power lower bounds unsatisfactory.
![Page 11: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/11.jpg)
OPF on trees: take 2
Theorem 2 (Lavaei, T. & Zhang 12):
Convex relaxation for OPF is tight if
1) the network is a tree
2) angle differences along lines are “reasonable”
More importantly:
Proof is entirely geometric and from first principles.
![Page 12: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/12.jpg)
Example: Two Bus Network
𝑉 1𝑉 2
𝑃1 𝑃2
min 𝑓 (𝑃1 ,𝑃2)𝑃1
𝑃2
Pareto-Front
Pareto-Front = Pareto-Front of its Convex Hull
𝑃1𝑃2
𝑔+ 𝑗𝑏
𝑃1=𝑔−𝑔 cos𝜃−𝑏sin𝜃𝑃2=𝑔−𝑔 cos𝜃+𝑏sin 𝜃
convex rank relaxation
=> convex rank relaxation is tight.
injection region
![Page 13: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/13.jpg)
Add constraints
• Two bus network
• Loss
• Power upper bounds
• Power lower bounds
𝑃1
𝑃2
𝑉 1 𝑉 2
𝑃1 𝑃2
𝑃1 𝑃2
![Page 14: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/14.jpg)
Add constraints
• Two bus network
• Loss
• Power upper bounds
• Power lower bounds
𝑃1
𝑃2
𝑉 1 𝑉 2
𝑃1 𝑃2
𝑃1 𝑃2
![Page 15: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/15.jpg)
Add constraints
• Two bus network
• Loss
• Power upper bounds
• Power lower bounds
𝑉 1 𝑉 2
𝑃1 𝑃2
𝑃1
𝑃2
This situation is avoided by adding angle constraints
𝑃1 𝑃2
![Page 16: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/16.jpg)
Angle Constraints
• Angle difference is often constrained in practice– Thermal limits, stability, …
• Only a partial ellipse where
all points are Pareto
optimal.• Power lower bounds
𝑃1
𝑃2jµj < tan¡ 1(
bg)
eg.bg= 3) jµj < 71o
![Page 17: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/17.jpg)
Angle constraints
• Angle difference is often constrained in practice– Thermal limits, stability, …
• Only a partial ellipse where
all points are Pareto
optimal• Power lower bounds
𝑃1
𝑃2
No SolutionProblem isInfeasible
Or
![Page 18: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/18.jpg)
Injection Region of Tree Networks
• Injections are sums of line flows• Injection region =
monotine linear transformation of the flow region• Pareto front of injection region is preserved under
convexification if same property holds for flow region.• Does it?
2
𝑃12
𝑃21
𝑃13
𝑃31
1
3
𝑃1=𝑃12+𝑃13𝑃2=𝑃21𝑃3=𝑃31
𝑃1𝑃2𝑃3
P1
P2 P3
![Page 19: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/19.jpg)
• One partial ellipse per line • Trees: line flows are decoupled
Flow Region
2
𝑃12
𝑃21 𝑃31
1
3
Flow region=Product of n-1 ellipses
𝑃13
Pareto-Front of Flow Region
Pareto-Front of its Convex Hull¿
![Page 20: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/20.jpg)
Application: Voltage regulation
Feeder
0 2 4 60.85
0.9
0.95
1
1.05
1.1
Distance from feeder
Vol
tage
Mag
nitu
de (
pu)
Voltage Profile
Use power electronics to regulate voltage via controlling reactive power.
Current capacitor banks • Too slow• Small operating range
(Lam,Zhang, Dominguez-Garcia & T., 12)
0 2 4 60.85
0.9
0.95
1
1.05
1.1
Distance from feeder
Voltage
Mag
nitude
0 2 4 60.85
0.9
0.95
1
1.05
1.1
Distance from feeder
Vo
lta
ge
Ma
gn
itu
de
0 2 4 60.85
0.9
0.95
1
1.05
1.1
Distance from feeder
Vo
lta
ge
Ma
gn
itu
de
Solar
EV
Desired
![Page 21: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/21.jpg)
• The reactive powers can be used to regulate voltages
Power Electronics
Solar Inverter
𝑃
𝑄P-Q Region
http://en.wikipedia.org/wiki/File:Solar_inverter_1.jpg
![Page 22: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/22.jpg)
Voltage Regulation Problem
• Can formulate as an loss minimization problem
System Loss
Voltage Regulation
Active Power
Reactive PowerControl
Previous results can be extended to reactive power constraints
![Page 23: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/23.jpg)
Random Solar Injections
• Solar Injections are random
Random Bus ActivePower Constraints
Net Load=Load - Solar
Random
8 am 6 pm
∈[𝐿𝑜𝑎𝑑−𝑆𝑜𝑙𝑎𝑟 ,𝐿𝑜𝑎𝑑]
![Page 24: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/24.jpg)
Reactive Power Constraints
𝑄1
𝑄2
Reactive
Rotation
• Upper bound• Lower bound
Active
𝑃1
𝑃2
We provide conditions on system parameters s.t. lower bounds are not tight
![Page 25: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/25.jpg)
Distributed Algorithm
• Convex relaxation gives an SDP, does not scale
• No infrastructure to transfer all data to a central node
• We exploit the tree structure to derive a distributed algorithm
• Local communication along physical topology.
![Page 26: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/26.jpg)
Example
• Each node solves its sub-problem with– Its bus power constraints– Lagrangian multipliers for its
neighbors
3 4
2
1
2
1𝜆
3 4
2
1𝜆
𝜆 𝜆 3
2
4
2𝜆 𝜆
• Update Lagrangian multipliers• Only new Lagrangian multipliers
are communicated
![Page 27: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/27.jpg)
Simulations
• 37 Bus Network
• 2.4 KV• Bus~ 10 households• Nominal Loads• Fixed capacitor banks• 20% Penetration
8am 6pm
8am 6pm
![Page 28: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/28.jpg)
Summary
• Geometrical view of power flow
• Optimization problems on tree networks can be convexified
• Applied to design an optimal distributed algorithm for voltage regulation.
![Page 29: Geometry of Power Flows on Trees with Applications to the Voltage Regulation Problem](https://reader038.fdocuments.in/reader038/viewer/2022110102/56813a10550346895da1e7b2/html5/thumbnails/29.jpg)
References
http://arxiv.org/abs/1107.1467 (Theorem 1)
http://arxiv.org/abs/1204.4419 (Theorem 2)
http://arxiv.org/abs/1204.5226 (the voltage regulation problem)