A Local Relaxation Approach for the Siting of Electrical Substations

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A Local Relaxation Approach for the Siting of
Electrical Substations
Walter Murray and Uday Shanbhag Systems
Optimization Laboratory Department of Management
Science and Engineering Stanford University, CA
94305
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SSO - Review
Service area
Washington State
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SSO - Review

• Colour
• Black
• substation
• Other
• Kw Load

Service area each grid block is 1/2 mile by 1/2
mile
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SSO - Review

• Model distribution lines and substation
  locations and
• Determine the optimal substation capacity
  additions
• To serve a known load at a minimum cost

Service area each grid block is 1/2 mile by 1/2
mile
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SSO - Review
Characteristics

• More substations
• Higher capital cost
• Lower transmission cost

Capital costs 4,000,000 for a 28 MW substation
Service area each grid block is 1/2 mile by 1/2
mile
Cost of losses 3,000 per kw of losses
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Variables
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Problem of Interest
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Admittance Matrix
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A Multiscale Problem
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SSO Algorithm
DETERMINE INITIAL DISCRETE FEASIBLE SOLUTION
INITIAL NUMBER OF SS
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Finding an Initial Feasible SolutionGlobal
Relaxation
Modified Objective
Continuous relaxation
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Finding an Initial Feasible SolutionGlobal
Relaxation
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Search Direction
Substation Positions
Candidate Positions
Good Neighbor
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Search DirectionLocal Relaxation
QP Subproblem
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Search Step Center of Gravity
Center of Gravity
Center of Gravity
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Optimal Number of Substations
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Sample Load Distributions
Snohomish PUD Distribution
Gaussian Distribution
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Comparison with MINLP Solvers
Note n and z represent the number of
substations and the optimal cost. In the SBB
column, z represents the cost for early
termination (1000 bb) nodes.
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Time (scaled) vs. Number of Integers (scaled)
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Large-Scale Solutions
Note n0 and z0 represent the initial number of
substations and the initial cost.
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Uniform Load Distribution
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Different Starting Points
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Quality of SolutionInitial Voltage
Initial Voltage
Load Distribution
Most Load Nodes Have Lower Voltages
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Quality of SolutionFinal Voltage
Final Voltage
Load Distribution
Most Load Nodes Have High Voltages
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Conclusions and Comments

• A very fast algorithm has been developed to find
  the optimal location in a large electrical
  network.
• The algorithm is embedded in a GUI developed by
  Bergen Software Services International (BSSI).
• Fast algorithm enables further embellishment of
  model to include
• Contingency constraints
• Varying impedance across network
• Varying substation sizes

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Acknowledgements

• Robert H. Fletcher, Snohomish PUD, Washington
• Patrick Gaffney, BSSI, Bergen, Norway.

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• Appendix

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Lower Bounds Based on MIPs and Convex Relaxations
Note We obtain two sets of bounds. The first is
based on a solution of mixed-integer linear
programs and the second is based on solving a
continuous relaxation (convex QP).
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Comparison with MINLP Solvers
Note n and z represent the number of
substations and the optimal cost. In the SBB
column, z represents the cost for early
termination (1000 bb) nodes.
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SSO - Review
Complexities

• Varying sizes of substations
• Transmission voltages
• Contingency constraints
• Is the solution feasible if one substation fails?

Constraints
Service area each grid block is 1/2 mile by 1/2
mile
Load-flow equations (Kirchoffs laws) Voltage
bounds Voltages at substations specified Current
at loads is specified
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SSO - Review
Characteristics
Cost function
New equipment Losses in the network Maintenance
costs
Constraints
Load and voltage constraints Reliability and
substation capacity constraints
Decision variables
Installation / upgrading of substations
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Variables
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Admittance Matrix Y
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Admittance Matrix
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A Local Relaxation Approach for the Siting of
Electrical Substations
Multiscale Optimization Methods and
Applications University of Florida at
Gainesville February 26th 28th, 2004 Walter
Murray and Uday Shanbhag Systems Optimization
Laboratory Department of Management Science and
Engineering Stanford University, CA 94305