Black – substation Other

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
SSO - Review
Service area
Washington State
SSO - Review
Colour:
•Black –
substation
•Other –
Kw Load
Service area: each grid block is 1/2 mile by 1/2 mile
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
SSO - Review
Characteristics:
More substations:
Higher capital cost
Lower transmission cost
Capital costs:
$4,000,000 for a 28 MW
substation
Cost of losses:
$3,000 per kw of losses
Service area: each grid block is 1/2 mile by 1/2 mile
Variables
Problem of Interest
Admittance Matrix
A Multiscale Problem
SSO Algorithm
DETERMINE INITIAL DISCRETE
FEASIBLE SOLUTION
INITIAL NUMBER OF SS
DETERMINE SEARCH
DIRECTION
ADJUST #
OF SS
DETERMINE SEARCH STEP
TO GET IMPROVED SOLN
UPDATE POSITIONS
OF SS
WHILE IMPROVED SOLUTION
CAN BE FOUND
WHILE # OF SS
NOT CONVERGED
FINAL NUMBER AND POSITIONS OF
SUBSTATIONS
Finding an Initial Feasible Solution
Global Relaxation
Modified
Objective
Continuous relaxation
Finding an Initial Feasible Solution
Global Relaxation
Search Direction
Substation
Positions
Candidate
Positions
K 9 1
Good
Neighbor
Search Direction
Local Relaxation
QP Subproblem
Search Step
Center of Gravity
Center of Gravity
Center of Gravity
Optimal Number of Substations
Sample Load Distributions
Gaussian Distribution
Snohomish PUD Distribution
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 b&b) nodes.
Time (scaled) vs. Number of Integers (scaled)
Large-Scale Solutions
Note: n0 and z0 represent the initial number of substations and the initial cost.
Uniform Load Distribution
Different Starting Points
Quality of Solution
Initial Voltage
Initial Voltage
Quality of Solution
Final Voltage
Final Voltage
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
Acknowledgements
 Robert
H. Fletcher, Snohomish PUD,
Washington
 Patrick Gaffney, BSSI, Bergen, Norway.
Appendix
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).
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 b&b) nodes.
Complexities:
SSO - Review
– 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 (Kirchoff’s laws)
Voltage bounds
Voltages at substations specified
Current at loads is specified
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
Variables
Admittance Matrix : Y
Admittance Matrix
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