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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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SSOSSO -- ReviewReviewService area
Washington State
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SSOSSO -- ReviewReview
Colour:
Black
substation
Other Kw Load
Service area: each grid block is 1/2 mile by 1/2 mile
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SSOSSO -- ReviewReview Model distribution lines
and substation locations
and Determine the optimal
substation capacityadditions
To serve a known load ata minimum costService area: each grid block is 1/2 mile by 1/2 mile
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SSOSSO -- ReviewReview
More substations:Higher capital cost
Lower transmission cost
Characteristics:
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
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VariablesV
ariables
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Problem of InterestProblem of Interest
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Admittance MatrixAdmittance Matrix
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A MultiscaleP
roblemA MultiscaleP
roblem
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SSO AlgorithmSSO AlgorithmDETERMINE INITIAL DISCRETE
FEASIBLE SOLUTION
INITIAL NUMBER OF SS
DETERMINE SEARCH
DIRECTION
DETERMINE SEARCH STEP
TO GET IMPROVED SOLN
FINAL NUMBER AND POSITIONS OF
SUBSTATIONS
WHILE # OF SS
NOT CONVERGED
ADJUST #
OF SS
WHILE IMPROVED SOLUTION
CAN BE FOUND
UPDATE POSITIONS
OF SS
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Finding an Initial Feasible SolutionFinding an Initial Feasible Solution
Global RelaxationGlobal Relaxation
Continuous relaxation
Modified
Objective
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Finding an Initial Feasible SolutionFinding an Initial Feasible Solution
Global RelaxationGlobal Relaxation
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Search DirectionSearch Direction
Substation
Positions
Candidate
Positions
Good
Neighbor
19 K
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Search DirectionSearch Direction
Local RelaxationLocal Relaxation
QP Subproblem
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Optimal Number of SubstationsOptimal Number of Substations
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Sample Load DistributionsSample Load Distributions
Gaussian DistributionGaussian Distribution Snohomish PUD DistributionSnohomish PUD Distribution
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Comparison with MINLP SolversComparison with MINLP Solvers
Note:Note: nn andandz*z*represent the number of substations and the optimal cost.represent the number of substations and the optimal cost.
In the SBB column,In the SBB column,zzrepresents the cost for early termination (1000 b&b) nodes.represents the cost for early termination (1000 b&b) nodes.
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Time (scaled) vs. Number of Integers (scaled)Time (scaled) vs. Number of Integers (scaled)
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LargeLarge--Scale SolutionsScale Solutions
Note:Note: nn00 andandzz00 represent the initial number of substations and the initial cost.represent the initial number of substations and the initial cost.
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Uniform Load DistributionUniform Load Distribution
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Different StartingP
ointsDifferent StartingP
oints
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Quality of SolutionQuality of Solution
Initial VoltageInitial Voltage
Initial Voltage
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Final Voltage
Quality of SolutionQuality of Solution
Final VoltageFinal Voltage
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Conclusions and CommentsConclusions 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 byBergen 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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AcknowledgementsAcknowledgements Robert H. Fletcher, Snohomish PUD,
Washington
Patrick Gaffney, BSSI, Bergen, Norway.
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Lower BoundsLower Bounds
Based on MIPs and Convex RelaxationsBased on MIPs and Convex Relaxations
Note: We obtain two sets of bounds. The first is based on a solution of mixedNote: We obtain two sets of bounds. The first is based on a solution of mixed--integerinteger
linear programs and the second is based on solving a continuous relaxation (convexlinear programs and the second is based on solving a continuous relaxation (convex
QP).QP).
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Comparison with MINLP SolversComparison with MINLP Solvers
Note:Note: nn andandz*z*represent the number of substations and the optimal cost.represent the number of substations and the optimal cost.
In the SBB column,In the SBB column,zzrepresents the cost for early termination (1000 b&b) nodes.represents the cost for early termination (1000 b&b) nodes.
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SSOSSO -- ReviewReview Varying sizes of substations
Transmission voltages
Contingency constraints:
Is the solution feasible if onesubstation fails?
Complexities:
Constraints:
Load-flow equations (Kirchoffs laws)
Voltage boundsVoltages at substations specified
Current at loads is specified
Service area: each grid block is 1/2 mile by 1/2 mile
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Cost function:
SSOSSO -- ReviewReview
New equipment
Losses in the networkMaintenance costs
Constraints:
Load and voltage constraintsReliability and substation capacity
constraintsDecision variables:
Installation / upgrading of
substations
Characteristics:
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VariablesV
ariables
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Admittance Matrix : YAdmittance Matrix : Y
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