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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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