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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. SSO - Review. Service area. Washington State. SSO - Review. Colour:

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slide1

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
SSO - Review

Service area

Washington State

sso review9
SSO - Review
  • Colour:
  • Black –
    • substation
  • Other –
    • Kw Load

Service area: each grid block is 1/2 mile by 1/2 mile

sso review10
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 review11
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
sso algorithm

DETERMINE SEARCH

DIRECTION

UPDATE POSITIONS

OF SS

DETERMINE SEARCH STEP

TO GET IMPROVED SOLN

ADJUST #

OF SS

WHILE IMPROVED SOLUTION

CAN BE FOUND

WHILE # OF SS

NOT CONVERGED

FINAL NUMBER AND POSITIONS OF

SUBSTATIONS

SSO Algorithm

DETERMINE INITIAL DISCRETE

FEASIBLE SOLUTION

INITIAL NUMBER OF SS

finding an initial feasible solution global relaxation
Finding an Initial Feasible SolutionGlobal Relaxation

Modified

Objective

Continuous relaxation

search direction
Search Direction

Substation

Positions

Candidate

Positions

Good

Neighbor

slide21

Search Step

Center of Gravity

Center of Gravity

Center of Gravity

sample load distributions
Sample Load Distributions

Snohomish PUD Distribution

Gaussian Distribution

comparison with minlp solvers
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.

large scale solutions
Large-Scale Solutions

Note: n0and z0represent the initial number of substations and the initial cost.

quality of solution initial voltage
Quality of SolutionInitial Voltage

Initial Voltage

Load

Distribution

Most Load Nodes

Have Lower Voltages

slide31

Quality of SolutionFinal Voltage

Final Voltage

Load

Distribution

Most Load Nodes

Have High Voltages

conclusions and comments
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
Acknowledgements
  • Robert H. Fletcher, Snohomish PUD, Washington
  • Patrick Gaffney, BSSI, Bergen, Norway.
lower bounds based on mips and convex relaxations
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 solvers37
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.

sso review38
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 (Kirchoff’s laws)
  • Voltage bounds
  • Voltages at substations specified
  • Current at loads is specified
sso review39
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
slide44

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