A Scalable Semidefinite Relaxation Approach to Grid Scheduling · • Semidefinite programming (SDP) relaxation: • Decomposed SDP: A collection of overlapping subsets are obtained

Ramtin Madani, UT Arlington

A Scalable Semidefinite Relaxation Approach to Grid Scheduling

Ramtin Madani

Department of Electrical EngineeringThe University of Texas at Arlington

Ali Davoudi

Department of Electrical EngineeringThe University of Texas at Arlington

Alper Atamtürk

Industrial Engineering and Operations ResearchUniversity of California, Berkeley

Ramtin Madani, UT Arlington 2

Generators Loads

PowerSystem

Determine:

1. On/off status of generators,2. Power injections,3. Voltages.

Day-ahead Scheduling


Generators Loads

PowerSystem

Determine:


Subject to:

1. Unit constraints,2. Network constraints.



Generators Loads

PowerSystem

Determine:


Subject to:


Proposed Approach: Third-order Semidefinite Programming (TSDP).• In-between the SDP and SOCP relaxations.



Generators Loads

PowerSystem

Determine:


Subject to:



Distinctive Features:• Compatible with Nonlinearity: Joint Unit Commitment AC Optimal Power Flow,



Generators Loads

PowerSystem

Determine:


Subject to:




• Massively Scalable: 24-hour problem with 4000 units in a 13000-bus network,



Generators Loads

PowerSystem

Determine:


Subject to:




• Massively Scalable: 24-hour problem with 4000 units in a 13000-bus network,

• Small Gap: Less than 2% away from global optimality.



Experimental Results24-hour problem based on IEEE and European grid data:


Experimental Results

Feasible solutions with less than 2% distance from the globally optimality

24-hour problem based on IEEE and European grid data:





• Several relaxation schemes have been proposed for UC and OPF:

SDPRelaxation

SOCPRelaxation





• Several relaxation schemes have been proposed for UC and OPF:

SDPRelaxation

SOCPRelaxation

TSDP Relaxation

An ideal balance between strength and complexity.


Problem Formulation

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• Consider a problem with time periods and units:


Problem Formulation

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• The on/off status of unit at time :


Problem Formulation

4



• Active power injection of unit at time :


Problem Formulation

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• Reactive power injection of unit at time :


Problem Formulation

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• Operating cost of unit at time :


Problem Formulation

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OverallCost







Problem Formulation

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UnitConstraints







Problem Formulation

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• Unit feasible sets:


Problem Formulation

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NetworkConstraints








Problem Formulation

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• Network feasible sets:


Problem Formulation

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Discrete Parameters









Problem Formulation

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Nonlinear Equations

Discrete Parameters









Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

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Unit Constraints

5


Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Network Constraints

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Convex Relaxation

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Convex Relaxation

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Convex Relaxation

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Convex Relaxation

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• A convex relaxation can be created by replacing and with theirconvex surrogates.


Convex Relaxation

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Convex Relaxation

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Convex Relaxation

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The proposed relaxation






Relaxation of Unit Constraints

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• Non-convex constraints:



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• Lifting: Define a number of auxiliary variables:



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• The new variables and are defined to linearize the non-convex constraint.



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• Whereas and are used to create a stronger relaxation.



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• Whereas and are used to create a stronger relaxation.

• The goal is to design a set of linear and conic inequalities that partially describe theconvex hull of all feasible variables:



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• Linearized version of the cost equations:



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• Binary requirement can be captured through the “McCormick constraints”:



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• 2 third-order Semidefinite (TSDP) inequalities:



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• 24 linear inequalities:



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• 4 additional inequalities can be designed in terms of the minimum down time constraints:



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• 4 additional inequalities can be designed in terms of the minimum down time constraints:

• 4 additional inequalities can be designed in terms of the minimum up time constraints:


Relaxation of Network Constraints

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• The network constraints can be cast linearly w.r.t. .



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• Semidefinite programming (SDP) relaxation:



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• Decomposed SDP: A collection of overlapping subsets are obtainedfrom a graph-theoretic analysis of the network:



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• Decomposed SDP: A collection of overlapping subsets are obtainedfrom a graph-theoretic analysis of the network:

• Second-order cone programming (SOCP):



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SDPRelaxation

SOCPRelaxation

• SDP relaxation is computationally expensive while SOCP has poor performance:



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SDPRelaxation

SOCPRelaxation

TSDP Relaxation


• The proposed TSDP relaxation:



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SDPRelaxation

SOCPRelaxation

TSDP Relaxation


• The proposed TSDP relaxation:

• More scalable than SDP and more likely to result in a feasible solution compared to SOCP.


Convex Relaxation

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Convex Relaxation

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• Third-order semidefinite relaxation:


Convex Relaxation

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• Third-order semidefinite relaxation:


Conclusions• Third-order semidefinite relaxation:

• Joint UC-AC-OPF.

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4-bus power network:

Feasible region of voltage angles:




• Advantages:• Compatible with the AC model of networks,• Massively scalable,• Near-globally optimal feasible points.

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• Advantages:• Compatible with the AC model of networks,• Massively scalable,• Near-globally optimal feasible points.

• Future directions:• Incorporation of uncertainties and security

consideration,• Implementation on GPU: Linear algebra on

3 x 3 matrices has closed-form solutions.

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Thank you

Documents

A Scalable Semidefinite Relaxation Approach to Grid Scheduling · • Semidefinite programming (SDP) relaxation: • Decomposed SDP: A collection of overlapping subsets are obtained