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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
Ramtin Madani, UT Arlington 2
Generators Loads
PowerSystem
Determine:
1. On/off status of generators,2. Power injections,3. Voltages.
Subject to:
1. Unit constraints,2. Network constraints.
Day-ahead Scheduling
Ramtin Madani, UT Arlington 2
Generators Loads
PowerSystem
Determine:
1. On/off status of generators,2. Power injections,3. Voltages.
Subject to:
1. Unit constraints,2. Network constraints.
Proposed Approach: Third-order Semidefinite Programming (TSDP).• In-between the SDP and SOCP relaxations.
Day-ahead Scheduling
Ramtin Madani, UT Arlington 2
Generators Loads
PowerSystem
Determine:
1. On/off status of generators,2. Power injections,3. Voltages.
Subject to:
1. Unit constraints,2. Network constraints.
Proposed Approach: Third-order Semidefinite Programming (TSDP).• In-between the SDP and SOCP relaxations.
Distinctive Features:• Compatible with Nonlinearity: Joint Unit Commitment AC Optimal Power Flow,
Day-ahead Scheduling
Ramtin Madani, UT Arlington 2
Generators Loads
PowerSystem
Determine:
1. On/off status of generators,2. Power injections,3. Voltages.
Subject to:
1. Unit constraints,2. Network constraints.
Proposed Approach: Third-order Semidefinite Programming (TSDP).• In-between the SDP and SOCP relaxations.
Distinctive Features:• Compatible with Nonlinearity: Joint Unit Commitment AC Optimal Power Flow,
• Massively Scalable: 24-hour problem with 4000 units in a 13000-bus network,
Day-ahead Scheduling
Ramtin Madani, UT Arlington 2
Generators Loads
PowerSystem
Determine:
1. On/off status of generators,2. Power injections,3. Voltages.
Subject to:
1. Unit constraints,2. Network constraints.
Proposed Approach: Third-order Semidefinite Programming (TSDP).• In-between the SDP and SOCP relaxations.
Distinctive Features:• Compatible with Nonlinearity: Joint Unit Commitment AC Optimal Power Flow,
• Massively Scalable: 24-hour problem with 4000 units in a 13000-bus network,
• Small Gap: Less than 2% away from global optimality.
Day-ahead Scheduling
Ramtin Madani, UT Arlington 3
Experimental Results24-hour problem based on IEEE and European grid data:
Ramtin Madani, UT Arlington 3
Experimental Results
Feasible solutions with less than 2% distance from the globally optimality
24-hour problem based on IEEE and European grid data:
Ramtin Madani, UT Arlington 3
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
Ramtin Madani, UT Arlington 3
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
TSDP Relaxation
An ideal balance between strength and complexity.
Ramtin Madani, UT Arlington
Problem Formulation
4
• Consider a problem with time periods and units:
Ramtin Madani, UT Arlington
Problem Formulation
4
• Consider a problem with time periods and units:
• The on/off status of unit at time :
Ramtin Madani, UT Arlington
Problem Formulation
4
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
Ramtin Madani, UT Arlington
Problem Formulation
4
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
Ramtin Madani, UT Arlington
Problem Formulation
4
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
Ramtin Madani, UT Arlington
Problem Formulation
4
OverallCost
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
Ramtin Madani, UT Arlington
Problem Formulation
4
UnitConstraints
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
Ramtin Madani, UT Arlington
Problem Formulation
4
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
• Unit feasible sets:
Ramtin Madani, UT Arlington
Problem Formulation
4
NetworkConstraints
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
• Unit feasible sets:
Ramtin Madani, UT Arlington
Problem Formulation
4
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
• Unit feasible sets:
• Network feasible sets:
Ramtin Madani, UT Arlington
Problem Formulation
4
Discrete Parameters
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
• Unit feasible sets:
• Network feasible sets:
Ramtin Madani, UT Arlington
Problem Formulation
4
Nonlinear Equations
Discrete Parameters
• Consider a problem with time periods and units:
• The on/off status of unit at time :
• Active power injection of unit at time :
• Reactive power injection of unit at time :
• Operating cost of unit at time :
• Unit feasible sets:
• Network feasible sets:
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Unit Constraints
5
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Network Constraints
6
Ramtin Madani, UT Arlington
Convex Relaxation
7
• Consider a problem with time periods and units:
Ramtin Madani, UT Arlington
Convex Relaxation
7
• Consider a problem with time periods and units:
Ramtin Madani, UT Arlington
Convex Relaxation
7
• Consider a problem with time periods and units:
Ramtin Madani, UT Arlington
Convex Relaxation
7
• Consider a problem with time periods and units:
• A convex relaxation can be created by replacing and with theirconvex surrogates.
Ramtin Madani, UT Arlington
Convex Relaxation
7
• Consider a problem with time periods and units:
• A convex relaxation can be created by replacing and with theirconvex surrogates.
• Unit feasible sets:
Ramtin Madani, UT Arlington
Convex Relaxation
7
• Consider a problem with time periods and units:
• A convex relaxation can be created by replacing and with theirconvex surrogates.
• Unit feasible sets:
• Network feasible sets:
Ramtin Madani, UT Arlington
Convex Relaxation
7
The proposed relaxation
• Consider a problem with time periods and units:
• A convex relaxation can be created by replacing and with theirconvex surrogates.
• Unit feasible sets:
• Network feasible sets:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
8
• Non-convex constraints:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
8
• Non-convex constraints:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
8
• Non-convex constraints:
• Lifting: Define a number of auxiliary variables:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
8
• Non-convex constraints:
• Lifting: Define a number of auxiliary variables:
• The new variables and are defined to linearize the non-convex constraint.
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
8
• Non-convex constraints:
• Lifting: Define a number of auxiliary variables:
• The new variables and are defined to linearize the non-convex constraint.
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
8
• Non-convex constraints:
• Lifting: Define a number of auxiliary variables:
• The new variables and are defined to linearize the non-convex constraint.
• Whereas and are used to create a stronger relaxation.
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
8
• Non-convex constraints:
• Lifting: Define a number of auxiliary variables:
• The new variables and are defined to linearize the non-convex constraint.
• 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:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
9
• Linearized version of the cost equations:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
9
• Linearized version of the cost equations:
• Binary requirement can be captured through the “McCormick constraints”:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
9
• Linearized version of the cost equations:
• Binary requirement can be captured through the “McCormick constraints”:
• 2 third-order Semidefinite (TSDP) inequalities:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
9
• Linearized version of the cost equations:
• Binary requirement can be captured through the “McCormick constraints”:
• 2 third-order Semidefinite (TSDP) inequalities:
• 24 linear inequalities:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
9
• Linearized version of the cost equations:
• Binary requirement can be captured through the “McCormick constraints”:
• 2 third-order Semidefinite (TSDP) inequalities:
• 24 linear inequalities:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
9
• Linearized version of the cost equations:
• Binary requirement can be captured through the “McCormick constraints”:
• 2 third-order Semidefinite (TSDP) inequalities:
• 24 linear inequalities:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
10
• 4 additional inequalities can be designed in terms of the minimum down time constraints:
Ramtin Madani, UT Arlington
Relaxation of Unit Constraints
10
• 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:
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
• The network constraints can be cast linearly w.r.t. .
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
• The network constraints can be cast linearly w.r.t. .
• Semidefinite programming (SDP) relaxation:
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
• The network constraints can be cast linearly w.r.t. .
• Semidefinite programming (SDP) relaxation:
• Decomposed SDP: A collection of overlapping subsets are obtainedfrom a graph-theoretic analysis of the network:
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
11
• The network constraints can be cast linearly w.r.t. .
• Semidefinite programming (SDP) relaxation:
• Decomposed SDP: A collection of overlapping subsets are obtainedfrom a graph-theoretic analysis of the network:
• Second-order cone programming (SOCP):
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
12
SDPRelaxation
SOCPRelaxation
• SDP relaxation is computationally expensive while SOCP has poor performance:
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
12
SDPRelaxation
SOCPRelaxation
TSDP Relaxation
• SDP relaxation is computationally expensive while SOCP has poor performance:
• The proposed TSDP relaxation:
Ramtin Madani, UT Arlington
Relaxation of Network Constraints
12
SDPRelaxation
SOCPRelaxation
TSDP Relaxation
• SDP relaxation is computationally expensive while SOCP has poor performance:
• The proposed TSDP relaxation:
• More scalable than SDP and more likely to result in a feasible solution compared to SOCP.
Ramtin Madani, UT Arlington
Convex Relaxation
13
• Consider a problem with time periods and units:
Ramtin Madani, UT Arlington
Convex Relaxation
13
• Consider a problem with time periods and units:
• Third-order semidefinite relaxation:
Ramtin Madani, UT Arlington
Convex Relaxation
13
• Consider a problem with time periods and units:
• Third-order semidefinite relaxation:
Ramtin Madani, UT Arlington
Conclusions• Third-order semidefinite relaxation:
• Joint UC-AC-OPF.
14
4-bus power network:
Feasible region of voltage angles:
Ramtin Madani, UT Arlington
Conclusions• Third-order semidefinite relaxation:
• Joint UC-AC-OPF.
• Advantages:• Compatible with the AC model of networks,• Massively scalable,• Near-globally optimal feasible points.
14
4-bus power network:
Feasible region of voltage angles:
Ramtin Madani, UT Arlington
Conclusions• Third-order semidefinite relaxation:
• Joint UC-AC-OPF.
• 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.
14
4-bus power network:
Feasible region of voltage angles:
Ramtin Madani, UT Arlington
Thank you
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