Stochastic AC Optimal Power Flow (OPF): A Data-Driven Approach Ilys - PowerPoint PPT Presentation

Stochastic AC Optimal Power Flow (OPF): A Data-Driven Approach Ilyès Mezghani, Sidhant Misra, Deepjyoti Deka April 23, 2020

Context and personal background • PhD student in OR and Energy at CORE/LIDAM, UCLouvain since 2016 ( https://uclouvain.be/fr/node/4474 ). • Under the supervision of Anthony Papavasiliou ( https://perso.uclouvain.be/anthony.papavasiliou ). • Part of the UCLouvain Engie Chair ( http://uclengiechair.be/ ). • Work done during summer 2019 at LANL, Theory Division. Group: Advanced Network Science Initiative ( https://lanl-ansi.github.io/ ). • Paper to appear in PSCC2020: https://arxiv.org/abs/1910.09144 . 1/31

Motivation The increase in renewable generation and load flexibility comes with new challenges. Source: https://www.genscape.com/blog → Need for more reliable decisions . 2/31

Research question Lot of historical data collected by power grid operators for the same static power network. How can historical data and network information be used efficiently to ensure reliable decision making on the grid? 3/31

Agenda 1. Introduction 2. Problem Formulation 3. A Data-Driven Scenario-Based Approach 4. Numerical Experiment 5. Conclusion & Future Work 4/31

Optimal Power Flow Goal of the problem: find the cheapest way to generate enough power to satisfy the demand without violating technical constraints. Sets: Variables: • Buses � , • p / q real/reactive generation, • Lines � = � f ⋃︁ � r , • f p / f q real/reactive power flow, • Generators � . • v /θ voltage magnitude/angle. 5/31

OPF and Power Flow (PF) Recourse Deterministic OPF PF Recourse ∑︂ c ( p g ) min (1) g ∈� 6/31

OPF and Power Flow (PF) Recourse Deterministic OPF PF Recourse ∑︂ c ( p g ) min (1) g ∈� ∑︂ ∑︂ f p ij = p g − P i − G s ∀ i ∈ � s . t . i v 2 (2) i g ∈� i ( i , j ) ∈� ∑︂ ∑︂ f q ij = q g − Q i + B s i v 2 ∀ i ∈ � (3) i g ∈� i ( i , j ) ∈� 6/31

OPF and Power Flow (PF) Recourse Deterministic OPF PF Recourse ∑︂ c ( p g ) min (1) g ∈� ∑︂ ∑︂ f p ij = p g − P i − G s ∀ i ∈ � s . t . i v 2 (2) i g ∈� i ( i , j ) ∈� ∑︂ ∑︂ f q ij = q g − Q i + B s i v 2 ∀ i ∈ � (3) i g ∈� i ( i , j ) ∈� f p ij = G i v 2 i − G ij v i v j cos ( θ i − θ j ) − B ij v i v j sin ( θ i − θ j ) ∀ ( i , j ) ∈ � (4) f q ij = − B i v 2 i + B ij v i v j cos ( θ i − θ j ) − G ij v i v j sin ( θ i − θ j ) ∀ ( i , j ) ∈ � (5) ij ) 2 + ( f q ij ) 2 ≤ S 2 ( f p ∀ ( i , j ) ∈ � (6) ij 6/31

OPF and Power Flow (PF) Recourse Deterministic OPF PF Recourse ∑︂ c ( p g ) min (1) g ∈� ∑︂ ∑︂ f p ij = p g − P i − G s ∀ i ∈ � s . t . i v 2 (2) i g ∈� i ( i , j ) ∈� ∑︂ ∑︂ f q ij = q g − Q i + B s i v 2 ∀ i ∈ � (3) i g ∈� i ( i , j ) ∈� f p ij = G i v 2 i − G ij v i v j cos ( θ i − θ j ) − B ij v i v j sin ( θ i − θ j ) ∀ ( i , j ) ∈ � (4) f q ij = − B i v 2 i + B ij v i v j cos ( θ i − θ j ) − G ij v i v j sin ( θ i − θ j ) ∀ ( i , j ) ∈ � (5) ij ) 2 + ( f q ij ) 2 ≤ S 2 ( f p ∀ ( i , j ) ∈ � (6) ij θ ij ≤ θ i − θ j ≤ θ ij ∀ ( i , j ) ∈ � f (7) p ≤ p ≤ p , q ≤ q ≤ q , v ≤ v ≤ v (8) → Non-linear, non convex optimization problem. 6/31

OPF and Power Flow (PF) Recourse Deterministic OPF PF Recourse 1. Fix ( p , v ) for PV buses . ∑︂ c ( p g ) min (1) g ∈� 2. Find ( p , q , f p , f q , v , θ ) by ∑︂ ∑︂ f p ij = p g − P i − G s ∀ i ∈ � s . t . i v 2 (2) i g ∈� i ( i , j ) ∈� solving (2)-(5). ∑︂ ∑︂ f q ij = q g − Q i + B s i v 2 ∀ i ∈ � (3) i g ∈� i ( i , j ) ∈� f p ij = G i v 2 i − G ij v i v j cos ( θ i − θ j ) − B ij v i v j sin ( θ i − θ j ) ∀ ( i , j ) ∈ � (4) → System of non-linear f q ij = − B i v 2 i + B ij v i v j cos ( θ i − θ j ) − G ij v i v j sin ( θ i − θ j ) ∀ ( i , j ) ∈ � (5) equalities (easy to solve). ij ) 2 + ( f q ij ) 2 ≤ S 2 ( f p ∀ ( i , j ) ∈ � (6) ij θ ij ≤ θ i − θ j ≤ θ ij ∀ ( i , j ) ∈ � f (7) p ≤ p ≤ p , q ≤ q ≤ q , v ≤ v ≤ v (8) → Non-linear, non convex optimization problem. 6/31

OPF and Power Flow (PF) Recourse Ideally: find ( p 0 , v 0 ) and an ’adjustment policy’ able to react in case of perturbations . Is this feature possible to ensure? If so, how? We suggest: • a formulation of Stochastic AC OPF (SACOPF). • attacking the problem with a practical iterative approach. 7/31

Stochastic AC OPF We’ll only assume load disturbances. Ω denotes the uncertainty set. For ω ∈ Ω , the feasible set for OPF is:  OPF ( ω ) = { ( p , q , f p , f q , v , θ ) satisfying ∑︂ ∑︂ p g − ( P i + μ ω, p ij = ) − G s ∀ i ∈ � , f p i v 2 i i g ∈� i ( i , j ) ∈� ∑︂ ∑︂ q g − ( Q i + μ ω, q ij = ) + B s ∀ i ∈ � , f q i v 2 i i g ∈� i ( i , j ) ∈� and (4) − (8) } 8/31

Stochastic AC OPF Since ( p , v ) need to be used for recourse, we suggest the following formulation: ∑︂ ( p 0 ( Ω ) , v 0 ( Ω )) = argmin c g ( p 0 g ) (9) g ∈� s . t . ( p ω , q ω , f p ,ω , f q ,ω , v ω , θ ω ) ∈  OPF ( ω ) ∀ ω ∈ Ω (10) (︄∑︂ )︄ p ω = p 0 + μ p ,ω ∀ ω ∈ Ω α (11) i i ∈� v ω = v 0 ∀ ω ∈ Ω (12) (11) and (12) define the adjustment policy. Note that α is a parameter, α g ≈ 1 ∀ g ∈ � . |� | 9/31

How to tackle SACOPF? One main issue concerning Ω : • finite but huge if based on historical data. • inifite if based on a probability distribution. The idea is to intelligently reduce Ω to Ω N = { ω 1 , . . . , ω N } , with N small enough, and compute ( p 0 ( Ω N ) , v 0 ( Ω N )) in a way that ensures feasibility for all (or most of) ω ∈ Ω . 10/31

Agenda 1. Introduction 2. Problem Formulation 3. A Data-Driven Scenario-Based Approach 4. Numerical Experiment 5. Conclusion & Future Work 11/31

General Idea Initialization Choose Ω 0 and compute p 0 ( Ω 0 ) , v 0 ( Ω 0 ) Sampling Test the robustness of p 0 ( Ω N ) , v 0 ( Ω N ) on a large number of samples � of Ω . Scenario Selection Choose n scenarios to add to Ω N . Ω N = Ω N ∪ { ω 1 , . . . , ω n } Stochastic Solution Solve SACOPF with Ω N to get new p 0 ( Ω N ) , v 0 ( Ω N ) . 12/31

Toy Example One simple way to apply the approach: • Ω 0 = { ω 0 } where ω 0 = ( μ p = 0 , μ q = 0 ) . • Add n random scenarios to Ω 0 . Test case 73_ieee : 73 bus-system, 51 loads . We assume max/min +/- 3% uniform perturbation of each load . n Infeasibility PF Recourse 0 1,000/1,000 9 595/1,000 19 250/1,000 29 323/1,000 39 80/1,000 49 122/1,000 13/31

Practical Approach Initialization Choose Ω 0 and compute p 0 ( Ω 0 ) , v 0 ( Ω 0 ) Sampling Test the robustness of p 0 ( Ω N ) , v 0 ( Ω N ) on a large number of samples � of Ω . Data-Driven Selection MaxViol , NbConstr , Hybrid Scenario Selection Choose n scenarios to add to Ω N . Ω N = Ω N ∪ { ω 1 , . . . , ω n } Enhancement Stochastic Solution Solve SACOPF with Ω N to get new p 0 ( Ω N ) , v 0 ( Ω N ) . 14/31

Data-Driven Selection We should use sampling information to choose the scenarios to add to Ω N . ω 2 ω 1 Sampling Test the robustness of p 0 ( Ω N ) , v 0 ( Ω N ) on a large number of samples � of Ω . ω 0 Scenario Selection ω 4 ω 3 Choose n scenarios to add to Ω N . Ω N = Ω N ∪ { ω 1 , . . . , ω n } 15/31

Data-Driven Selection How to select ’bad’ scenarios? Example: we want to add 3 of these samples to Ω N . • Sample s 1 . Constraints violated: [QgUp10,FlowLim3,VDown12] . Max violation: 7.5% . • Sample s 2 : Constraints violated: [QgUp10,VDown12] . Max violation: 5.0% . • Sample s 3 . Constraint violated: [QgUp10] . Max violation: 15.0% . • Sample s 4 . Constraints violated: [QgUp10,FlowLim3,VDown12] . Max violation: 2.2% . • Sample s 5 . Constraint violated: [QgDown11] . Max violation: 9.0% . 16/31

Data-Driven Selection How to select ’bad’ scenarios? Max Viol. Example: we want to add 3 of these samples to Ω N . • Sample s 1 . Constraints violated: [QgUp10,FlowLim3,VDown12] . 3 Max violation: 7.5% . • Sample s 2 : Constraints violated: [QgUp10,VDown12] . Max violation: 5.0% . • Sample s 3 . Constraint violated: [QgUp10] . 1 Max violation: 15.0% . • Sample s 4 . Constraints violated: [QgUp10,FlowLim3,VDown12] . Max violation: 2.2% . • Sample s 5 . Constraint violated: [QgDown11] . 2 Max violation: 9.0% . 16/31

Data-Driven Selection How to select ’bad’ scenarios? Number of constraints. Example: we want to add 3 of these samples to Ω N . • Sample s 1 . Constraints violated: [QgUp10,FlowLim3,VDown12] . 1 Max violation: 7.5% . • Sample s 2 : Constraints violated: [QgUp10,VDown12] . 2 Max violation: 5.0% . • Sample s 3 . Constraint violated: [QgUp10] . 3 Max violation: 15.0% . • Sample s 4 . Constraints violated: [QgUp10,FlowLim3,VDown12] . Max violation: 2.2% . • Sample s 5 . Constraint violated: [QgDown11] . Max violation: 9.0% . 16/31