Model Order Reduction of Energy Networks with a Focus on Hyperbolic - PowerPoint PPT Presentation

Model Order Reduction of Energy Networks with a Focus on Hyperbolic Systems ICERM Virtual Workshop March 23rd -27th Sara Grundel March 24, 2020

Future Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 2/42

Future How could 2050 look like? Renewable Energies Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 2/42

Future Renewable Energies Mobility largely electric Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 2/42

Future Renewable Energies Mobility largely electric Housing efficient and smart Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 2/42

Future Renewable Energies Mobility largely electric Housing efficient and smart Closed Carbon Cycle Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 2/42

Future Renewable Energies Mobility largely electric Housing efficient and smart Closed Carbon Cycle Synthetic Fuels Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 2/42

Modeling Simulation Optimization Global Optimal Solutions of the entire energy system Each subsystem has its own simulation tool Efficient and fast simulation of each subsystem is wanted and probably needed! ⇒ Complexity and Dimension Reduction Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 3/42

Modeling Simulation Optimization Global Optimal Solutions of the entire energy system Each subsystem has its own simulation tool Efficient and fast simulation of each subsystem is wanted and probably needed! ⇒ Complexity and Dimension Reduction Power Grid - different Levels Smart Home - Control Centers Gas transportation and storage networks Energy conversion Focus on gas distribution networks in this talk Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 3/42

Funding and Collaborators Federal Ministry for Economic Affairs and Energy Manuel Baumann Michael Herty Christian Himpe Philipp Sauerteig Petar Mlinari´ c Martin Stoll Neeraj Sarna Karl Worthmann Yue Qiu Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 4/42

Table of Content 1. Introduction/Motivation 2. Gas network model/PDAE 3. Discretization/Modeling of the isothermal Euler equation 4. Model Order Reduction based on ODEs 5. Feature Tracking Reduced Order Modelling for hyperbolic systems 6. Other Examples of Complexity Reduction in the context of the energy system Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 5/42

Gas transportation network PDAE hyperbolic PDE on the pipe ODEs or algebraic equations on other components algebraic node conditions Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 6/42

Gas Transport in the Pipe Isothermal Euler Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Density Friction Gravity Transient Nonlinear Linear Continuity Elevation Configurable ∂ρ ∂t = − S − 1 ∂q ∂x ∂q ∂t = − S ∂p ∂x − Sgρ∂h f g q | q | ∂x − 2 DS ρ p = γρz Gas State Compressibility Mass-Flux Pressure Nonlinear Transient Density Configurable Momentum Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 7/42

Gas Network System Overall structure is a directed graph G = ( N , E ) . At each node in N algebraic conditions are prescribed. The edges are the pipes described by the Euler equations. The resulting system looks like M ∂ t φ ( x, t ) = K φ ( x, t ) + f ( φ ( x, t ) , u ( t ) , t ) which discretized is x = Kx + Bu + f ( x, t ) 1 , M ˙ where φ ( x, t ) is a vector of pressure and flux values at and x ( t ) at different spatial points Depending on the network, the algebraic conditions used and the discretization schemes the matrices M, K, B and the function f can vary. In u ( t ) the input functions are collected. 1 Benner, G., Himpe, Huck,Streubel, Gas Network Benchmark Models, Springer, 2018 Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 8/42

Challenges of (P)DAEs existence of solutions index concepts space discretization solver for the discretized PDAE (time integration) model order reduction (nonlinear, DAE, uncertain and parameterized) parameter optimization uncertainty quantification optimal control/ optimization Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 9/42

Table of Content 1. Introduction/Motivation 2. Gas network model/PDAE 3. Discretization/Modeling of the isothermal Euler equation 4. Model Order Reduction based on ODEs 5. Feature Tracking Reduced Order Modelling for hyperbolic systems 6. Other Examples of Complexity Reduction in the context of the energy system Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 10/42

Isothermal Euler and Discretization Naive Approach ∂p ∗ ∂t = − 1 q R − q L γzS ∆ x Basic equation ∂q ∗ q ∗ | q ∗ | ∂t = − S p R − p L − f g γz p ∗ ∆ x 2 DS ∂p ∂t = − 1 ∂q γzS ∂x Decoupled approach ∂q ∂t = − S ∂p ∂x − f g γz q | q | 2 DS p w ± = 1 2( q ± √ γzSp ) √ γz ∂ x w ± = 1 1 ∂ t w ± ± 2 f ( q, p ) Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 11/42

One Pipe - Speed and Accuracy Simulation of a Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∆ h 1000 300 50 10 Midpoint Discretization 68.36932 68.36932 68.36932 68.36932 Left/Right Discretization 68.36541 68.36834 68.36912 68.36928 Decoupled Discretization 68.36932 68.36932 68.36932 68.36932 True Value 68.36932 68.36932 68.36932 68.36932 Table: Accuracy of the stationary solution ∆ h 250 250 100 100 10 Solver ode15s IMEX ode15s IMEX IMEX Midpoint Discretization 4.97 0.02 35.9 0.03 0.18 LeftRight Discretization 1.29 0.01 2.67 0.02 0.11 Decoupled Discretization 1.22 0.01 1.93 0.02 0.09 Table: Speed of a simple simulation Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 12/42

Dynamic Simulation Midpoint ∆ h = 300 Mass Flow @ Demand 71 62 Pressure @ Supply data 1 data 1 70 . 5 61 70 60 69 . 5 69 59 0 200 400 600 800 1 , 000 0 200 400 600 800 1 , 000 Mass Flow @ Supply Pressure @ Demand 62 data 1 data 1 68 . 1 61 68 60 59 67 . 9 0 200 400 600 800 1 , 000 0 200 400 600 800 1 , 000 Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 13/42

Dynamic Simulation Left/Right ∆ h = 300 Mass Flow @ Demand 71 62 Pressure @ Supply data 1 data 1 70 . 5 61 70 60 69 . 5 69 59 0 200 400 600 800 1 , 000 0 200 400 600 800 1 , 000 Mass Flow @ Supply Pressure @ Demand 62 data 1 data 1 68 . 1 61 68 60 59 67 . 9 0 200 400 600 800 1 , 000 0 200 400 600 800 1 , 000 Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 14/42

Comparison Numerical simulation of a pressure drop at the inlet of a pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Flow @ Demand 70 31 Pressure @ Supply 68 30 . 5 66 30 64 29 . 5 62 60 29 0 50 100 0 50 100 Mass Flow @ Supply Pressure @ Demand 75 mid mid 0 70 end end new new 65 − 500 60 − 1 , 000 55 0 50 100 0 50 100 Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 15/42

Nonlinear Decoupling Euler equation ∂ t ρ ( t, x ) + 1 S ∂ x q ( t, x ) = 0 , q ( t, x ) | q ( t, x ) | S ∂ t q ( t, x ) + ∂ x p ( t, x ) = − f g 1 2 dS 2 ρ ( t, x ) � 1 � ρ With w ± ( t, x ) = 1 0 λ ± ( s ) ds � where λ ± ( ρ ) = ± � S q + ∂ ρ p ( ρ ) we get 2 ∂ t w ± ( t, x ) + λ ± ∂ x w ± ( t, x ) = − 1 f g 2 dS 2 ( ρu )( w + , w − )( t, x ) | u ( w + , w − )( t, x ) | . 2 S. Grundel, M. Herty, Hyperbolic Discretization via Riemann Invariants submitted Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 16/42

Oszillations of mass flux at the inlet in steady state Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 17/42

A network with cycles Figure: Topology of the diamond network Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 18/42

Numerical Simulation on the diamond network Mass Flow @ Demand 71 50 Pressure @ Supply 70 . 5 40 70 69 . 5 30 69 0 20 40 60 80 100 0 20 40 60 80 100 50 71 Mass Flow @ Supply Pressure @ Demand mid mid 70 . 5 end end 40 new new 70 69 . 5 30 69 0 20 40 60 80 100 0 20 40 60 80 100 Sara Grundel, grundel@mpi-magdeburg.mpg.de hyperbolic MOR on energy networks 19/42