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

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Future

How could 2050 look like? Renewable Energies

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Future

Renewable Energies Mobility largely electric

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Future

Renewable Energies Mobility largely electric Housing efficient and smart

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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

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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

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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

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Funding and Collaborators

Federal Ministry for Economic Affairs and Energy

Manuel Baumann Christian Himpe Petar Mlinari´ c Neeraj Sarna Yue Qiu Michael Herty Philipp Sauerteig Martin Stoll Karl Worthmann

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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

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Gas transportation network

PDAE

hyperbolic PDE on the pipe ODEs or algebraic equations on other components algebraic node conditions

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Gas Transport in the Pipe

Isothermal Euler Equation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Density Transient Continuity Gravity Linear Elevation Friction Nonlinear Configurable ∂ρ ∂t = −S−1 ∂q ∂x ∂q ∂t = −S ∂p ∂x − Sgρ∂h ∂x − fg 2DS q|q| ρ p = γρz Mass-Flux Transient Momentum Gas State Pressure Density Compressibility Nonlinear Configurable

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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∂tφ(x, t) = Kφ(x, t) + f(φ(x, t), u(t), t) which discretized is M ˙ x = Kx + Bu + f(x, t)1, 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.

1Benner, G., Himpe, Huck,Streubel, Gas Network Benchmark Models, Springer, 2018

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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

• ptimal control/ optimization

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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

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Isothermal Euler and Discretization

Basic equation

∂p ∂t = − 1 γzS ∂q ∂x ∂q ∂t = −S ∂p ∂x − fgγz 2DS q|q| p

Naive Approach

∂p∗ ∂t = − 1 γzS qR − qL ∆x ∂q∗ ∂t = −S pR − pL ∆x − fgγz 2DS q∗|q∗| p∗

Decoupled approach

w± = 1 2(q ± √γzSp) ∂tw± ± 1 √γz ∂xw± = 1 2f(q, p)

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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

• de15s

IMEX

• de15s

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

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Dynamic Simulation Midpoint ∆h = 300

200 400 600 800 1,000 69 69.5 70 70.5 71 Pressure @ Supply data1 200 400 600 800 1,000 59 60 61 62 Mass Flow @ Demand data1 200 400 600 800 1,000 67.9 68 68.1 Pressure @ Demand data1 200 400 600 800 1,000 59 60 61 62 Mass Flow @ Supply data1

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Dynamic Simulation Left/Right ∆h = 300

200 400 600 800 1,000 69 69.5 70 70.5 71 Pressure @ Supply data1 200 400 600 800 1,000 59 60 61 62 Mass Flow @ Demand data1 200 400 600 800 1,000 67.9 68 68.1 Pressure @ Demand data1 200 400 600 800 1,000 59 60 61 62 Mass Flow @ Supply data1

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Comparison

Numerical simulation of a pressure drop at the inlet of a pipe

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50 100 60 62 64 66 68 70 Pressure @ Supply 50 100 29 29.5 30 30.5 31 Mass Flow @ Demand 50 100 −1,000 −500 Mass Flow @ Supply mid end new 50 100 55 60 65 70 75 Pressure @ Demand mid end new

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Nonlinear Decoupling

Euler equation

∂tρ(t, x) + 1 S ∂xq(t, x) = 0, 1 S ∂tq(t, x) + ∂xp(t, x) = − fg 2dS2 q(t, x)|q(t, x)| ρ(t, x) With w±(t, x) = 1

2

1

S q +

ρ

0 λ±(s)ds

• where λ±(ρ) = ±
• ∂ρp(ρ) we get

∂tw±(t, x) + λ±∂xw±(t, x) = −1 2 fg 2dS2 (ρu)(w+, w−)(t, x)|u(w+, w−)(t, x)|.

• S. Grundel, M. Herty, Hyperbolic Discretization via Riemann Invariants

submitted

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Oszillations of mass flux at the inlet in steady state

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A network with cycles

Figure: Topology of the diamond network

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Numerical Simulation on the diamond network

20 40 60 80 100 69 69.5 70 70.5 71 Pressure @ Supply 20 40 60 80 100 30 40 50 Mass Flow @ Demand 20 40 60 80 100 30 40 50 Mass Flow @ Supply mid end new 20 40 60 80 100 69 69.5 70 70.5 71 Pressure @ Demand mid end new

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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

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Input-Output Systems

(Possibly nonlinear) Input-Output System: ˙ x(t) = f(x(t), u(t)) y(t) = g(x(t), u(t)) Input: u : R → RM State: x : R → RN Output: y : R → RQ Vector Field: f : RN × RM → RN Output Functional: g : RN × RM → RQ N ≫ 1, M ≪ N, Q ≪ N Input-to-Output Mapping: u: :

ξ

→: : x: :

η

→: : y Is there a low(er) dimensional mapping η ◦ ξ : u → y? Find transformation T such that T(x) is sorted by I/O importance. System-theoretic approach: Quantify and balance ξ and η.

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Empirical Gramians

Empirical Controllability and Observability Gramians [Lall et al’99]: WC = ∞ ξ(t)ξ∗(t) dt, WO = ∞ η∗(t)η(t) dt Balancing WC and WO yields transformation T [Laub et al’87]. Truncating T leads to reducing projection T1. Purely data-driven computation via ξ and η. Also applicable to unstable, parametric or implicit systems [Himpe’16]. For linear (A,B,C) systems, this is balanced truncation [Moore’81]. Projection-Based Reduced Order Model: ˙ xr(t) = T1f(T −1

1

xr(t), u(t)) ˜ y(t) = g(T −1

1

xr(t), u(t))

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Model Order Reduction Online Time

100 200 300 400 500 600 700 100 101 102 State Dimension Relative Online Time Proper Orthogonal Decomposition (Structured) Dynamic Mode Decomposition (Structured) Empirical Balanced Truncation (Structured) Empirical Cross Gramian (Structured) Empirical Nonsymmetric Cross Gramian (Structured)

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Model Order Reduction L∞ Error

100 200 300 400 500 600 700 10−15 10−12 10−9 10−6 10−3 100 State Dimension Relative Output Error Proper Orthogonal Decomposition (Structured) Dynamic Mode Decomposition (Structured) Empirical Balanced Truncation (Structured) Empirical Cross Gramian (Structured) Empirical Nonsymmetric Cross Gramian (Structured)

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Hyperbolic Systems and Classical Model Order Reduction

Basic equation ∂p ∂t = − 1 γzS ∂q ∂x ∂q ∂t = −S ∂p ∂x − fgγz 2DS q|q| p Decoupled equation and solution w± = 1 2(q ± √γzSp) ∂tw± ± 1 √γz ∂xw± = 1 2f(q, p)

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Hyperbolic Systems and Classical Model Order Reduction

Basic equation ∂p ∂t = − 1 γzS ∂q ∂x ∂q ∂t = −S ∂p ∂x − fgγz 2DS q|q| p Decoupled equation and solution w± = 1 2(q ± √γzSp) ∂tw± ± 1 √γz ∂xw± = 0 w±(x, t) = w0(x ∓ 1 √γz t)

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Hyperbolic Systems and Classical Model Order Reduction

Basic equation ∂p ∂t = − 1 γzS ∂q ∂x ∂q ∂t = −S ∂p ∂x − fgγz 2DS q|q| p Decoupled equation and solution w± = 1 2(q ± √γzSp) ∂tw± ± 1 √γz ∂xw± = 0 w±(x, t) = w0(x ∓ 1 √γz t) 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 0.8 1 t=0 t=1 t=2 t=3

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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

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Similar Work

Rowley, Marsden, Reconstruction equations and the karhunen-loeve expansion for systems with symmetry, 2000 Gerbeau, Lombardi, Approximated lax pairs for the reduced order integration

• f nonlinear evolution equation, 2014

Peherstorfer, Model reduction for transport-dominated problaems via online adaptive basis and adaptive sampling, 2018 Nair, Balajewicz, Transported snapshot model order reduction approach for parametric, steady-state fluid flows containing parameter-dependent shocks, 2019 Rim, Mandli, Displacement interpolation using monotone rearrangement, 2018 Welper, Interpolation of functions with parameter dependent jumps by transformed snapshots, 2017 ...

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Hyperbolic Partial Differential Equation

The Model

∂tu(·, ·, µ) = L(u(·, ·, µ), µ) on Ω × [0, T], u(·, 0, µ) = u0(·, µ) on Ω G(u(·, ·, µ), µ) = 0 on ∂Ω × [0, T] µ ∈ P ⊂ R with P being a bounded parameter domain, T is the final time u0(·, µ) is the initial data Ω ⊂ Rd is a bounded and open spatial domain G(·, µ) prescribes some boundary conditions u(·, t, µ) ∈ X and u(x, t, µ) ∈ R. L(·, µ) : R → R is of the form L(·, µ) = −∇x · f(·, µ)

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Solving the full order model

Finite dimensional approximation space

L2(Ω) ⊃ X N = span{φi : φi = 1 √ ∆x ✶Ix

i , i ∈ {1, . . . , N}}.

Above, ✶A represents a characteristic function of the set A ⊂ R, and N = nx. Using X N, we express the evolution equation for the FOM as uN(·, tk+1, µ) = uN(·, tk, µ) + ∆t × LN(uN(·, tk, µ), µ), ∀k ∈ {1, . . . , K − 1}, where LN(·, µ) : X N → X N is an approximation of the original L(·, µ).

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Core of the reduction problem

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2

Figure: Snapshots taken from (a) M and (b) Mµ.

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Core of the reduction problem

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2

Figure: Snapshots taken from (a) M and (b) Mµ.

Consider the manifold M := {f(·, µ) : µ ∈ P} ⊂ L2(R), where f(·, µ) is a step function that scales and shifts to the right, and is given as f(x, µ) :=

• 1 + µ,

x ≤ µ 0, x > µ , µ ∈ P := [0, 1]. Furthermore consider Mµ :={f(ϕ(·, µ, ˆ µ), ˆ µ) : ϕ(·, µ, ˆ µ) = x − (µ − ˆ µ), ˆ µ ∈ P}, ={αf(·, µ) : α ∈ [1, 2]}.

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Core of the reduction problem

0.2 0.4 0.6 0.8 1 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 0.5 1 1.5 2

Figure: Snapshots taken from (a) M and (b) Mµ.

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Reduced Solution manifold

For each time discretization point tk we look for the solution in an approximation

• f this manifold;

Mµ,tk := {uN(ϕ(·, µ, ˆ µ, tk), tk, ˆ µ) : ˆ µ ∈ P}, namely uN(·, tk, µ) ≈ un(·, tk, µ) ∈ X n

µ,tk,

where X n

µ,tk := span{ψj µ,tk : ψj µ,tk = uN(ϕM(·, µ, ˆ

µj, tk), tk, ˆ µj), j ∈ {1, . . . , M}}. We compute a solution in X n

µ,tk using residual-minimisation. Writing the

finite-volume scheme from above as a residual minimisation problem provides uN(·, tk+1, µ) = arg min

w∈X N Res(w, uN(·, tk, µ))RN ,

∀k ∈ {0, . . . , K − 1}, un(·, tk+1, µ) = arg min

w∈X n

µ,tk+1

Res(Πµ,tk+1w, Πµ,tkun(·, tk, µ))RN .

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Algorithms Summary

Offline Phase: Algorithm for model reduction

• 1. Compute the FOM for all µ ∈ {ˆ

µj}j=1,...,M using the time-evolution scheme and a finite volume scheme.

• 2. Compute all the snapshots of the spatial transforms

{ϕ(x, ˆ µj, ˆ µl, tk)}j,l=1,...,M for all k ∈ {1, . . . , K}.

• 3. Perform the offline phase of hyper-reduction.

Online Phase: Algorithm for model reduction

• 1. For a given µ, approximate {ϕ(x, µ, ˆ

µj, tk)}j=1,...,M using polynomial interpolation.

• 2. Perform the online phase of hyper-reduction.
• 3. Compute un(·, tk, µ) for all k ∈ {1, . . . , K} using residual-minimisation and

hyper-reduction.

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Numerical Experiments

Example

Two dimensional Burger’s equation with parameterised initial data ∂tu(·, ·, µ) + 1 2∂xu(·, ·, µ)2 + 1 2∂yu(·, ·, µ)2 = 0, on Ω × [0, T]. We choose P = [1, 3], Ω = [0, 1] and T = 0.8. The initial data is given as u0(x, µ) =    µ × exp

• −1/
• 1 −
• x−δ1

δ2

2 ,

x−δ1 δ2

< 1 0, else . We set δ1 = (0.5, 0.5)T and δ2 = 0.2.

• N. Sarna, S.Grundel, Model Reduction of Time-Dependent Hyperbolic

Equations using Collocated Residual Minimisation and Shifted Snapshots submitted

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Quantitative Results

We compare S-ROM (snapshots based linear ROM) and SS-ROM (shifted snapshots based non-linear ROM).

1 1.5 2 2.5 3 480 490 500 510 520 530 1 1.5 2 2.5 3 10-3 10-2 10-1

Figure: Results for test-2. (a) Speed-up resulting from SS-ROM; and (b) EROM resulting from S-ROM and SS-ROM. Fig-(b) has a y-axis on a log-scale.

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Qualitative Results

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6

staircase effect

0.3 0.32 0.34 0.36 0.38 0.4 0.02 0.04 0.06

Figure: Results for test-2. FOM and the ROM along the cross-section x = y for µ = 2.6 and t = 0.8.

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Limitations

Two Feature Initial Condition

u0(x) =            µ exp

• −1/
• 1 −
• x−δ1

δ2

2 ,

• x−δ1

δ2

• < 1

− exp

• −1/
• 1 −
• x+δ1

δ2

2 ,

• x+δ1

δ2

• < 1

0, else

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Limitations

• 1.5
• 1
• 0.5

0.5 1 1.5

• 0.4
• 0.2

0.2 0.4 0.6 Figure: Results that show the limitation of SS-ROM. Computed with one-dimensional Burger’s equation with the initial data as given in (36). The solutions are for µ = 2 and t = 1.

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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

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Preconditioning

Table: Computational time for the 1st Newton iteration

h #DF tS1 IDR(4) backslash 40 1.03e+05 3.85 0.25 0.13 20 2.01e+05 8.12 0.52 0.36 10 3.97e+05 17.84 1.06 1.18 5 7.91e+05 38.44 2.13 1054.62 2.5 1.58e+06 81.42 4.34

• FVM and Iterative Solvers2,

2Yue, Grundel, Stoll, Benner Efficient Numerical Methods for Gas Network Modeling and

Simulation, arXiv

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Dynamic Power Flow

Generator model (swing equation) Mi¨ δi(t) + Di ˙ δi(t) = Pi −

N

• j=1

aij sin(δi(t) − δj(t)), 1 20,000 40,000 0.2 0.4 0.6 POD + k-means partition ID 5 10 1 2 3 t δi, ˆ δi Mlinari´ c, P., Grundel S., and Benner P. Decision and Control (CDC), 54th Annual Conference on. IEEE, 2015. Jongsma, H.-J.; Mlinari´ c, P.; Grundel, S.; Benner, P.; Trentelman, H. L.: Mathematics of Control, Signals, and Systems 30 (1), 6 (2018)

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Coupled Microgrids

We consider four microgrids (MGs), connected via transportation lines. Grundel S., Sauerteig P., Worthmann K., Surrogate models for coupled microgrids Progress in Industrial Mathematics at ECMI 2018,

MG1 MG2 MG3 MG4

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 0.1 0.2 0.3 0.4 0.5 0.6 0.7 true fit 5 10 15 20 25 0.05 0.1 0.15 0.2 5 10 15 20 25 0.5 1 1.5

Error in local approximation 10−2 but in global problem 10−1

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Bidirectional Optimization of coupled microgrids

Figure: Impact of mapping error (top) and approximation via RBF and NN (bottom) on the costs within 48 consecutive time steps.

• M. Baumann, S. Grundel, P. Sauerteig, and K. Worthmann. Surrogate

models in bidirectional optimization of coupled microgrids. at-Automatisierungstechnik, 67(12), 1035-1046, 2019.

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Thank you for your attention

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