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It’s all about power: definition, modeling and control of swarm type direct current microgrids

Lia Strenge

T alk, Workshop on Industrial and Applied Mathematics 2016 September 1, 2016

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Outline

1

Introduction

2

Modeling and simulation frameworks

3

Possible interfaces to applied mathematics

4

Modeling

5

Control

6

Ongoing and future work

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 2/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Outline

1

Introduction

2

Modeling and simulation frameworks

3

Possible interfaces to applied mathematics

4

Modeling

5

Control

6

Ongoing and future work

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 3/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Solar Home Systems in Bangladesh

Figure: Solar Home System (SHS) (left); SHSs in Raipura, Bangladesh (right)

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 4/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

SHS Definition

An electrical energy provision system is called Solar Home System (SHS) if it satisfies the following conditions.

1

It is composed by a photovoltaic (PV) panel, a battery storage, a charge controller and domestic loads.

2

The voltage level is 12-220 volt direct current (DC), where 12 volt is more common.

3

It usually operates independently as islanded system. It can be connected to other solar home systems through the swarm concept.

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 5/ 40

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The swarm type low voltage direct current microgrid

Figure: Swarm electrification. Tiers from the World Bank Multi-Tier Framework for Energy Access.

Source: S. Groh and M. Koepke. A system complexity approach to swarm electrification. International Symposium for Next Generation Infrastructure, Vienna, Austria, 2014.

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 6/ 40

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Swarm concept [Str15] Definition

The swarm concept is a bottom-up electrification scheme which

1

interconnects existing generation, storage and consumption units to form an electrical power grid, typically where a cluster

• f stand-alone energy provision systems is installed, e.g., SHSs.

This setting is called swarm type cluster;

2

allows for plug-and-play operation, i.e., each unit can connect to or disconnect from the grid;

3

grows organically, i.e., the network topology changes [...] arise spontaneously when a new household is connected. It interconnects swarm type clusters with each other or with existing true islanded power systems called microgrids, minigrids or nanogrids in practice through a point of common

• coupling. This setting is called swarm type microgrid;

4

grows towards and eventually reaches the main power grid, which is usually operated by a national power entity, in order to draw or feed-in power.

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 7/ 40

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The swarm type low voltage direct current microgrid

Figure: Swarm electrification. Tiers from the World Bank Multi-Tier Framework for Energy Access.

Source: S. Groh and M. Koepke. A system complexity approach to swarm electrification. International Symposium for Next Generation Infrastructure, Vienna, Austria, 2014.

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 8/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Problem statement

PhD thesis (objective): How can a swarm type low voltage DC microgrid be modularly modeled, controlled and simulated in order to contribute to the technology development? → I am in the first year comparing different modeling and simulation frameworks for control and did some preliminary work in my master thesis

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 9/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Outline

1

Introduction

2

Modeling and simulation frameworks

3

Possible interfaces to applied mathematics

4

Modeling

5

Control

6

Ongoing and future work

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 10/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Modeling frameworks, qualitative comparison

Framework → Criteria ↓ ODEs DAEs DESs HS BG pH Pre-knowledge

• f the author

high high medium medium low low Access to gurus very high high very high very high low low Existing theory for control very high advanc. advanc. advanc. ? ? Usage in (con- trol) application very high medium high increas. ? ? Analytical han- dling easy

• k

for s-free diffic. diffic. ? ? Numerical han- dling easy index dep. easy diffic. ? ? Bidirectional power flow diffic. diffic. easy easy easy? easy

ODE - ordinary differential equ., DAE - differential-algebraic equ., DES - discrete event sys., HS - hybrid sys., BG - bond graph, pH - port-Hamiltonian

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 11/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Simulation frameworks

Programm → Criteria ↓ Dymola Matlab/ Simulink Octave Scilab/ Xcos, Power Devs 20- sim Phython Cost [Euro] > 1000 >> 2000 free free > 1000 free Open Source No* No Yes Yes Not any- more Yes Platform W L,M,W L L L,W W,M,L?

* language Modelica yes. (L)inux, (M)ac OS, (W)indows

Other suggestions?

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 12/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Outline

1

Introduction

2

Modeling and simulation frameworks

3

Possible interfaces to applied mathematics

4

Modeling

5

Control

6

Ongoing and future work

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 13/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Maths and engineering research: a clash of paradigms?

Assumptions (Applied, numerical) mathematics wants to advance in theory: from general to specific Practical engineering wants to build new machines and use modeling and simulation for component design and parameter selection: from specific to general Result for me Theoretical engineering with focus on control wants to apply the theory: general statements for a specific class of models Questions What are the incentives to bridge the communication gaps? Are those gaps a result of community paradigms? (A math PhD student has to publish in math journals and an engineering PhD student in engineering journals)

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 14/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

T

• uched areas of applied mathematics

control systems based on ordinary differential equations differential-algebraic equations (descriptor systems) graph theory switched systems

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The standard feedback control loop

We have r : I → R as reference/setpoint, e : I → R as control error, u : I → R as input or control, y : I → R as output, ym : I → R as measured output. Controller System Disturbances u Measurements r e y − ym

Figure: Standard control loop: single-input-single-output (SISO).

Source: www.texample.net/tikz/examples/control-system-principles/.

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 16/ 40

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Differential-algebraic equations (DAEs) Definition

A set of equations of the form 0 = F(t, x(t), ˙ x(t)), (short 0 = F(t, x, ˙ x)) (1), with F: I × Dx × D ˙

x → Cne; Dx, D ˙ x ⊆ Cns are suitable open sets;

ne, ns ∈ N; I ⊆ R is a compact interval; is called a set of differential-algebraic equations (DAE). Furthermore x: I → Cns are called the state variables or unknown variables. t ∈ I is called the independent variable. If in addition to (1) an initial condition x(t0) = x0 with t0 ∈ I, x0 ∈ Cns (2) exists, then (1), (2) is called initial value problem (IVP) and x0 is called the initial value. [Ste13] If a set of equations can be written in the form ˙ xd(t) = ˜ f(t, xd(t), xa(t)) 0 = ˜ g(t, xd(t), xa(t)) with (˜ f T, ˜ gT)T : I × Rnd × Rna → Rne; nd, na, ne ∈ N; it is called a semi-explicit DAE. xd are called differential state variables (or differential states) and xa algebraic state variables (or algebraic states). In addition, we have nd + na =: ns as total number of states.

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 17/ 40

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DAEs and the strangeness-index Hypothesis

Consider F ∈ Cμ(D, Rne) with D = I × Dx × D ˙

• x. Let there exist μ, d, a ∈ N0, such

that Lμ = ∅ and for every z0 ∈ Lμ there exist a sufficiently small neighborhood B(z0) ⊂ Lμ, such that the following properties hold.

1

We have rank(Mμ(z0)) = (μ + 1)ne − a on Lμ and there exists a matrix function Z2 with orthonormal columns and maximal rank(Z2) = a on Lμ, such that ZT

2Mμ = 0 on Lμ. Locally, we have that

Z2 ∈ Cμ(B(z0), R(μ+1)ne×a).

2

We have rank(ZT

2 ¯

Nμ) = a on Lμ, where ¯ Nμ = Nμ[Ine 0 . . . 0]T, and there exists T1 with orthonormal columns and maximal rank(T1) = d, d = ne − a, on Lμ, such that ZT

2 ¯

NμT1 = 0 on Lμ. Locally, we have that T1 ∈ Cμ(B(z0), Rne×d).

3

We have rank(F ˙

x(t, x, ˙

x)T1(zμ)) = d on Lμ and there exists Z1 ∈ Rne×d with orthonormal columns and maximal rank(Z1) = d on Lμ, such that rank(ZT

1F ˙ xT1) = d on Lμ.

The smallest μ for which the hypothesis holds, is called the s-index (strangeness-index) of F. If μ = 0, then F(t, x, ˙ x) = 0 is called s-free.

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 18/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Outline

1

Introduction

2

Modeling and simulation frameworks

3

Possible interfaces to applied mathematics

4

Modeling

5

Control

6

Ongoing and future work

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 19/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

The swarm type low voltage DC microgrid model

Figure: Generic DC microgrid adapted from S. Anand and G.B. Fernandes.

Reduced-Order Model and Stability Analysis of Low-Voltage DC Microgrid. IEEE Transactions on Industrial Electronics, 60(11):5040-5049, 2013.

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Network as oriented undirected graph

Vertices → (grid) nodes where the SHSs are connected to the grid. Edges → connections or power lines between the SHSs. For the connection ¯ jk of two grid nodes j and k with j, k ∈ {1, 2, ..., N} =: N ⊂ N, j < k containing power line h ∈ {1, 2, ..., m} =: M ⊂ N, m ∈ {η ∈ N | η ≤ N(N − 1)/2}, we define the edge-vertice incidence matrix M :=    m1,1 ... m1,N . . . ... . . . mm,1 ... mm,N    ∈ Rm×N, with mh,j = 1 and mh,k = −1 and all others mh,κ = 0 with power line h ∈ M not connecting the SHS grid node κ ∈ N. We order the connections as pq, pr, qr with p, q, r ∈ N, p < q < r ≤ N}. Remark: M is singular!

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Electrical equivalence circuit

fα Lc,h ic,h Rc,h Ccap,α vgn,α icap,α Ccap,β vgn,β icap,β fβ producing SHS consuming SHS igh,α gnα igh,β gnβ vc,h Lc,λ ic,λ

Figure: Electrical equivalence circuit for two connected SHSs [Str15]

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Modeling the SHS

We define fj(t): = gP,j(t) , j ∈ NP gC,j(t) , j ∈ NC , with gP,j : I → R≥0 as producer current in ampere and gC,j : I → R<0 as consumer current in ampere, j ∈ N. The SHS as producer For the SHS as producer, we get a controlled current source gP,α(t) = uα(t), with uα : I → R as input, α ∈ NP. The SHS as consumer Constant power load (CPL): gC,β(t) = Pgh,given,β vgn,β(t) with β ∈ NC Constant current load (CCL): gC,β(t) = igh,C,given,β < 0, which has the (mathematical) advantage of being a linear equation.

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Model equations: uncontrolled system

We reorder the equations and get the uncontrolled model for t ∈ I.

Uncontrolled model (∗)

Lc dic(t) dt = vc(t) − Rcic(t), ˆ Ccap dˆ vgn(t) dt = ˆ icap(t), = ˆ Mˆ vgn(t) − vc(t), = ˆ MTic(t) −ˆ igh(t), = −ˆ icap(t) −ˆ igh(t) + gP(t) gC(t)

• ,

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 24/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Outline

1

Introduction

2

Modeling and simulation frameworks

3

Possible interfaces to applied mathematics

4

Modeling

5

Control

6

Ongoing and future work

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 25/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Control: Droop control as standard control scheme

1

Control objectives

2

Droop control

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Control objectives

1

Bounded voltage deviation in steady state from the reference voltage of the grid | vgn,j(t) − vgn,ref,j | ≤ eV, ∀t ∈ I, j ∈ N , eV ∈ R>0, limt→∞ | vgn,j(t) − vgn,ref,j | ≤ eV,∞ ≤ eV, j ∈ N .

2

Power sharing of the producing SHSs, i.e., limt→∞Pgh,α(t) = limt→∞Pgh,˜

α(t)∀α, ˜

α ∈ NP. Next step: Introduce droop control and check if the closed-loop model meets the control objectives. For which droop coefficients?

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Droop control

Decentralized control scheme for the producing SHSs introducing a virtual resistance ddroop,α ∈ R>0 in ohm, α ∈ NP. uα(t) = gP,α(t) = 1 ddroop,α (vgn,ref,α − vgn,α) = kdroop,α(rα − yα) [Kdroop O] system (disturbances) u (measurements) r = vgn,ref e y = ˆ vgn − ym = y

Figure: Closed-loop model with droop control in standard feedback control loop representation without measurement dynamics or disturbances, Kdroop = diag(kdroop,α) and vgn,ref = vector(vgn,ref,j), j ∈ N

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Closed-loop model

Adding droop control and the nonlinear constant power load model gC,β(t) = Pgh,given,β vgn,β(t) , β ∈ NC to the uncontrolled model (∗), we

• btain the nonlinear DAE (∗∗) for t ∈ I.

Closed-loop model (∗∗)

Lc dic(t) dt = vc(t) − Rcic(t), ˆ Ccap dˆ vgn(t) dt = ˆ icap(t), = ˆ Mˆ vgn(t) − vc(t), = ˆ MTic(t) −ˆ igh(t), = −ˆ icap,P(t) −ˆ igh,P(t) + Kdroop(vgn,ref − ˆ vgn,P(t)), = −ˆ vgn,C(t) ◦ˆ icap,C(t) − ˆ vgn,C(t) ◦ˆ igh,C(t) + Pgh,given.

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Closed-loop model

Adding droop control and the nonlinear constant power load model gC,β(t) = Pgh,given,β vgn,β(t) , β ∈ NC to the uncontrolled model (∗), we

• btain the nonlinear DAE (∗∗) for t ∈ I.

Closed-loop model (∗∗)

Lc dxd,1 dt = xa,1 − Rcxd,1, (2) ˆ Ccap dxd,2 dt = xa,2, (3) = ˆ Mxd,2 − xa,1, (4) = ˆ MTxd,1 − xa,3, (5) = −xa,P,2 − xa,P,3 + Kdroopvgn,ref,P − Kdroopxd,P,2, (6) = −xd,C,2 ◦ xa,C,2 − xd,C,2 ◦ xa,C,3 + Pgl,given. (7)

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Condition for a strangeness-free DAE Lemma

The closed-loop model (∗∗) is strangeness-free with d = m + N, a = m + 2N if vgn,β(t) > 0, β ∈ NC, ∀t ∈ I.

Proof.

We omit the time argument of the states hereafter and write (∗∗) in the form ˙ xd = ˜ f(xd, xa), 0 = ˜ g(xd, xa) with xd := (iT

c , ˆ

vT

gn)T,

xa := (vT

c ,ˆ

iT

cap,ˆ

iT

gh)T and (˜

f T, ˜ gT)T as right side of (∗∗). We need to show that ˜ gxa is regular.

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Proof Proof.

˜ gxa =    −Im Om×N Om×N Om×N ON×N −IN Op×m Ol×m

• −Ip

Op×l Ol×p −diag(ˆ vgn,C)

• −Ip

Op×l Ol×p −diag(ˆ vgn,C)

• 

  Let ˜ P be a suitable permutation matrix of dimension m + 2N.

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Proof (II) Proof.

We get that ˜ P˜ gxa =    −Im Om×N Om×N Op×m Ol×m

• −Ip

Op×l Ol×p −diag(ˆ vgn,C)

• −Ip

Op×l Ol×p −diag(ˆ vgn,C)

• Om×N

ON×N −IN    . ˜ P˜ gxa is regular since it is an upper triangular matrix with full diagonal for diag(ˆ vgn,C) > 0∀t ∈ I. Hence, the DAE (∗∗) is strangeness-free.

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T echnical interpretation and conclusion

Preliminary results for droop control (based on exemplary simulation) Constant offset of maximum 2.4 volt, i.e., 5 percent Equal power sharing achieved for simple network topologies Conclusion We have a modular mathematical representation of the swarm type low voltage DC microgrid allowing for control design and simulation Open issues until a possible contribution to the technology dev. Mode switching between producer and consumer (prosumer) Inclusion of Pgh,max in order to evaluate proportional power sharing

, Definition, modeling and control of swarm type direct current microgrids | Lia Strenge | September 1, 2016 34/ 40

Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Outline

1

Introduction

2

Modeling and simulation frameworks

3

Possible interfaces to applied mathematics

4

Modeling

5

Control

6

Ongoing and future work

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Introduction Modeling and simulation frameworks Interfaces to applied mathematics Modeling Control Ongoing and future work

Work in progress

Simulation study on parameter variations for linear and nonlinear model and a fixed network topology T est bed development to verify modeling assumptions by experimental data

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Switching: mode changes in the SHS (I - time-based)

We have a partition of the time interval I = μ

ν=0 Iν, with

Iν = [tν, tν+1), Iμ = [tμ, ∞) and T := {t0, t1, t2, ..., tν, ..., tμ}.

Assumption

We assume a fixed q(0) ∈ {0, 1}N for all t ∈ I0. At t = t1, the first power flow change including a mode change from producer to consumer or vice versa of at least one SHS j ∈ N takes place with a sign change of the corresponding igh,j and therefore the switching of the corresponding q(0),j. Hence, we have a new q(1) ∈ {0, 1}N for all t ∈ I1. Analogously, at t = tν ∈ T, the ν-th mode power flow change including a sign change of at least one igh,j, j ∈ N takes place and the corresponding q(ν−1),j switch(es). We have q(ν) ∈ {0, 1}N for all t ∈ Iν. Hence, we have q: T → {0, 1}N; q = q(tν) = q(ν). Hence, fj(t, qj): = gP,j(t) , qj = 1 gC,j(t) , qj = 0 , with gP,j : I → R≥0 as producer current in ampere and gC,j : I → R<0 as consumer current in ampere, j ∈ N.

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Future work

Analytic derivation of boundaries for stable operation of the swarm type DC microgrid Measurements to verify modeling assumptions by experimental data

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Bibliography I

• A. Steinbrecher.

Differential-Algebraic Equations. TU Berlin, Numerical Mathematics, Lecture Notes, 2013. Lia Strenge. Modeling and simulation of a droop controlled swarm type low voltage DC microgrid in a DAE framework. Master’s thesis, T echnische Universität Berlin, 2015.

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Summary and discussion

Summary Strangeness-free closed-loop model without switching Bi-directional power flow by discrete event systems or hybrid system theory Thank you for your attention!

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Strangeness-free differential-algebraic equations (DAEs)

Advantages of strangeness-free DAEs Closer to physical intuition compared to a manually derived

• rdinary differential equation (ODE) in fewer variables

A large part of the ODE theory has been transferred to strangeness-free DAEs Decreased numerical differentiation and hence increased numerical stability compared to problems of higher s-index No hidden constraints

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Research projects at Control Systems Group

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