power balancing in a dc microgrid elevator system through...

Power balancing in a DC microgrid elevator system through constrainedoptimization

Thanh Hung PHAM, Ionela PRODAN and Laurent LEFEVRE

Grenoble INP (Institut National Polytechnique de Grenoble),LCIS (Laboratoire de Conception et d’Integration des Systemes), Valence, France,

[email protected],[email protected]

This work was supported by a mobility project of the Romanian National Authority forScientific Research and Innovation, CNCS - UEFISCDI, project number

PN-III-P1-1.1-MCT-2016-0037, within PNCDI III

T.H. Pham, I. Prodan, L.Lefevre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Integration des Systemes), Valence, France, [email protected],[email protected])Power balancing in a DC microgrid December 8th, 2016 1 / 27

Introduction

Outline

1 Introduction

2 DC microgrid modelingPort-Hamiltonian system on graphsDC microgrid elevator system modeling

3 Battery scheduling by optimization-based controlEnergy-preserving discrete-time modelScheduling formulation

4 SimulationSimulation software and numerical dataNominal scenarioPerturbation-affected scenario

5 Conclusions

6 Reference


Introduction

DC microgrid elevator system

Battery

Three-phase

electrical

network

Solar panel

Synschronous

machine

Mechanical

system

DC/DC

converter

DC/DC

converter

AC/DC

converter

AC/DC

converter

DC microgrid elevator system, Pham et al. (2015)


Introduction

IntroductionGeneral goal: Constrained optimization control for for efficiently managing the DC microgridoperation.

Battery

Three-phase

electrical

network

Solar panel

Synschronous

machine

Mechanical

system

DC/DC

converter

DC/DC

converter

AC/DC

converter

AC/DC

converter

DC microgrid elevator system, Pham et al. (2015)

State of the art:

bus voltage control (Alamir et al. (2014);Zonetti et al. (2015))⇒ do not optimize electricity cost,

logic rules (Xu and Chen (2011))⇒ high storage capacity and not efficient,

offline optimization-based control approach(Lifshitz and Weiss (2014))⇒ lack of the robustness,

Economic MPC (Parisio et al. (2016);Touretzky and Baldea (2016))⇒ looses relevant details in what regardsthe physical power-preserving connection,neglects the nonlinear storage dynamic andthe system dissipation.

Solution:

Port-Hamiltonian (PH) formulation for the modeling (van der Schaft and Maschke (2013)),

Energy-preserving time discretization model (Talasila et al. (2006)),

Centralized economic Model Predictive Control (MPC) design (Rawlings and Mayne (2009)).


DC microgrid modeling

Outline

1 Introduction




5 Conclusions

6 Reference


DC microgrid modeling Port-Hamiltonian system on graphs

Bond graph and Port-Hamiltonian systemBond Graph

+

_

+

_

+

_

+

_

+ _+ _

Example: Bond Graph for simple series and parallel DC electricalcircuit.

AdvantageExplicit description of the exchange of power, ofthe dissipation and of the energy storage formulti-physics system.

Dirac structure and PH system

Dirac structure and port-Hamiltonian systems.

Constrained input-output representationFor all PH system, there exists λ(t) such that

e(t) = Jf(t) + Gλ(t),

0 = GT f(t),

e(t) =

∇H(x)eR(t)eE (t)

, f(t) =

−x(t)fR(t)fE (t)

,J = −JT

x(t) : state vectorf(t) : flow vector (current, voltage, speed, force, ...)e(t) : effort vector (voltage, current, force, speed, ...)H(x) : Hamiltonian (energy function)∇H(x) : gradient of H(x)


DC microgrid modeling Port-Hamiltonian system on graphs

PH system on graphs for RC circuit

RC circuit graphConsider a circuit including:

Ne edges: NS capacitors, NR resistors, NE

external elements,

Nv vertices: nodes between the edges.

The incidence matrix B ∈ RNv×Ne :

Bij =

1, if node i is a head vertex of edge j,−1, if node i is a end vertex of edge j,

0, else.

PH system on graphs formulatione(t) = −BT vp(t),

0 = Bf(t),

e(t) =

∇H(x)vR(t)vE (t)

, f(t) =

−x(t)iR(t)iE (t)

,x(t) : capacitor charge (state vector)iR (t), iE (t) : currentvR (t), vE (t) : voltagevp(t) : potential of the vertices

RC electrical circuit: example

+

_

+_

2 3

1

+

_

2

1

+

_

+

_

Edge order: C-R-E

+

_

Energy stored in the capacitor:

H(x) =1

2

x(t)2

C.


DC microgrid modeling DC microgrid elevator system modeling

DC microgrid elevator system model

Battery

Three-phase

electrical

network

Solar panel

Synschronous

machine

Mechanical

system

DC/DC

converter

DC/DC

converter

AC/DC

converter

AC/DC

converter

v(t) : voltagei(t) : currentP(t) : powerd(t) : converter duty cycle

xi (t) : i th state variable (charge)∂xi

H : partial derivative of H with respect to xi (voltage)

xi (t) : time derivative of xi (current)R : resistor

+_

+

_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

+

__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid



Model of the components

+_

+

_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

+

__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid

External grid is a current source ie(t):

ie,min ≤ ie(t) ≤ ie,max .

Load is a power profile Pl (t):

il (t)vl (t) = Pl (t).

Renewable source is a power profile Pr (t):

ir (t)vr (t) = Pr (t).

Battery admits the stored energy:

H(x) , x(t)T Q1 +1

2x(t)T Q2x(t),

Battery charge limitation:

0.5xmax ≤ x(t) ≤ xmax ,

Battery current limitation:

imin ≤ ib,R2(t) ≤ imax .



Model of the components

+_

+

_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

+

__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid

DC/DC converter respects the power-preservingrelation:

d(t)ic1(t) = −ic2(t),

vc1(t) = d(t)vc2(t),

with the positive duty cycle:

d(t) > 0.

Resistor network includes

the resistors of battery,

the resistors of transmission lines.

The Ohm’ law is:

vR(t) = −RiR(t),

where R is positive diagonal matrix.



DC microgrid network

+_

+

_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

+

__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid

Microgrid network:v(t) = −BT vp(t),

0 = Bi(t),

Ground potential (node 1): vp1(t) , 0,Incidence matrix:

B =

1T

2 1T3 1T

2 0

I2 0 0 B1

0 I3 0 B2

0 0 I2 B3

,

Current and voltage of the energy sources

iE (t) ,

il (t)ie(t)ir (t)

, vE (t) ,

vl (t)ve(t)vr (t)

,Current and voltage of the elements:

i(t) ,

−x(t)iE (t)ic (t)iR(t)

, v(t) ,

∇H(x)

vE (t)vc (t)vR(t)

.T.H. Pham, I. Prodan, L.Lefevre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Integration des Systemes), Valence, France, [email protected],[email protected])Power balancing in a DC microgrid December 8th, 2016 11 / 27


Global DC microgrid model

+_

+

_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

+

__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid

Dynamics:

[−x(t)iE (t)

]= L(d)

[∇H(x)

vE (t)

],

0 = A1(d)

[∇H(x)

vE (t)

],

Pl (t) = vl (t)il (t), Pr (t) = vr (t)ir (t).

Constraints:

ie,min ≤ ie(t) ≤ ie,max ,

0.5xmax ≤ x(t) ≤ xmax ,

imin ≤ A2(d)

[∇H(x)

vE (t)

]≤ imax ,

0 < d(t)

The interconnection matrices B1,B2,B3, resistor matrix R, and duty cycle d(t)⇒ L(d), A1(d), A2(d).


Battery scheduling by optimization-based control

Outline

1 Introduction




5 Conclusions

6 Reference


Battery scheduling by optimization-based control Energy-preserving discrete-time model

Energy-preserving discrete-time modelThe discrete-time model preserves:

the DC network v(j) = −BT vp(j),

0 = Bi(j),

the linear form of the Ohm’ law

vR(j) = −RiR(j),

the power-preserving relation of DC/DC converterd(t)ic1(j) = −ic2(j),

vc1(j) = d(t)vc2(j),

the stored energy in the battery (the chain rule)

H(j)− H(j − 1)

h= ∇H(j)ˇx(j).

⇒ The energy conservation property:

H(x(j))− H(x(j − 1)) = ie(j)ve(j)h − vR(j)T R−1vR(j)h +

jh∫(j−1)h

(Pl (τ) + Pr (τ))dτ.


Battery scheduling by optimization-based control Scheduling formulation

Scheduling formulation

Low level

control

Assume that the load voltage is forced to adesired value vref ∈ R:

vl (t) = vref .

The electricity cost:

C(t+jh|t) = price(t+jh|t)·ie(t+jh|t)·ve(t+jh|t).

with the electricity price price(t).Control laws is defined by:

ie(t|t) = argminie (t)

N∑j=1

γC(t + jh|t),

subject to:discrete-time dynamic,discrete-time constraints.

⇒ The optimization problem is nonlinear both incost and in constraints.⇒ IPOPT solver (Biegler and Zavala (2009)).


Simulation

Outline

1 Introduction




5 Conclusions

6 Reference


Simulation Simulation software and numerical data

Simulation software and numerical data

The simulation is implemented by using Yalmip (Lofberg (2004)), IPOPT (Wachter (2002)) andMatlab 2015a.

Name Notation ValueClosed-loop sampled time [s] 36Scheduling time step h [s] 1800Prediction horizon N 48Weighting parameter γ ∈ (0, 1) 0.5Battery parameters Q1 [V ] [ 13 13 ]T

Q2 [V /C ] diag 0.3036, 0.2024Battery constraints xmax [Ah] [ 73.2 109.8 ]T

ib,min [A] -20ib,max [A] 20

Grid constraints ie,min [A] -8ie,max [A] 8

Load voltage reference vref [V ] 380Resistors R [Ω] diag 0.012, 0.015, 0.31, 0.29, 0.23, 0.19


Simulation Simulation software and numerical data

Simulation software and numerical data

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

500

1,000

Pl[W

]

Load power profile

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0

200

400

600

Pr[W

]

Renewable power profile

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240.12

0.13

0.14

0.15

0.16

Time [h]

pric

e[e

ur/k

Wh] Electricity price profile


Simulation Nominal scenario

Nominal scenario

+_

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_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

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__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid

The battery State of Charge (SoC)

SoC1 ,x1

x1,max,

SoC2 ,x2

x2,max,

SoC ,x1 + x2

x1,max + x2,max.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.6

0.8

1

SoC

1[%

]

State of charge 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.6

0.8

1

SoC

2[%

]

State of charge 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.6

0.8

1

SoC

[%]

State of charge

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24−10

−5

0

5

10

Time [h]

i b,R

2[A

]

Battery current


Simulation Nominal scenario

Nominal scenario

+_

+

_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

+

__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24−500

0

500

1,000

Time [h]

Ele

ctri

calpow

er[W

]

Electrical power of the DC microgrid components

storage unit: vc2(t) · ic2(t)load: −Pl(t)external grid: ve(t) · ie(t)renewable: Pr(t)


Simulation Perturbation-affected scenario

Perturbation-affected scenario

+_

+

_

+ _

Load power

source

+ _

Renewable

power source

+_

+_

+

__

+

+

_

+

_

+_

+_

1

11

1

2

3

4

5 6

7

External

grid

Assumption:

Pl (t) ∈ Pl (t) [1− εlmin, 1 + εlmax ] ,

Pr (t) ∈ Pr (t) [1− εrmin, 1 + εrmax ] ,

with the simulation values:

εlmin = εlmax = 0.2,

εrmin = εrmax = 0.2.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240.4

0.6

0.8

1

SoC

1[%

]

State of charge 1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.6

0.8

1

SoC

2[%

]

State of charge 2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.6

0.8

1

SoC

[%]

State of charge

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24−10

−5

0

5

10

Time [h]

i b,R

2[A

]

Battery current


Simulation Perturbation-affected scenario

Perturbation-affected scenario

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24−500

0

500

1,000

1,500

Time [h]

Ele

ctri

calpow

er[W

]

Electrical power of the DC microgrid components under perturbation


9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13−500

0

500

1,000

1,500

Time [h]

Ele

ctri

calpow

er[W

]

Electrical power of the DC microgrid components under perturbation



Conclusions

Outline

1 Introduction




5 Conclusions

6 Reference


Conclusions

Conclusion

Contributions:

the DC microgrid is modeled through Port Hamilonian formulations with the advantage ofexplicitly taking into account the power conservation of the system interconnections;

the constrained optimization problem proposed which finds the optimum balance betweenbattery usage and the profit gained from electricity management;

the simulation results for the energy management of a particular DC microgrid elevatorsystem which validate the proposed approach.

Future work:

stability by considering the properties and specific form of Port Hamiltonian formulations;

robustness by taking explicitly in consideration the disturbances;

improvements in the cost function formulation and constraints, etc.

extension of this approach by taking explicitly into account different times scales in thecontrol design scheme.


Reference

Outline

1 Introduction




5 Conclusions

6 Reference


Reference

Reference

Mazen Alamir, M.A. Rahmani, and D. Gualino. Constrained control framework for a stand-alone hybrid. Applied Energy, 118:192–206, 2014.

Lorenz T Biegler and Victor M Zavala. Large-scale nonlinear programming using ipopt: An integrating framework for enterprise-wide dynamic optimization.Computers & Chemical Engineering, 33(3):575–582, 2009.

D. Lifshitz and G. Weiss. Optimal control of a capacitor-type energy storage system. IEEE Transactions on Automatic Control, 60(1):216–220, May 12 2014.

Johan Lofberg. YALMIP: A toolbox for modeling and optimization in MATLAB. In IEEE International Symposium on Computer Aided Control SystemsDesign, pages 284–289. IEEE, September 2004.

A. Parisio, E. Rikos, and L. Glielmo. Stochastic model predictive control for economic/environmental operation management of microgrids: an experimentalcase study. Journal of Process Control, 43:24–37, 2016.

T. H Pham, I. Prodan, D. Genon-Catalot, and L. Lefevre. Port-Hamiltonian model and load balancing for DC-microgrid lift systems. In 5th IFAC Workshopon Lagrangian and Hamiltonian Methods for Non Linear Control, Lyon, France, July 2015. IFAC.

J.B. Rawlings and D.Q. Mayne, editors. Model Predictive Control: Theory and Design. Nob Hill Publishing, 2009.

V. Talasila, J. Clemente-Gallardo, and A.J. van der Schaft. Discrete port-hamiltonian systems. Systems & Control Letters, 55:478–486, August 2006.

C. R. Touretzky and M. Baldea. A hierarchical scheduling and control strategy for thermal energystorage systems. Energy and Buildings, 110(8):94–107,2016.

A. J. van der Schaft and B. M. Maschke. Port-hamiltonian system on graphs. SIAM Journal on Control and Optimization, 51(2):906–937, 2013.

Andreas Wachter. An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering. PhD thesis, CarnegieMellon University, 2002.

Lie Xu and Dong Chen. Control and Operation of a DC Microgrid with Variable Generation and Energy Storage. IEEE Transactions on power delivery, 26(4):2513–2522, October 2011.

Daniele Zonetti, Romeo Ortega, and Abdelkrim Benchaib. Modeling and control of hvdc transmission systems from theory to practice and back. ControlEngineering Practice, 45:133–146, 2015.


Reference

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