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
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 2 / 27
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)
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 3 / 27
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)).
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 4 / 27
DC microgrid modeling
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
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 5 / 27
DC microgrid modeling Port-Hamiltonian system on graphs
Bond graph and Port-Hamiltonian systemBond Graph
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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)
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 6 / 27
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
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2 3
1
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2
1
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Edge order: C-R-E
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_
Energy stored in the capacitor:
H(x) =1
2
x(t)2
C.
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 7 / 27
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
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Load power
source
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Renewable
power source
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+_
1
11
1
2
3
4
5 6
7
External
grid
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 8 / 27
DC microgrid modeling DC microgrid elevator system modeling
Model of the components
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Load power
source
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Renewable
power source
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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 .
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 9 / 27
DC microgrid modeling DC microgrid elevator system modeling
Model of the components
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+ _
Load power
source
+ _
Renewable
power source
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_
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_
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+_
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.
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 10 / 27
DC microgrid modeling DC microgrid elevator system modeling
DC microgrid network
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Load power
source
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Renewable
power source
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1
11
1
2
3
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5 6
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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
DC microgrid modeling DC microgrid elevator system modeling
Global DC microgrid model
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Load power
source
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Renewable
power source
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1
11
1
2
3
4
5 6
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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).
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 12 / 27
Battery scheduling by optimization-based control
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
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 13 / 27
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.
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 14 / 27
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)).
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 15 / 27
Simulation
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
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 16 / 27
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
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 17 / 27
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
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 18 / 27
Simulation Nominal scenario
Nominal scenario
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Load power
source
+ _
Renewable
power source
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1
11
1
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4
5 6
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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
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 19 / 27
Simulation Nominal scenario
Nominal scenario
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Load power
source
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Renewable
power source
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1
11
1
2
3
4
5 6
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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)
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 20 / 27
Simulation Perturbation-affected scenario
Perturbation-affected scenario
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Load power
source
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Renewable
power source
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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
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 21 / 27
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
storage unit: vc2(t) · ic2(t)load: −Pl(t)external grid: ve(t) · ie(t)renewable: Pr(t)
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
storage unit: vc2(t) · ic2(t)load: −Pl(t)external grid: ve(t) · ie(t)renewable: Pr(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 22 / 27
Conclusions
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
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 23 / 27
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.
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 24 / 27
Reference
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
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 25 / 27
Reference
Reference
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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 26 / 27
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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 27 / 27