Power balancing in a DC microgrid elevator system through constrained - - PowerPoint PPT Presentation

Power balancing in a DC microgrid elevator system through constrained

• ptimization

Thanh Hung PHAM, Ionela PRODAN and Laurent LEFEVRE

Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration des Syst` emes), Valence, France, thanh-hung.pham@lcis.grenoble-inp.fr,phamthanhhung1204@gmail.com

This work was supported by a mobility project of the Romanian National Authority for Scientific Research and Innovation, CNCS - UEFISCDI, project number PN-III-P1-1.1-MCT-2016-0037, within PNCDI III

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 1 / 27

Introduction

Outline

1

Introduction

2

DC microgrid modeling Port-Hamiltonian system on graphs DC microgrid elevator system modeling

3

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

4

Simulation Simulation software and numerical data Nominal scenario Perturbation-affected scenario

5

Conclusions

6

Reference

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration 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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 3 / 27

Introduction

Introduction

General goal: Constrained optimization control for for efficiently managing the DC microgrid

• peration.

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,

• ffline 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 regards the physical power-preserving connection, neglects the nonlinear storage dynamic and the 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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 4 / 27

DC microgrid modeling

Outline

1

Introduction

2

DC microgrid modeling Port-Hamiltonian system on graphs DC microgrid elevator system modeling

3

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

4

Simulation Simulation software and numerical data Nominal scenario Perturbation-affected scenario

5

Conclusions

6

Reference

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 5 / 27

DC microgrid modeling Port-Hamiltonian system on graphs

Bond graph and Port-Hamiltonian system

Bond Graph + _ + _ + _ + _ + _ + _

Example: Bond Graph for simple series and parallel DC electrical circuit.

Advantage Explicit description of the exchange of power, of the dissipation and of the energy storage for multi-physics system. Dirac structure and PH system

Dirac structure and port-Hamiltonian systems.

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

• e(t)

= Jf(t) + Gλ(t), = GT f(t), e(t) =   ∇H(x) eR(t) eE (t)   , f(t) =   −˙ x(t) fR(t) fE (t)   ,

J = −JT x(t) : state vector f(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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration 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 graph Consider 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 formulation

• e(t)

= −BT vp(t), = 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) : current vR (t), vE (t) : voltage vp(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 .

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration 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) : voltage i(t) : current P(t) : power d(t) : converter duty cycle xi (t) : ith 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 1 1 1 2 3 4 5 6 7 External grid

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 8 / 27

DC microgrid modeling DC microgrid elevator system modeling

Model of the components

+ _ + _ + _ Load power source + _ Renewable power source + _ + _ + _ _ + + _ + _ + _ + _ 1 1 1 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 2 x(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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 9 / 27

DC microgrid modeling DC microgrid elevator system modeling

Model of the components

+ _ + _ + _ Load power source + _ Renewable power source + _ + _ + _ _ + + _ + _ + _ + _ 1 1 1 1 2 3 4 5 6 7 External grid

DC/DC converter respects the power-preserving relation:

• 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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 10 / 27

DC microgrid modeling DC microgrid elevator system modeling

DC microgrid network

+ _ + _ + _ Load power source + _ Renewable power source + _ + _ + _ _ + + _ + _ + _ + _ 1 1 1 1 2 3 4 5 6 7 External grid

Microgrid network:

• v(t)

= −BT vp(t), = Bi(t), Ground potential (node 1): vp1(t) 0, Incidence matrix: B =      1T

2

1T

3

1T

2

I2 B1 I3 B2 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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 11 / 27

DC microgrid modeling DC microgrid elevator system modeling

Global DC microgrid model

+ _ + _ + _ Load power source + _ Renewable power source + _ + _ + _ _ + + _ + _ + _ + _ 1 1 1 1 2 3 4 5 6 7 External grid

Dynamics:                −˙ x(t) iE (t)

• = L(d)
• ∇H(x)

vE (t)

• ,

= 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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 12 / 27

Battery scheduling by optimization-based control

Outline

1

Introduction

2

DC microgrid modeling Port-Hamiltonian system on graphs DC microgrid elevator system modeling

3

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

4

Simulation Simulation software and numerical data Nominal scenario Perturbation-affected scenario

5

Conclusions

6

Reference

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration 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 model

The discrete-time model preserves: the DC network

• v(j)

= −BT vp(j), = Bi(j), the linear form of the Ohm’ law vR(j) = −RiR(j), the power-preserving relation of DC/DC converter

• d(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 = ˇ ∇H(j)ˇ ˙ x(j). ⇒ The energy conservation property: H(ˇ x(j)) − H(ˇ x(j − 1)) = ˇ ie(j) ˇ ve(j)h − ˇ vR(j)T R−1 ˇ vR(j)h +

jh

• (j−1)h

(Pl(τ) + Pr(τ))dτ.

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration 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 a desired 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) = argmin

ie(t) N

• j=1

γC(t + jh|t), subject to: discrete-time dynamic, discrete-time constraints. ⇒ The optimization problem is nonlinear both in cost and in constraints. ⇒ IPOPT solver (Biegler and Zavala (2009)).

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 15 / 27

Simulation

Outline

1

Introduction

2

DC microgrid modeling Port-Hamiltonian system on graphs DC microgrid elevator system modeling

3

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

4

Simulation Simulation software and numerical data Nominal scenario Perturbation-affected scenario

5

Conclusions

6

Reference

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration 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 (L¨

• fberg (2004)), IPOPT (W¨

achter (2002)) and Matlab 2015a. Name Notation Value Closed-loop sampled time [s] 36 Scheduling time step h [s] 1800 Prediction horizon N 48 Weighting parameter γ ∈ (0, 1) 0.5 Battery parameters Q1 [V ] [ 13 13 ]T Q2 [V /C] diag {0.3036, 0.2024} Battery constraints xmax [Ah] [ 73.2 109.8 ]T ib,min [A]

• 20

ib,max [A] 20 Grid constraints ie,min [A]

• 8

ie,max [A] 8 Load voltage reference vref [V ] 380 Resistors R [Ω] diag {0.012, 0.015, 0.31, 0.29, 0.23, 0.19}

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 17 / 27

Simulation Simulation software and numerical data

Simulation software and numerical data

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 1,000

¯ Pl [W]

Load power profile

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 200 400 600

¯ Pr [W]

Renewable power profile

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0.12 0.13 0.14 0.15 0.16

Time [h] price [eur/kWh]

Electricity price profile

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 18 / 27

Simulation Nominal scenario

Nominal scenario

+ _ + _ + _ Load power source + _ Renewable power source + _ + _ + _ _ + + _ + _ + _ + _ 1 1 1 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 .

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

SoC1 [%]

State of charge 1

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

SoC2 [%]

State of charge 2

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

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 5 10

Time [h] ib,R2 [A]

Battery current T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 19 / 27

Simulation Nominal scenario

Nominal scenario

+ _ + _ + _ Load power source + _ Renewable power source + _ + _ + _ _ + + _ + _ + _ + _ 1 1 1 1 2 3 4 5 6 7 External grid

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 −500 500 1,000

Time [h] Electrical power [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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 20 / 27

Simulation Perturbation-affected scenario

Perturbation-affected scenario

+ _ + _ + _ Load power source + _ Renewable power source + _ + _ + _ _ + + _ + _ + _ + _ 1 1 1 1 2 3 4 5 6 7 External grid

Assumption: Pl(t) ∈ Pl(t) [1 − ǫlmin, 1 + ǫlmax] , Pr(t) ∈ Pr(t) [1 − ǫrmin, 1 + ǫrmax] , with the simulation values: ǫlmin = ǫlmax = 0.2, ǫrmin = ǫrmax = 0.2.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0.4 0.6 0.8 1

SoC1 [%]

State of charge 1

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

SoC2 [%]

State of charge 2

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

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 5 10

Time [h] ib,R2 [A]

Battery current T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 21 / 27

Simulation Perturbation-affected scenario

Perturbation-affected scenario

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 −500 500 1,000 1,500

Time [h] Electrical power [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 −500 500 1,000 1,500

Time [h] Electrical power [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.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 22 / 27

Conclusions

Outline

1

Introduction

2

DC microgrid modeling Port-Hamiltonian system on graphs DC microgrid elevator system modeling

3

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

4

Simulation Simulation software and numerical data Nominal scenario Perturbation-affected scenario

5

Conclusions

6

Reference

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration 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 of explicitly taking into account the power conservation of the system interconnections; the constrained optimization problem proposed which finds the optimum balance between battery usage and the profit gained from electricity management; the simulation results for the energy management of a particular DC microgrid elevator system 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 the control design scheme.

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 24 / 27

Reference

Outline

1

Introduction

2

DC microgrid modeling Port-Hamiltonian system on graphs DC microgrid elevator system modeling

3

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

4

Simulation Simulation software and numerical data Nominal scenario Perturbation-affected scenario

5

Conclusions

6

Reference

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 25 / 27

Reference

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Mellon 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. Control Engineering Practice, 45:133–146, 2015. T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 26 / 27

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THANK YOU FOR YOUR ATTENTION

T.H. Pham, I. Prodan, L.Lef` evre (Grenoble INP (Institut National Polytechnique de Grenoble), LCIS (Laboratoire de Conception et d’Int´ egration Power balancing in a DC microgrid December 8th, 2016 27 / 27