Voltage Stabilization in MVDC Microgrids Using Passivity...

Voltage Stabilization in MVDC Microgrids Using Passivity-Based Nonlinear Control

57th IEEE Conference on Decision and ControlDecember 17-19, 2018, Fontainebleau, Miami Beach, USA

A. Martinelli*, P. Nahata†, G. Ferrari-Trecate†

* Department of Electronics, Information and Bioengineering, Politecnico di Milano(now with Automatic Control Laboratory, ETH Zürich)

† Automatic Control Laboratory, École Polytechnique Fédérale de Lausanne

INTRODUCTION• Microgrids

• Control Problem

• Passivity Theory

MVDC MICROGRID STABILIZATION• Design of Nonlinear Local Regulators

• Microgrid Global Stability through Passivity of Closed-Loop Agents

FINAL CONSIDERATIONS• Simulations

• Main Results and Future Developments

Outline

2A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 2 / 17

Microgrids


DEFINITION (MICROGRID): Electricnetwork composed by loads anddistributed generation units (DGUs).

Microgrids



WHY MICROGRIDS?• Electrify remote areas, islands, or

large buildings

• Improve resilience to faults andpower quality in power networks

• Easy integration with renewableenergy sources

Microgrids



WHY MICROGRIDS?• Electrify remote areas, islands, or

large buildings

• Improve resilience to faults andpower quality in power networks

• Easy integration with renewableenergy sources

VOLTAGE STABILITY: Key problem in islanded microgrids1

1[P. Dragičević et al., 2016; L. Meng et al., 2017]

Control Problem


GOAL: Decentralized control architecture, with

• Scalable control design of local controllers• Asymptotic stability and tracking of voltage references in spite of couplings

Σ1

Σ3 Σ4

Σ2𝒞1

𝒞3 𝒞4

𝒞2

Passivity Theory


DEFINITION (PASSIVITY):The square dynamical system

Σ ∶ ቊሶ𝑥 = 𝑓 𝑥, 𝑢

𝑦 = ℎ 𝑥, 𝑢

is said to be strictly passive if there exists a continuously differentiable positivesemidefinite function 𝑉(𝑥), called the storage function, such that

𝑢𝑇𝑦 =𝜕𝑉

𝜕𝑥𝑓 𝑥, 𝑢 + 𝜓 𝑥 , ∀ 𝑥, 𝑢,

for some positive definite function 𝜓 𝑥 .

Passivity Theory


DEFINITION (PASSIVITY):The square dynamical system

Σ ∶ ቊሶ𝑥 = 𝑓 𝑥, 𝑢

𝑦 = ℎ 𝑥, 𝑢

is said to be strictly passive if there exists a continuously differentiable positivesemidefinite function 𝑉(𝑥), called the storage function, such that

𝑢𝑇𝑦 =𝜕𝑉

𝜕𝑥𝑓 𝑥, 𝑢 + 𝜓 𝑥 , ∀ 𝑥, 𝑢,

for some positive definite function 𝜓 𝑥 .

WHY PASSIVITY?• Compositional framework for analyzing complex systems• Strong relationship with Lyapunov stability• Design control actions based on system's energetic considerations


• Control Problem






Outline


MV Microgrid Model


Medium Voltage (MV) DC mG model:Bipartite graph, where nodes are• DGUs with Boost converter dynamics• RL lines

MV Microgrid Model



Apply the state-space averaging method to converters’ switched dynamics

൝𝐶𝑖 ሶ𝑉𝑖 = 𝑑𝑖

∗𝐼𝑖 − 𝐼𝑙𝑜𝑎𝑑,𝑖 + 𝐼𝑛𝑒𝑡,𝑖

𝐿𝑖 ሶ𝐼𝑖 = −𝑑𝑖∗𝑉𝑖 − 𝑅𝑖𝐼𝑖 + 𝑉𝑖𝑛,𝑖

{ 𝐿𝐿𝑗 ሶ𝐼𝐿𝑗 = 𝑉𝑛𝑒𝑡,𝑗 − 𝑅𝐿𝑗𝐼𝐿𝑗

State-space Model and Equilibria


DGUi

Linej

• 𝑉𝑖 , 𝐼𝑖 , 𝐼𝐿𝑗 : state variables

• 𝑑𝑖∗ ∈ (0,1) : duty cycle (input)

• 𝐼𝑛𝑒𝑡,𝑖 , 𝑉𝑛𝑒𝑡,𝑗 : coupling terms

Apply the state-space averaging method to converters’ switched dynamics

൝𝐶𝑖 ሶ𝑉𝑖 = 𝑑𝑖

∗𝐼𝑖 − 𝐼𝑙𝑜𝑎𝑑,𝑖 + 𝐼𝑛𝑒𝑡,𝑖

𝐿𝑖 ሶ𝐼𝑖 = −𝑑𝑖∗𝑉𝑖 − 𝑅𝑖𝐼𝑖 + 𝑉𝑖𝑛,𝑖

{ 𝐿𝐿𝑗 ሶ𝐼𝐿𝑗 = 𝑉𝑛𝑒𝑡,𝑗 − 𝑅𝐿𝑗𝐼𝐿𝑗


DGUi

Linej

Bilinear system

LEMMA (UNIQUE GLOBAL EQUILIBRIUM):The equilibrium of the global dynamics ሶ𝐱 = 𝐴𝐱 + 𝑄, when the control variablesand the exogenous terms are constant, exists and is unique: ത𝐱 = 𝐴−1𝑄.

• 𝑉𝑖 , 𝐼𝑖 , 𝐼𝐿𝑗 : state variables

• 𝑑𝑖∗ ∈ (0,1) : duty cycle (input)

• 𝐼𝑛𝑒𝑡,𝑖 , 𝑉𝑛𝑒𝑡,𝑗 : coupling terms

State-space mG Model and Equilibria

Design of Nonlinear Local Regulators


Shift of variables: deviation from equilibrium values

෨𝑉𝑖 = 𝑉𝑖 − ത𝑉𝑖 , ሚ𝐼𝑖 = 𝐼𝑖 − ҧ𝐼𝑖 , ሚ𝐼𝐿𝑗 = 𝐼𝐿𝑗 − ҧ𝐼𝐿𝑗 .

Considered control laws2:

𝑑𝑖∗ 𝑡 = sat ҧ𝑑𝑖

∗ + ෨𝜙𝑖 𝑡

ሶ෨𝜙𝑖 𝑡 = 𝛾𝑖 𝑉𝑟𝑒𝑓,𝑖 ሚ𝐼𝑖 − ҧ𝐼𝑖 ෨𝑉𝑖 − 𝛾𝑖𝐾𝑊𝑖 sat ҧ𝑑𝑖∗ + ෨𝜙𝑖 𝑡 − ҧ𝑑𝑖

∗ .

2[J. Moreno-Valanzuela and O. Garcia-Alarcon, 2017]






𝑑𝑖∗ 𝑡 = sat ഥ𝒅𝒊

∗ + ෩𝝓𝒊 𝑡

ሶ෨𝜙𝑖 𝑡 = 𝜸𝒊 𝑉𝑟𝑒𝑓,𝑖 ሚ𝐼𝑖 − ҧ𝐼𝑖 ෨𝑉𝑖 − 𝜸𝒊𝑲𝑾𝒊 sat ҧ𝑑𝑖∗ + ෨𝜙𝑖 𝑡 − ҧ𝑑𝑖

∗ .


Dynamic compensation termConstant open-loop control

Design parameters






𝑑𝑖∗ 𝑡 = sat ഥ𝒅𝒊

∗ + ෩𝝓𝒊 𝑡

ሶ෨𝜙𝑖 𝑡 = 𝜸𝒊 𝑉𝑟𝑒𝑓,𝑖 ሚ𝐼𝑖 − ҧ𝐼𝑖 ෨𝑉𝑖 − 𝜸𝒊𝑲𝑾𝒊 sat ҧ𝑑𝑖∗ + ෨𝜙𝑖 𝑡 − ҧ𝑑𝑖

∗ .

• Scalability: computation of ҧ𝑑𝑖∗ requires local info only

• Plug-and-Play: in addition/removal of DGUs, only neighbouring regulatorsneed to be retuned


Dynamic compensation termConstant open-loop control

Design parameters

Passivity of Closed-Loop Agents


PROPOSITION 1 (PASSIVITY OF DGUs):Consider the following positive definite storage function4

𝒱𝑖 =1

2𝐶𝑖 ෨𝑉𝑖

2 +1

2𝐿𝑖 ሚ𝐼𝑖

2 + 𝛾𝑖−1

෨𝜙𝑖2

2−න

0

෩𝜙𝑖

(( ҧ𝑑𝑖∗ + ෨𝜙𝑖) − 𝑠𝑎𝑡( ҧ𝑑𝑖

∗ + ෨𝜙𝑖))𝑑 ෨𝜙𝑖 .

Then, DGUs are passive with respect to input ሚ𝐼𝑛𝑒𝑡,𝑖 and output ෨𝑉𝑖 .

PROPOSITION 2 (PASSIVITY OF LINES)3:Consider the following positive definite storage function

𝒱𝑗 =1

2𝐿𝐿𝑗 ሚ𝐼𝐿𝑗

2 .

Then, Lines are passive with respect to input ෨𝑉𝑛𝑒𝑡,𝑗 and output ሚ𝐼𝐿𝑗 .

3[Brogliato et al., 2007] 4[J. Moreno-Valanzuela and O. Garcia-Alarcon, 2017]

Passivity of Closed-Loop Agents


PROPOSITION 1 (PASSIVITY OF DGUs):Consider the following positive definite storage function4

𝒱𝑖 =1

2𝐶𝑖 ෨𝑉𝑖

2 +1

2𝐿𝑖 ሚ𝐼𝑖

2 + 𝛾𝑖−1

෨𝜙𝑖2

2−න

0

෩𝜙𝑖

(( ҧ𝑑𝑖∗ + ෨𝜙𝑖) − 𝑠𝑎𝑡( ҧ𝑑𝑖

∗ + ෨𝜙𝑖))𝑑 ෨𝜙𝑖 .

Then, DGUs are passive with respect to input ሚ𝐼𝑛𝑒𝑡,𝑖 and output ෨𝑉𝑖 .

PROPOSITION 2 (PASSIVITY OF LINES)3:Consider the following positive definite storage function

𝒱𝑗 =1

2𝐿𝐿𝑗 ሚ𝐼𝐿𝑗

2 .

Then, Lines are passive with respect to input ෨𝑉𝑛𝑒𝑡,𝑗 and output ሚ𝐼𝐿𝑗 .

𝜕𝒱

𝜕𝑥𝑓 𝑥,𝑢

ሚ𝐼𝑛𝑒𝑡,𝑖 ෨𝑉𝑖 = ሶ𝒱𝑖 + 𝜓𝑖෨𝑉𝑖 , ሚ𝐼𝑖 , ෨𝜙𝑖 , 𝜓𝑖 ≥ 0.

𝜕𝒱

𝜕𝑥𝑓 𝑥,𝑢

෨𝑉𝑛𝑒𝑡,𝑗 ሚ𝐼𝐿𝑗 = ሶ𝒱𝑗 + 𝜓𝑗 ሚ𝐼𝐿𝑗 , 𝜓𝑗 ≥ 0.

3[Brogliato et al., 2007] 4[J. Moreno-Valanzuela and O. Garcia-Alarcon, 2017]

Microgrid Global Stability (I)


Multiple passive systems interconnected in a skew-symmetric fashion

Assume:

• Passive and control affine agents

• Electrical graph weakly connected

• Coupled together in a skew-symmetric fashion

Then consider as a global Lyapunov function the following

𝒲 ො𝐱 =𝑖𝒱𝑖( ො𝑥)

Several results on interconnected passive systems5. We exploit our recent result6:

5[N. Chopra, 2012; P. Dragičević et al., 2016] 6[P. Nahata et al., 2018]

Microgrid Global Stability (II)


𝐸 = ො𝐱 ∶ ሶ𝒲 ො𝐱 = 𝟎

= ො𝑥𝑖 ∶ 𝜓𝑖 ො𝑥 = 0, ∀𝑖 .

Then global trajectories converge to the largest invariant set 𝑀 contained in

⟹ ሶ𝒲 ො𝐱 = −𝑖𝜓𝑖 ≤ 0.

Microgrid Global Stability (II)


𝐸 = ො𝐱 ∶ ሶ𝒲 ො𝐱 = 𝟎

= ො𝑥𝑖 ∶ 𝜓𝑖 ො𝑥 = 0, ∀𝑖 .

THEOREM (STABILITY OF THE MG): For the controlled MVDC model,

𝑀 = 0

and hence global asymptotic stability of the origin is guaranteed.

Then global trajectories converge to the largest invariant set 𝑀 contained in

⟹ ሶ𝒲 ො𝐱 = −𝑖𝜓𝑖 ≤ 0.

Energetic Interpretation of mG Stability


The passivity condition is expressed by an equation of the form

𝑢𝑇𝑦 = ሶ𝒱 𝑥 + 𝜓 𝑥 , 𝜓 𝑥 ≥ 0,

that represents the power balance for a controlled DGU, where

Accumulated energy < Supplied energy


• Control Problem






Outline


Control Simulation


Main Results and Developments


MAIN RESULTS:

• Set of nonlinear control laws that passivate DGU dynamics

• Decentralized architecture allows PnP operation- Scalable control design

- Explicit synthesis of the regulators

• General result, independent from specific topology

FUTURE DEVELOPMENTS:

• Evaluate control performances

• Extend the control architecture by adding an integral action.

Q&A


Thank you for your attention

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