voltage stabilization in mvdc microgrids using passivity...
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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
3A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 3 / 17
DEFINITION (MICROGRID): Electricnetwork composed by loads anddistributed generation units (DGUs).
Microgrids
4A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 3 / 17
DEFINITION (MICROGRID): Electricnetwork composed by loads anddistributed generation units (DGUs).
WHY MICROGRIDS?• Electrify remote areas, islands, or
large buildings
• Improve resilience to faults andpower quality in power networks
• Easy integration with renewableenergy sources
Microgrids
5A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 3 / 17
DEFINITION (MICROGRID): Electricnetwork composed by loads anddistributed generation units (DGUs).
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
6A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 4 / 17
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
7A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 5 / 17
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
8A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 5 / 17
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
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
9A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 6 / 17
MV Microgrid Model
10A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 7 / 17
Medium Voltage (MV) DC mG model:Bipartite graph, where nodes are• DGUs with Boost converter dynamics• RL lines
MV Microgrid Model
11A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 7 / 17
Medium Voltage (MV) DC mG model:Bipartite graph, where nodes are• DGUs with Boost converter dynamics• RL lines
MV Microgrid Model
12A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 7 / 17
Medium Voltage (MV) DC mG model:Bipartite graph, where nodes are• DGUs with Boost converter dynamics• RL lines
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
13A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 8 / 17
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
14A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 8 / 17
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
15A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 9 / 17
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]
Design of Nonlinear Local Regulators
16A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 9 / 17
Shift of variables: deviation from equilibrium values
෨𝑉𝑖 = 𝑉𝑖 − ത𝑉𝑖 , ሚ𝐼𝑖 = 𝐼𝑖 − ҧ𝐼𝑖 , ሚ𝐼𝐿𝑗 = 𝐼𝐿𝑗 − ҧ𝐼𝐿𝑗 .
Considered control laws2:
𝑑𝑖∗ 𝑡 = sat ഥ𝒅𝒊
∗ + ෩𝝓𝒊 𝑡
ሶ෨𝜙𝑖 𝑡 = 𝜸𝒊 𝑉𝑟𝑒𝑓,𝑖 ሚ𝐼𝑖 − ҧ𝐼𝑖 ෨𝑉𝑖 − 𝜸𝒊𝑲𝑾𝒊 sat ҧ𝑑𝑖∗ + ෨𝜙𝑖 𝑡 − ҧ𝑑𝑖
∗ .
2[J. Moreno-Valanzuela and O. Garcia-Alarcon, 2017]
Dynamic compensation termConstant open-loop control
Design parameters
Design of Nonlinear Local Regulators
17A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 9 / 17
Shift of variables: deviation from equilibrium values
෨𝑉𝑖 = 𝑉𝑖 − ത𝑉𝑖 , ሚ𝐼𝑖 = 𝐼𝑖 − ҧ𝐼𝑖 , ሚ𝐼𝐿𝑗 = 𝐼𝐿𝑗 − ҧ𝐼𝐿𝑗 .
Considered control laws2:
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
2[J. Moreno-Valanzuela and O. Garcia-Alarcon, 2017]
Dynamic compensation termConstant open-loop control
Design parameters
Passivity of Closed-Loop Agents
18A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 10 / 17
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
19A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 10 / 17
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)
20A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 11 / 17
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)
21A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 12 / 17
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)
22A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 12 / 17
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
23A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 13 / 17
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
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
24A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 14 / 17
Control Simulation
25A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 15 / 17
Control Simulation
26A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 15 / 17
Main Results and Developments
27A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 16 / 17
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
28A. Martinelli, P. Nahata, G. Ferrari-Trecate Dec. 19th 2018 @Fontainebleau, Miami Beach, USA Slide 17 / 17
Thank you for your attention