Assuring Transient Safety in Inverter-Based

Microgrids via Local Control Adjustments

Soumya KunduSenior Engineer, PNNL

January 15, 2021

2

Acknowledgement

SponsorsDOE GMLC, OE Microgrid

CollaboratorsKaran Kalsi (PNNL)Sai Pushpak Nandanoori (PNNL)Sijia Geng (U Michigan)Ian Hiskens (U Michigan)We Du (PNNL)Frank Tuffner (PNNL)Kevin Schneider (PNNL)

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Power Electronics (PE) Interfaced Grid

§ Existing operational framework is insufficient to deal with the evolving challenges of extreme high (100%) power electronics (PE)-interfaced grid§ Lower of inertia§ Larger transients at fast (electromagnetic) timescales§ Higher uncertainties in power generation§ Reduced stability and safety margins

… need transformational change to achieve extreme high >75% PE penetration

*Source: EU MIGRATE Report

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Emerging Technologies: Grid-Forming Inverters

January 15, 2021

Multi-loop droop-control (P-𝜔, Q-V)

main grid

• Grid-forming inverters § Provides virtual inertia; acts as a voltage

source; stable synchronization via inner control loops; black-start, and more …

• Multi-loop droop-control regulates voltage and frequency by controlling power (P,Q)

(P-𝜔 droop)(Q-V droop)

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Hierarchical and Distributed Framework

• Hierarchical and distributed operations to coordinate many distributed energy resources (DERs) over the network

• Individual resources (e.g., inverters) received control set-points to track

Example: Optimal Power-Flow- DERs receive set-points; in turn regulates voltage and frequency

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Multi-timescales Problem

State-of-the-art operational practices in inverter-based microgrids lack the spatiotemporal granularity required to proactively prevent transient safety and

stability violations which are often local and fast-evolving in nature

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Problem 1: Local Parametric Stability Constraints

Choice of droop coefficients is critical to stability

January 15, 2021

• Small-signal stability of the system depends on various parameters, including, for example, droop coefficient values

Problem 2: Local Transient Safety Constraints

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Local disturbances (e.g., solar fluctuations) cause unsafe excursions in voltages

unsafe excursions in voltages

desired transient safe control

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• Parametric Stability Certificates

• Transient Safety Certificates

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Small-Signal Model of Electromagnetic Transients

(inverter dynamics)

(load dynamics)

§ Inverter dynamics (ignoring relatively faster internal states)§ Electromagnetic transients cannot be ignored§ Apply dynamic phasors to model line and load dynamics

(line dynamics)

Parametric small-signal model

,

January 15, 2021

• Identify parameter values for which the system is stable

• Check:§ All left-half-plane eigenvalues

• Brute-force eigenvalue analysis is computationally intractable

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Parametric Stability Constraints (Main Idea)

(Lyapunov Reformulation)• Existence of positive

(semi-)definite matrices• Find:

• Generally not easy to solve

• Tractable problem for microgrid (fortunately)• Exploits network structure and sparsity!

§ Construct a block-diagonal P§ Relax Q to block-diagonal structure§ Decompose into small sub-matrices

January 15, 2021Nandanoori, Kundu, Du, Tuffner and Schneider, "Distributed Small-Signal Stability Conditions for Inverter-Based Unbalanced Microgrids," TPWRS 2020.

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Final Results• Closed-form (algebraic) expressions

§ Relate between line parameters, load, frequency and voltage droop values

• Distributed in nature § Constraints associated with every line

• Computationally efficient§ Scales linearly with the number of lines§ Alternative approaches at best scale 𝑂 𝑁!

with the number of buses (𝑁)

• Provides sufficient conditions of stability § Estimated stability regions are subsets of

the exact (true) stability region

G0 + 2⌧(G+ G̃) � 0

� (G0 + 2⌧G) � 0

2⌧B +G0 + 2⌧G� ⌧G̃ � 0[⇤p]ik � 2[B0]ik + 4⌧ [B]ik � 0

2[⇤p]ik � 4[B0]ik � [G0]ik � 2⌧⇣[G]ik + [G̃]ik

⌘� 0

4⌧ [⇤q]ik + [G0]ik + 2⌧ [G]ik � 4[B0]ik � 0

2⌧ [⇤q]ik + 2⌧⇣[B]ik + [B̃]ik

⌘� [G0]ik � 2⌧ [G]ik � 3⌧ [G̃]ik � 0

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January 15, 2021Nandanoori, Kundu, Du, Tuffner and Schneider, "Distributed Small-Signal Stability Conditions for Inverter-Based Unbalanced Microgrids," TPWRS 2020.

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IEEE 34 bus system: Frequency droop vs. Inverter Sizes

80 90 100 110 1200

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2

3

4

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Stability Boundary - PSCADStability Boundary - Eigen value analysisStability Boundary - Estimated

0 0.5 1 1.5 2 2.5 359.2

59.4

59.6

59.8

60

60.2

60.4GFM 1GFM 2

GFM 3GFM 4

GFM 5GFM 6

GFM 7

0 0.5 1 1.5 2 2.5 352

54

56

58

60

62

64

66

68GFM 1GFM 2

GFM 3GFM 4

GFM 5GFM 6

GFM 7

xx

• Stability is verified by creating a load disturbance of 0.1 pu at 0.6 seconds • Stable case: frequency is stabilized back to its post disturbance value• Unstable case: frequency deviates from its post disturbance value causing system instability

January 15, 2021

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• Parametric Stability Certificates

• Transient Safety Certificates

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Safety Filter: The Concept

• Safety filters are deployed locally at the inverter terminals, and act as gatekeepers for allowable (safe) set-points

§ State-inclusive bounds on the allowable control set-points

• In a robust design, guarantees transient safety constraint satisfaction under bounded uncertainties in the network

Decouple network-level objectives from local transient safety constraints

Kundu and Kalsi, "Transient Safety Filter Design for Grid-Forming Inverters," ACC 2020.

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Mathematically

• Grid-forming inverter dynamics

• Transient safety satisfaction under bounded disturbances

(local dynamics) (neighbor interactions)(control set-points)

(identify)

Such that, implies

under bounded disturbances from the neighborsKundu and Kalsi, "Transient Safety Filter Design for Grid-Forming Inverters," ACC 2020.

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Control Barrier Functions

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Safety vs. Stability

January 15, 2021

Stability: Lyapunov Certificates

• Convergences of states

Safety: Barrier Certificates

• Constraints on states

… finding these functions for generic nonlinear systems is not always trivial

(Dynamical System)

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Polynomial Systems: Sum-of-Squares

• Sum of squared polynomials:

§ Gramm matrix representation and equivalence with SDPs:

§ (Putinar’s) Positivstellensatz: deal with semi-algebraic conditions!!

ü MATLAB tools (example): SOSTOOLS, SeDuMi.

January 15, 2021

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

AAACpXicbVFNb9NAEF2brxI+msKRy6oRpUghshFSe2lVwaUXpBaatFLWWOvN2Fl1vTa748qR8T/jV3Dj37BJTEVbRlrp6c08vZm3SamkxSD47fn37j94+Gjjce/J02fPN/tbLya2qIyAsShUYS4SbkFJDWOUqOCiNMDzRMF5cvlp2T+/AmNloc9wUUKU80zLVAqOjor7P3fK3frtYcAQamxooWnLVGK4AFr/YMOGrSyaRFXQZnHoZlkG32kwZLMC7TCL9TXVMrMWMtbbYTJN6V+1gVnLoHbXWMqszHIeSyY1+7qE0zpqO3dbUJxzpG35jtkqjxt5ELbf9LXE2d3Qxf1BMApWRe+CsAMD0tVJ3P/l1hZVDhqF4tZOw6DEqOEGpVDQ9lhloeTikmcwdVDzHGzUrI5o6WvHzGhaGPc00hX7r6LhubWLPHGTOce5vd1bkv/rTStM96NG6rJC0GJtlFaKoovDfRmdSQMC1cIBLox0u1Ix5y5qdB/bcyGEt0++CybvR2EwCk8/DI4+dnFskFdkm+ySkOyRI3JMTsiYCG/bO/ZOvS/+G/+zf+ZP1qO+12lekhvlx38ATcjQmw==

• M. Putinar, “Positive polynomials on compact semi-algebraic sets,” 1993.• A. Papachristodoulou, et al, “SOSTOOLS: Sum of squares optimization toolbox for MATLAB,” 2013.• J. F. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” 1999.

Constructive method for Lyapunov and barrier functions – if polynomial!

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Power-Flows have Non-Polynomial Terms …

• Power system dynamics are non-polynomial ODEs§ … because of trigonometric terms (sine, cosine) in power-flow

• Lift the state-space to convert into polynomial representation

January 15, 2021

Obtain set of polynomial DAEs from non-polynomial ODEs

Pe,i(�) =X

j

EiEj (Gij cos(�i � �j) +Bij sin(�i � �j))

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Iterative Lyapunov & Barrier Function Computation

• Efficient iterative algorithms exist to compute the barrier and Lyapunov functions• Brief outline: warm-start Barrier Computation with Lyapunov Level-sets*

January 15, 2021Wang, Han, and Egerstedt, “Permissive barrier certificates for safe stabilization using sum-of-squares,” ACC 2018.

Kundu, Geng, Nandanoori, Hiskens and Kalsi, “Distributed Barrier Certificates for Safe Operation of Inverter-Based Microgrids”, ACC 2019.

Safety-constrained set as a subset of the region of attraction

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Local Safety Control Sets

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Local Nominal Safety Control Policy

• Microgrid as an interconnected polynomial system:

• Robust Safety:

January 15, 2021

(isolated dynamics) (interactions)

Kundu, Geng, Nandanoori, Hiskens and Kalsi, “Distributed Barrier Certificates for Safe Operation of Inverter-Based Microgrids”, ACC 2019.

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A Family of Safety Control Policies

• For any nominal control policy , the following family of state-feedback controls guarantee robust transient safety under bounded disturbances

Kundu and Kalsi, "Transient Safety Filter Design for Grid-Forming Inverters," ACC 2020.

We have a family of local control policies, instead of just one, that ensure robust safety guarantees

Challenge: the set has vanishing cardinality around origin (x=0)

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A Family of Safety Control Policies (Contd.)

• For any nominal control policy , the following family of state-feedback controls guarantee robust transient safety under bounded disturbances

• Expand the allowable safety control set by introducing a relaxation term

§ The relaxation term is >0 near origin, but approaches 0 at the safety set boundary

Kundu and Kalsi, "Transient Safety Filter Design for Grid-Forming Inverters," ACC 2020.

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Main Result: A Family of Safety Control Policies

• Summary: Under mild conditions on the distributed barrier functions1, there exist a family of state-feedback control policies, with non-vanishing cardinality that ensure robust safety guarantees.

• In other words, any control input of the following form is robustly safe:

where the existence of these two safe control policies are guaranteed:

Kundu and Kalsi, "Transient Safety Filter Design for Grid-Forming Inverters," ACC 2020.

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Numerical Example

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Visualization of Allowable Safe Control Set

Safe values of reactive power control input, as a function of the voltage deviation for

various values of the relaxation coefficient

Sets of values on the state-space over which a control set-point 𝑢 = 0 is deemed to be

safe, for varying relaxation coefficient γ

Higher values of (design parameter) ensures larger allowable safe set γ

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Numerical Example

CERTS MicrogridAn inverter terminal voltage violates the safety limits in absence of safety filters, but not in presence of it. The filtered reactive power control input and its allowable range, with the centrally dispatched setpoint at 𝑢 = 0.

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References

• Kundu and Kalsi. “Transient Safety Filter Design for Grid-Forming Inverters”, 2019 American Control Conference (ACC). IEEE, 2020.

• Nandanoori, Kundu, Du, Tuffner and Schneider. "Distributed small-signal stability conditions for inverter-based unbalanced microgrids." IEEE Transactions on Power Systems (2020).

• Schneider et al. "Preliminary Design Process for Networked Microgrids." Pacific Northwest National Laboratories, Tech. Rep (2020).

• Kundu, Geng, Nandanoori, Hiskens and Kalsi. "Distributed barrier certificates for safe operation of inverter-based microgrids." 2019 American Control Conference (ACC). IEEE, 2019.

• Kundu, Du, Nandanoori, Tuffner and Kalsi. "Identifying parameter space for robust stability in nonlinear networks: A microgrid application." ACC 2019.

Thank you

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