Low Inertia for Power System Frequency Control · Embracing Low Inertia for Power System Frequency...

Embracing Low Inertia for Power System Frequency ControlA Dynamic Droop Approach

Enrique Mallada

4th Grid Science Winter School and ConferenceJanuary 14, 2021

Acknowledgements

Yan Jiang Richard Pates Fernando PaganiniHancheng Min Petr VorobevEliza Cohn

Enrique Mallada JHUJan 14 2021 2

Dynamic Degradation

Jan 14 2021 Enrique Mallada JHU 3

In the United States:

[FERC, Nov. 16]

Dynamic Degradation


In the United States:

[FERC, Nov. 16]

Inverter‐based Control

Challenges• Measurements with noise and delays• Stability + robustness (plug & play)• Lack of incentives


Virtual Synchronous Generator

Controller

Telecom Analogy

Current approach: Use inverter‐based control to mimic generators response




Virtual Synchronous Generator

Controller

Telecom Analogy

Current approach: Use inverter‐based control to mimic generators response

It works, but perhaps there is

something better…




Dynamic Droop Control (iDroop)

Our approach: Design and tune of controllers rooted on sound control principles

Dynamic Droop

Design Objectives:

• Exploit power electronics capabilities• Improve Dynamic Performance• Minimize control effort• Stability and Robustness

Merits and Trade‐offs of Inertia





Pros: Provides natural disturbance rejection Cons: Hard to regain steady‐state


Merits and Trade‐offs of Low Inertia

Cons: Susceptible to disturbances Pros: Regains steady‐sate faster


• Performance Specification

•Limits of Virtual Inertia and Droop Control

•Dynamic Droop Control: iDroop

Roadmap to Low Inertia Frequency Control

IEEE Transactions on Automatic Control, October 2020

arXiv preprint arXiv:1910.04954 (2019)

IEEE Transactions on Automatic Control, July 2020





Power System Performance

Depends on several factors: generators, network, disturbancegood performance metric must identify the source of the degradation!


Frequency Response Control Effort

Performance Specification

Decomposition of Step Response


Synchronization Error

Center of Inertia: Kundur ‘94

System Frequency

Center of Inertia: Kundur ‘94

System Frequency

Step Disturbance Performance


RoCoF

Nadir

Nadir RoCoF

Deviation from Mean

Synchronization Cost

Steady‐state

Steady‐state



RoCoF

Nadir

Steady‐state

Sync. Cost

System Freq. :

Sync. Error :

Control Effort


Inverter Power

Power Rating

Power Rating Max Energy

Injected Energy

Steady‐State Effort

Steady‐State Effort



RoCoFSteady‐state

Sync. Cost

System Freq. :

Sync. Error :

Power Rating

Injected Energy:

Injected Power:

Steady‐State

Benchmark: Quantify control ability to eliminate overshoot in the Nadir

NadirMax Energy


• Limits of Virtual Inertia and Droop Control

• Controller Design: iDroop


Power Network Model


Laplacian Matrix

Step:

[Bergen Hill ‘81]

Bus DynamicsGenerator:


Model: Swing Equations Turbine

inverter

generator

ci

+ ! i

power imbalance

frequency

gi

piBus Dynamics

+

+

xiinverter

power injection

ui − pe,i

Bus DynamicsGenerator:


Model: Swing Equations Turbine

inverter

generator

ci

+ ! i

power imbalance

frequency

gi

piBus Dynamics

+

+

xiinverter

power injection

ui − pe,i

Bus DynamicsInverter:


inverter

generator

ci

+ ! i

power imbalance

frequency

gi

piBus Dynamics

+

+

xiinverter

power injection

ui − pe,i

Droop Control and Virtual Inertia:

Closed-loop Bus Dynamics:

Control of Low Inertia Pendulum

Cons: Susceptible to disturbances Pros: Regains steady‐sate faster



Pros:Provides disturbance rejection

Cons: Hard to regain steady‐state + excessive control effort


Virtual Mass Control:



Pros: Provides disturbance rejection, quickly restore steady‐state, with reasonable control effort.

Virtual Friction Control:

Cons?None, at least for pendulum



RoCoFSteady‐state

Sync. Cost

System Freq. :

Sync. Error :

Power Rating

Injected Energy:

Injected Power:

Steady‐State

Benchmark: Quantify control ability to eliminate overshoot in Nadir

NadirMax Energy

Modal Decomposition for Multi‐Rated Machines

Assumption: Let be the machine relative inertia ( ), and assume

Swing Equations + Turbine Virtual Inertia



Assumption: Let the machine relative inertia ( ), and assume



Assumption: Let be the machine relative inertia ( ), and

F -12 V Tu u

Change of Vars.

F -12V

w w

Change of Vars.

[Paganini M ‘17 , Guo Low 18’]Enrique Mallada JHUJan 14 2021 15


Assumption: Let be the machine relative inertia ( ), and

System Frequency

Sync Error

[Paganini M ‘17 , Guo Low 18’]

Eigenvalues of:


F -12 V Tu u

Change of Vars.

F -12V

w w

Change of Vars.



RoCoFSteady‐state

Sync. Cost

System Freq. :

Sync. Error :

Power Rating

Injected Energy:

Injected Power:

Steady‐State


NadirMax Energy

System Frequency w/ Virtual Inertia

Nadir Overshoot Elimination:


requires 𝝂 𝟎 in low inertia systems (low 𝒎)

Virtual Inertia Droop ControlNo Control

System

Freq.



RoCoFSteady‐state

Sync. Cost

System Freq. :

Sync. Error :

Power Rating

Injected Energy:

Injected Power:

Steady‐State


NadirMax Energy

Control Effort


System

Freq.

Control Effo

rt

Virtual InertiaNo Control Droop Control





Pros: Provides disturbance rejection, quickly restore steady‐state, with reasonable control effort.



Virtual Friction Control:

Cons? Large steady‐state effort in power systems

iDroop: Dynamic Droop Control

Instead of Virtual Inertia:


Virtual Inertia

iDroop: Dynamic Droop Control

Instead of Virtual Inertia:

We use


iDroop – Lead iDroop – Lag

Bus Dynamics /w iDroopInverter:

iDroop Control:

inverter

generator

ci

+ ! i

power imbalance

frequency

gi

piBus Dynamics

+

+

xiinverter

power injection

ui − pe,i

Closed-loop Bus Dynamics w/ iDroop:

iDroop




Dynamic Friction Control:



Dynamic Friction ControlNo Control



RoCoFSteady‐state

Sync. Cost

System Freq. :

Sync. Error :

Power Rating

Injected Energy:

Injected Power:

Steady‐State


NadirMax Energy

Overshoot Elimination w/ iDroop


Whenever and

System

Freq.

Control Effo

rt

Virtual Inertia Droop Control iDroop

Overshoot Elimination w/ iDroop


Whenever

first order step response

and

1ms + d

r − 1g

⌧s + 1

�s + δr − 1r

s + δ

- -1

ms + d

r − 1g

⌧s + 1

r − 1r + r − 1

g −r − 1

g

⌧s + 1

- -1

ms + d

r − 1r + r − 1

g

-

System Frequency w/ iDroop

iDroop can reduce system frequency step‐response to a first order system ‐ no Nadir!

Nadir Cancellation

1ms + d

r − 1g

⌧s + 1

�s + δr − 1r

s + δ

- -

first order step response

1ms + d

r − 1g

⌧s + 1

r − 1r + r − 1

g −r − 1

g

⌧s + 1

- -1

ms + d

r − 1r + r − 1

g

-

Step Disturbance

Base Case: No Inverter Control

Example: Icelandic Power Grid• Iceland power network: 189 buses, 35 generators, load 1.3GW (PSAT)


Icelandic Grid

Example: Icelandic Power Grid• Iceland power network: 189 buses, 35 generators, load 1.3GW (PSAT)• Droop equally set for inverters in all cases• Virtual inertia tuned for critically damped response 𝝂 𝝂𝒎𝒊𝒏• iDroop tuned for Nadir elimination


Icelandic Grid


0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100 0 50 100

Icelandic Grid

iDroop Benefit Summary• Overshoot Elimination in Nadir• Noise Attenuation• Disturbance Rejection• Reduce Inter‐area Oscillations


0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100 0 50 100

Icelandic Grid


Step Disturbance ‐ Frequency

iDroop Tuning

• Droop equally shared between gens. and inverters.• Virtual inertia tuned for critically damped response 𝝂 𝝂𝒎𝒊𝒏• iDroop tuned for Nadir elimination

Step Disturbance – Control EffortInertia parameters of iDroop and VI are set to achieve zero overshoot.

Zero Nadir Tuning

0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100 0 50 100

Parameter Uncertainty

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2-224

-222

-220

-218

-216

-214

-212

-210


Other Benefits of iDroop

Benefits Summary• Overshoot Elimination in Nadir• Noise Attenuation• Disturbance Rejection• Reduce Inter‐area Oscillations

iDroop – Lead


Thanks!Related Publications:• Paganini and M, “Global analysis of synchronization performance for power systems: bridging the theory‐practice gap,” IEEE TAC 2020

• Jiang, Pates, M, “Dynamic Droop Control for Low Inertia Power Systems,” IEEE TAC 2021• Jiang, Cohn, Vorobev, M “Dynamic Droop Approach for Storage‐based Frequency Control,” IEEE TPS,conditionally accepted (arXiv)

• Min, Paganini, M, “Accurate Reduced Order Models for Coherent Synchronous Generators,” L‐CSS 2020• Y. Jiang, E. Cohn, P. Vorobev, and E. M, ”Storage‐Based Frequency Shaping Control,” L‐CSS 2020

Enrique [email protected]

http://mallada.ece.jhu.eduHancheng Min Eliza Cohn


Richard Pates Fernando PaganiniPetr VorobevYan Jiang

Backup Slides


Power Engineering Metrics

Step Response Nadir, RoCoF, Steady‐State

60.0 Hz

59.3 Hz UFLSSetting

RoCoF

Point C: Nadir

Point B

Point A

based on classical control theory…

Eigenvalue Analysis [Verghese ‘89]Dom. Eig, Damping Ratios, Part. Fact.

Domain specific, capture system degradationRelation with cause? Eigenvalue sensitivity? More than one ”dominant” eig.?+-


System Theoretic Metrics

‐norm: ‐norm:

Close form solutions, qualitative analysis, computational methodsRestrictive assumptions, not direct connection with RoCoF, Nadir, step disturbances+-

[Tegling… ’15, Poolla… ’15, Grunberg… ’16, Simpson‐Porco…‘16,Wu et al ’16, Adreasson ’17, Coletta ‘17… ]



Assumption: Let the machine relative inertia ( ), and assume




Assumption: Let the machine relative inertia ( ), and


F -12 V T

F -12 V Tu u

wPwP

V TF12

w! w!

Change of Vars.

F -12V

w w

Change of Vars.



Assumption: Let the machine relative inertia (. ), and


pe,1

-u1

wP,1 w1

w! ,1

c0

p0

wP,k

uk

wk

w! ,k

-

pe,k

c0

p0

1s λk

wn

w! ,n

un

wP,n

-

pe,k

c0

p0

1s λn

1s 0

System Frequency

F -12 V T

F -12 V Tu u

wPwP

V TF12

w! w!

Change of Vars.

F -12V

w w

Change of Vars.

Sync Error


Eigenvalues of:

Step Disturbance

Base Case: No Inverter Control



Icelandic Grid

Step Disturbance ‐ Frequency


iDroop Tuning

• Droop equally shared between gens. and inverters.• Virtual inertia tuned for critically damped response 𝝂 𝝂𝒎𝒊𝒏• iDroop tuned for Nadir elimination

Step Disturbance – Control EffortInertia parameters of iDroop and VI are set to achieve zero overshoot.


Zero Nadir Tuning

0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100-0.02

-0.01

0

0.01

0.02

0 50 100-0.2

-0.1

0

0.1

0.2

0 50 100 0 50 100

Parameter Uncertainty

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2-224

-222

-220

-218

-216

-214

-212

-210


Step DisturbanceInertia parameters of iDroop and VI are set to achieve zero overshoot.


Zero Nadir Tuning

0 50 100-0.5

-0.4

-0.3

-0.2

-0.1

0

SW DC VI iDroop0

0.5

1

Step DisturbanceInertia parameters of iDroop and VI are set to achieve zero overshoot.


System Frequency Synchronization Cost

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