small signal modelling and eigenvalue analysis of ... · • traditional π-section models of hvdc...

Small Signal Modelling and Eigenvalue Analysis ofMultiterminal HVDC Grids

Salvatore D'Arco, Jon Are Suul SINTEF Energi AS

SINTEF Energy Research

• Power systemstability is commonly assessed by eigenvalueanalysis• Enables analysis and mitigation of oscillatory behaviour or instability due to system

configuration,systemparameters andcontroller settings

• VSCHVDCsystems has different dynamics compared to traditional generators• Models of MMCHVDCterminals arecurrently under development

• State-spacemodels for HVDCsystems canbeusedfor multiplepurposes• Analysis, identification and mitigation of oscillations and small-signal instability

mechanisms inHVDCtransmission schemes• Analysis of controller tuningand interactionbetweencontrol loops inHVDCterminals• Integration in larger power systemmodels for assessment of howHVDCtransmission

will influenceoverall small signal stability andoscillationmodes

2

Eigenvalue based small signal analysis

SINTEF Energy Research 3

Protection and Fault Handling in Offshore HVDC gridsObjectives: Establish tools and guidelines to support the design of multi-terminal offshore HVDC grids in order to maximize system availability. Focus will be on limiting the effects of failures and the risks associated to unexpected interactions between components.

• Develop models of offshore grid components (cables, transformers, AC and DC breakers, HVDC converters) for electromagnetic transient studies.

• Define guidelines to reduce the risks of unexpected interactions between components during normal and fault conditions.

• Define strategies for protection and fault handling to improve the availability of the grid in case of failures.

• Demonstrate the effectiveness of these tools with numerical simulations (PSCAD, EMTP), real time simulations (RTDS, Opal-RT) and experimental setups.

• Expand the knowledgebase on offshore grids by completion of two PhD degrees / PostDoc at NTNU and one in RWTH.


Overview of models and methods for stability analysis

4

Detailed Circuit Model(including IGBT’s)

Mathematical model with discontinuous switching functions

Average model with continuous insertion

indices, and time-periodic solutions in

steady-state

Average model with continuous insertion

indices, and time-invariant solutions in

steady-state

Piecewise (linear) models

Linearized SSTP models

Impedance Representation (seq. domain)

Linearized SSTI models

• Computationally intensive, time-consuming EMT simulation studies for large signal stability.

• Search for a Lyapunov function to prove large-signal stability.

• Estimate of regions of attraction as a measure of the system large-signal stability robustness.

Lyapunov methods for piecewise linear models:• Common quadratic Lyapunov function,• Switched quadratic Lyapunov function• Multiple Lyapunov functions

Small-signal stability assessment via time-periodic theory:• Poincaré multipliers

Small-signal stability assessment via traditional eigenvalue-based methods• Eigenvalue plots• Parametric sensitivity• Participation factor analysis

Small-signal stability assessment via by means of Nyquist criteria.Impedance

Representation (dq domain)

SINTEF Energy Research 6

GSC

Pw1

Pg1,Qg1Onshore

AC Grid #1

DC CABLE

Vdc and Q Controller

AC VoltageControl

Vdc_g1

Vdc_g1*fw1*|Vac_w1*|

Vac_w1

Offshore Grid #1

WFC Offshore Onshore

Qg1*

θw1*

Pw1

DC NETWORK

AC VoltageControl

fw1*Vac_w1*

Vac_w1

Offshore Grid #1

WFC #1

Offshore Onshore

θw1*

GSC #1 Pg1,Qg1Onshore

AC Grid #1

(Vdc vs. P) and Q Controller

Vdc_g1* Qg1*

Vdc_g1

AC VoltageControl

fw12*Vac_w2*

Vac_w2

WFC #2

θw2*

Pw2

Offshore Grid #1


Frequency-Dependent State-Space modelling of HVDC cables

7

• The modelling approach is based on a lumped circuit and constant parameters– Parallel branches allow for capturing the frequency dependent behavior of the cable– Compatible with a state space representation in the same way as classical models with simple

π sections– Model order depends on the number of parallel branches and the number of π sections


State-space frequency-dependent π section modelling

8


Behavior in a point to point HVDC transmission scheme

9

Interaction modesAll modes

5 π sections5 parallel branches

5 π sectionsclassical

Eigenvalue trajectory for a sweep of dc voltage controller gain


Main conclusions related to cable modelling

10

• ULM is established for EMT simulations• Traditional π-section models of HVDC cables

are not suitable for dynamic simulation or stability-assessment of HVDC systems

• Single inductive branches imply significant under-representation of the damping in the system

• Frequency-dependent (FD) π-model for small-signal stability analysis

• For simplified models, representation of cables by equivalent resistance and capacitance can be sufficient

• Developed Matlab-code and software tool for generating FD-πmodels


• Advantages• Modularity• Scalability• Redundancy• Low losses• DC-capacitor is not

required• Disadvantages

• High number of switches• Large total capacitance• Complexity• Sub-module Capacitors

will have steady-state voltage oscillations and internal currents can have corresponding frequency components

11

3-phase MMC: Basic Topology

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

SMA

B

aL

aL aL aL

aL aL

DCi

m

DCi

avi b

vi cvi

gVAC source

n

abc

Arm Leg

,dc lV

Submodule

CA

B

ali

aui

bli

cli

bui

cui

dcV fL

fC

gLoV

iV

,dc uV

Main challenge for small-signal modelling


Classification of MMC Modelling for eigenvalue analysis

12

MMC Small-Signal Modelling

PhasorModelling:

U. Aberdeen / NCEPU/Manitoba

Approach

Two-Level VSC Equivalent

Strathclyde approach

Zhejiang Universityapproach

MMC storage

?

YESNO

Duisburg-Essen

University approach

MMC Complete Power Balance

Circulating Current?

YESNO

Compensated Modulation?

(Jovcic et al. & Li et al.)

NO

Energy Aggregation:

SINTEF'S Approach

YES

Dynamic Phasors

(Deore et al.)

IIT Mumbai

Approach

(Trinh et al.)(Liu et al.)(Adam et al.) (Bergna et al.)

Voltage Aggregation:

SINTEF'S Approach

(Bergna et al.)


Compensated vs. Uncompensated Modulation

13

Compensated Modulation

Uncompensated Modulation

* *

* *

vk ckuk

cu

vk cklk

cl

e unu

e unu

Σ

Σ

− +=

+=

* *

* *

vk ckuk

dc

vk cklk

dc

e unV

e unV

− +≈

+≈

Voltage reference signals are divided by the actual arm voltage

Voltage references are divided by a constant value

Insertion Indexes

Insertion Indexes

*

*ck ck

vk vk

u ue e

≈

≈

MMC output voltage component will be approximately equal to the reference

Non-linear relationship between reference and "real" driving voltages

( ) ( )( ) ( )

*

*

12

k k k k k k k kck ck

vk dc vkk k k k k k k k

w w w w w w w wu ue V ew w w w w w w w

Σ ∆ Σ ∆ Σ ∆ Σ ∆

Σ ∆ Σ ∆ Σ ∆ Σ ∆

+ + − − + − − = − + − − + + −

The control output is modified by the energy information in each arm/phase

Appropriate for energy-based modelling

Energy-based modelling is not suitable for this case


Main conclusions related to MMC modelling

16

• The internal energy storage dynamics of MMCs must be represented for obtaining accurate models

• Established models of 2-Level VSCs should not be used for studying fast dynamics in HVDC systems

• Models assuming ideal power balance between AC- and DC-sides can only be used for studying phenomena at very low frequency

• Two cases of MMC modelling• Compensated modulation with Energy-based modelling• Un-compensated modulation with Voltage-based modelling

Energy-based model

Voltage-based model


Generation of a small signal model for MT HVDC

• A modular approach was developed to generate the small signal model of MT HVDC transmission system

– Decompose the HVDC MT into predefined modular blocks (cable, converters)– Modules can be customized by modifying the parameters but not the structure of the

subsystem– Several blocks are developed for the converters reflecting the topology and the control– Steady state conditions (linearization points) for each block were precalculated as a function of

the input– Steady state conditions for the entire system were obtained by implementing a dc loadflow

17


Definition of subsystem interfaces

18

CONVERTER A

CONVERTER C

CONVERTER B

CONVERTER D

CABLE AB

CABLE BC CABLE BD

CABLE CD


Definition of subsystem interfaces

19

CONVERTER A

CONVERTER C

CONVERTER B

CONVERTER D

CABLE AB

CABLE BCCABLE BD

CABLE CD

ADDINGNODE

ADDINGNODE

ADDINGNODE


Workflow for generating the small signal model

20

Definition of the grid topology

Input components parameters

dc Load flow

Calculation steady state conditions for the

submodules

Calculation state space matrices for the

submodules

Assemble submodules matrices into system

matricesExport data

Data inputCalculation steady

state conditionsCalculation state space matrices


Screenshot of the GUI after generating the small signal model


• Modes associatedwith thecablearequitequickly damped• One oscillatory mode and one real pole are slightly dependent on operating conditions

Systemis stableandwell-damped in thefull rangeof expectedoperatingconditions

23

MMC-based point-to-point transmission scheme

-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0-4000

-3000

-2000

-1000

0

1000

2000

3000

4000combination

-120 -100 -80 -60 -40 -20 0-150

-100

-50

0

50

100

150

Trajectory of critical eigenvalue with power reference is varied from -1.0 to 1.0 pu

Eigenvalues of MMC HVDC point-to-point scheme


• Variables of small-signal model can accurately represent the nonlinear system model for variables at both terminals

24

Time-domain verification of point-to-point MMC scheme

DC vo

ltage

[pu]

Activ

e cur

rent

I d[p

u]

DC voltage controlled terminal

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.56

-0.54

-0.52

-0.5

-0.48

-0.46

-0.44

time (s)

REFERENCELINEAR

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.96

0.98

1

1.02

1.04

1.06

1.08

1.1

time (s)

REFERENCELINEAR


Aggregated participation factor analysis

26

• Approach proposed for identifying interactions in an interconnected system• An interaction mode is defined as an eigenvalue having participation ρ higher than

a threshold χ from both parts of the interconnected system• Interaction modes identified as shown below for χ = 0.20• Close correspondence can be identified between identified interaction modes

and eigenvalues that are significantly influenced by the interconnection

-1000 -800 -600 -400 -200 0-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 104 Eigenvalues - 3T system

real

imag

-1000 -800 -600 -400 -200 0-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 104 Eigenvalues - Interaction modes

real

imag

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

modes

cum

ulat

ive

part

icip

atio

n in

sys

tem

S1

,,

ii

i

ααη =

pp

,,

,

ii

iS

αα

γγ

ηρ

η∈

=∑


• More interaction modes compared to casewith 2LVSCs

• In total 14 eigenvalues - 12oscillatory modes (6 pairs) and tworeal poles.

• A first group is defined as those welldamped oscillatory modes (real partsmaller than -200).

• A second group of interaction modes isfoundmuchcloser to the imaginary axis

• Oscillatory mode (39-40)• Tworeal eigenvalues (48and49)

27

Interaction modes –MMC HVDC point-to-point scheme

-400 -350 -300 -250 -200 -150 -100 -50 0-4000

-3000

-2000

-1000

0

1000

2000

3000

4000eigenvalues of interarea modes

8

9

10

11

14

15

21

22

25

26

39404849

-12.5 -12 -11.5 -11 -10.5 -10 -9.5 -9 -8.5 -8-30

-20

-10

0

10

20

30eigenvalues of interarea modes

39

40

48 49


• For fast interaction modes:• Balanced participation from the

two converter stations • High participation from the cable

in the fastest modes• Slow interaction modes

• Dominant participation from the DC-voltage controlled terminal in oscillatory modes

• Low participation from the cable, especially for the two real poles

• Depending on the eigenvalue, one station will have a higher participation

28

Interaction modes –Aggregated participation factor analysis

Aggregated participation factor analysis of interarea modes oftheMMC-HVDCpoint-to-point scheme–blue:DCVoltagecontrollingstation–green:power controllingstation–brown:dc cable


• The fast oscillatory modes (8-9, 10-11, and 14-15)• Related to dc voltages at both

cable ends • Associated with cable dynamics

• Modes 21-22 and 25-26 • "DC-side" interactions• Almost no participation from the

AC-sides• Associated with the MMC

energy-sum w∑ and the circulating current ic,z.

29

Participation Factor Analysis of Interaction Modes

vod voq ild ilq gamd gamq iod ioq phid phiq vplld vpllq epspll dthetapll vdc vdcf rho pacm icz wSigmaKappaSigma Xiz0

0.05

0.1

0.15

0.2

0.25mode 8, with eigenvalue at -397.87635+3360.0213i

conv 1conv 2


0.05

0.1

0.15

0.2

0.25

0.3

0.35


conv 1conv 2


0.05

0.1

0.15

0.2


conv 1conv 2


0.05

0.1

0.15

0.2

0.25

0.3

0.35


conv 1conv 2

8-9

10-11

14-15

21-22


• Oscillatory modegivenby eigenvalues 39-40• Interaction modes associated with the

power flowcontrol in thesystem• Associated with the integrator state of

theDCvoltagecontroller,ρ• Real poles 48 and 49

• Associated with integrator states of thePI controllers for the circulating current,ξz,

• The interaction of both stations in theseeigenvalues is mainly due to the powertransfer through the circulating current.

• Small participation of the cable since thedynamics are slow and the equivalentparameters of the arm inductorsdominate over the equivalent DCparameters of the cable

30

Participation Factor Analysis of Interaction Modes


0.05

0.1

0.15

0.2

0.25

0.3


conv 1conv 2


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1mode 48, with eigenvalue at -10.2218

conv 1conv 2


0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9mode 49, with eigenvalue at -9.2989

conv 1conv 2


0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


conv 1conv 2

39-40

25-26

48

49


Main conclusions related to interaction analysis

33

• Small-signal eigenvalue analysis can be utilized to reveal the properties of modes and interactions in the system

• Participation and sensitivity of all oscillations and small-signal stability problems can be analyzed

• Suitable for system design, controller tuning and screening studies based on open models

• Aggregated participation factor analysis can reveal interaction between different elements or sub-systems


Technology for a better society

36

small signal modelling and eigenvalue analysis of ... · • traditional π-section models of hvdc...

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