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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
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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.
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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)
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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
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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
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State-space frequency-dependent π section modelling
8
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Behavior in a point to point HVDC transmission scheme
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Interaction modesAll modes
5 π sections5 parallel branches
5 π sectionsclassical
Eigenvalue trajectory for a sweep of dc voltage controller gain
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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
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• 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
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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.)
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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
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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
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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
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Definition of subsystem interfaces
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CONVERTER A
CONVERTER C
CONVERTER B
CONVERTER D
CABLE AB
CABLE BC CABLE BD
CABLE CD
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Definition of subsystem interfaces
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CONVERTER A
CONVERTER C
CONVERTER B
CONVERTER D
CABLE AB
CABLE BCCABLE BD
CABLE CD
ADDINGNODE
ADDINGNODE
ADDINGNODE
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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
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Screenshot of the GUI after generating the small signal model
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• 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
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• 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
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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∈
=∑
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• 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
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• 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
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• 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
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
0.25
0.3
0.35
0.4mode 10, with eigenvalue at -355.55027+3025.4561i
conv 1conv 2
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 14, with eigenvalue at -282.05785+2005.4132i
conv 1conv 2
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
0.25
0.3
0.35
0.4mode 21, with eigenvalue at -238.34277+1138.0026i
conv 1conv 2
8-9
10-11
14-15
21-22
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• 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
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
0.25
0.3
0.35mode 39, with eigenvalue at -12.3531+24.6583i
conv 1conv 2
vod voq ild ilq gamd gamq iod ioq phid phiq vplld vpllq epspll dthetapll vdc vdcf rho pacm icz wSigmaKappaSigma Xiz0
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
vod voq ild ilq gamd gamq iod ioq phid phiq vplld vpllq epspll dthetapll vdc vdcf rho pacm icz wSigmaKappaSigma Xiz0
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
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
0.25
0.3
0.35
0.4
0.45mode 25, with eigenvalue at -222.2334+683.0681i
conv 1conv 2
39-40
25-26
48
49
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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
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Technology for a better society
36
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