At the beginning of power systems was . . .
Virtual Inertia Emulation and
Placement in Power Grids
Séminaire d’Automatique du Plateau de Saclay
Laboratoire de Signaux et Systèmes du Supelec
Florian Dörfler
At the beginning was the synchronous machine:
M
d
ω(t) = Pgeneration (t) − Pdemand (t)
dt
Pgeneration
ω
change of kinetic energy = instantaneous power balance
Pdemand
Fact: the AC grid & all of power system operation
has been designed around synchronous machines.
2 / 35
Operation centered around bulk synchronous generation
50.02
f [Hz]
50.01
Primary Control
Distributed/non-rotational/renewable generation on the rise
Tertiary Control
50.00
f - Setpoint
49.99
49.98
Secondary Control
49.97
49.96
PP - Outage
Oscillation/Control
49.95
PS Oscillation
49.94
49.93
49.91
49.88
16:45:00
16:50:00
a
49.89
rti
ne
49.90
lI
ca
ni
ha
ec
M
49.92
Frequency Mettlen, Switzerland
16:55:00
17:00:00
17:05:00
17:10:00
17:15:00
8. Dezember 2004
Frequency Athens
Source: W. Sattinger, Swissgrid
3 / 35
Source: Renewables 2014 Global Status Report
4 / 35
A few (of many) game changers . . .
synchronous generator
Fundamental challenge: operation of low-inertia systems
new workhorse
scaling
We slowly loose our giant electromechanical low-pass filter:
M
d
ω(t) = Pgeneration (t) − Pdemand (t)
dt
change of kinetic energy = instantaneous power balance
Pgeneration
ω
Pdemand
location & distributed implementation
Almost all operational problems can principally be resolved . . . but one (?)
6 / 35
5 / 35
Low-inertia stability: # 1 problem of distributed generation
75 mHz Criterion Summary - Short View - Year 2001-2011
2001
2002
2003
2004
2005
2006
30000
2007
2008
2009
Virtual inertia emulation
devices commercially available, required by grid-codes or incentivized through markets
!""" #$%&'%(#!)&' )& *)+"$ ','#"-'. /)01 23. &)1 2. -%, 2456
2010
!"#$%&'"'() %* +$,(-.'() /'-#%(-'
.( 0.1$%2$.3- 4-.(2 5.$)6,7 !('$).,
25000
20000
Duration [s]
Events [-]
8.".-9 :%(.! "#$%&'# (&)*&+! ,---; :6$<,(,$,<,(, =%%77,! (&)*&+! ,---; ,(3
06>67 ?@ ?9,(3%$>,$! (&)*&+! ,---
15000
-JLB88BI@;MB DBNLB@> -J?LBIIB8 %@B<>! "#$%&'# (&)*&+! ,---. B<I "LBO D1 ":F'BBIB<P! "&'./+ (&)*&+! ,---
Virtual synchronous generators: A survey and new perspectives
10000
Hassan Bevrani a,b,⇑, Toshifumi Ise b, Yushi Miura b
5000
a
Dept. of Electrical and Computer Eng., University of Kurdistan, PO Box 416, Sanandaj, Iran
b
Dept. of Electrical, Electronic and Information Eng., Osaka University, Osaka, Japan
0
5676
!89:;8;<=><? />@=AB: !<;@=>B >< CD!EFGBH;I
+><I *JK;@ E;<;@B=>J<
!"#$%&'()*+,-+#'"(./#0*/1(2-33/*04($(5&*0-$1((
6#+*0&$(7*/8&9+9(:"(!&;0*&:-0+9(<#+*="(20/*$=+(
0/(6;/1$0+9(7/>+*(2";0+%;((
!""" #$%&'%(#!)&' )& *)+"$ ','#"-'. /)01 23. &)1 2. -%, 2456
?$-0@&+*(!+1&11+A(!"#$"%&'()))A(B*-#/()*$#C/&;A(*"+,-%'!"#$"%&'()))A($#9(?&11+;(D$1$*$#=+(
Number * 10
!789:;< "=>?<:;@7 (@7:9@? ':9<:8AB C@9
/'(DE/F( #9<7G=;GG;@7 'BG:8=G
Months of the year
# frequency violations in Nordic grid
Number * 10
Fig. 3.2:
(source: ENTSO-E)
Duration
H;8I8; JK>. (<=LI8?? F1 M@@:K. N9<;7 *1 %O<=. %7O98P H1 $@GQ@8. <7O (K9;G N1 M9;AK:
Frequency quality behaviour in Continental Europe during the last ten years. Source: Swissgrid
7898:%+1*%;'<'*=''4>?%@%58;8A8%$'%A11*?@%!"#$%&'("()"&*'+,,,@%98%/1"'21B%1*$%38%/#<<4.'"C@%
!"#$%&'("()"&'+,,,%
It can clearly be
how the accumulated time
continuously increases
with higher
same
inobserved
Switzerland
(source:
Swissgrid)
frequency deviations as well as the number of corresponding events.
3.1.2.
!"#$%&#'$%()*+'",'"%-#,.%/#",012%3#*',#4%
5,)"16'%
CAUSES
The unbundling process has separated power generation from TSO, imposing new
commercial rules in the system operating process. Generation units are considered as
simple balance responsible parties without taking dynamic behaviour into account: slow
or fast units. Following the principle of equality, the market has created unique rules for
settlement: energy supplied in a time frame versus energy calculated from schedule in
the same time frame. Energy is traded as constant power in time frame.
inertia is shrinking, time-varying, & localized, . . . & increasing disturbances
The market, being orientated on energy, has not developed rules for real time operation
as power. In consequence we are faced with the following unit behaviour (Figure 3.3):
Solutions in sight: none really . . . other than emulating virtual inertia
through fly-wheels, batteries, super caps, HVDC, demand-response, . . .
Load
evolution
which
must be
covered
Power
basepoint
scheduled
A: Fast units
response
B: Slow unit
response
Energy to be compensated - real
cause of frequency deterministic
deviations
7 / 35
Energy
Contracted
M
d
ω(t) = Pgeneration (t)−Pdemand (t)
dt
. . . essentially D-control
⇒ plug-&-play (decentralized & passive), grid-friendly, user-friendly, . . .
⇒ today: where to do it? how to do it properly?
8 / 35
Outline
Introduction
Novel Virtual Inertia Emulation Strategy
inertia emulation
Optimal Placement of Virtual Inertia
Conclusions
Classification & choice of actuators
Inertia emulation & virtual synchronous machines
1
2
d
naive D-control on ω(t): M dt
ω(t) = Pgeneration (t) − Pdemand (t)
more sophisticated emulation of virtual synchronous machine
Electrical Power and Energy Systems 54 (2014) 244–254
Contents lists available at ScienceDirect
Electrical Power and Energy Systems
journal homepage: www.elsevier.com/locate/ijepes
Virtual synchronous generators: A survey and new perspectives
Hassan Bevrani a,b,⇑, Toshifumi Ise b, Yushi Miura b
a
b
Dept. of Electrical and Computer Eng., University of Kurdistan, PO Box 416, Sanandaj, Iran
Dept. of Electrical, Electronic and Information Eng., Osaka University, Osaka, Japan
a r t i c l e
(source: Stephan Masselis)
3
i n f o
Article history:
Received 31 December 2012
Received in revised form 12 June 2013
Accepted 13 July 2013
a b s t r a c t
In comparison of the conventional bulk power plants, in which the synchronous machines dominate, the
distributed generator (DG) units have either very small or no rotating mass and damping property. With
growing the penetration level of DGs, the impact of low inertia and damping effect on the grid stability
and dynamic performance increases. A solution towards stability improvement of such a grid is to provide virtual inertia by virtual synchronous generators (VSGs) that can be established by using short term
energy storage together with a power inverter and a proper control mechanism.
The present paper reviews the fundamentals and main concept of VSGs, and their role to support the
power grid control. Then, a VSG-based frequency control scheme is addressed, and the paper is focused
on the poetical role of VSGs in the grid frequency regulation task. The most important VSG topologies
with a survey on the recent works/achievements are presented. Finally, the relevant key issues, main
technical challenges, further research needs and new perspectives are emphasized.
! 2013 Elsevier Ltd. All rights reserved.
everything in between . . . and much more . . .
⇒ by measuring AC current/voltage/power/frequency
each of these (& far more) have been proposed for virtual inertia emulation
Keywords:
Virtual inertia
Renewable energy
VSG
Frequency control
Voltage control
Microgrid
⇒ software model of virtual machine provides converter setpoints
⇒ actuation via modulation (switching) or DC injection (batteries etc.)
1. Introduction
9 / 35
The capacity of installed inverter-based distributed generators
(DGs) in power system is growing rapidly; and a high penetration
level is targeted for the next two decades. For example only in Japan, 14.3 GW photovoltaic (PV) electric energy is planned to be
A solution towards stabilizing such a grid is to provide additional inertia, virtually. A virtual inertia can be established for
DGs/RESs by using short term energy storage together with a
power electronics inverter/converter and a proper control mechanism. This concept is known as virtual synchronous generator
(VSG) [3] or virtual synchronous machine (VISMA) [4]. The units will
10 / 35
Challenges in real-world converter implementations
Averaged inverter model
1
Electrical Power and Energy Systems 54 (2014) 244–254
Contents lists available at ScienceDirect
Electrical Power and Energy Systems
Real Time Simulation of a Power System with
VSG Hardware in the Loop
journal homepage: www.elsevier.com/locate/ijepes
Vasileios Karapanos, Sjoerd de Haan, Member, IEEE, Kasper Zwetsloot
Faculty of Electrical Engineering, Mathematics and Computer Science
Delft University of Technology
Delft, the Netherlands
E-mails: [email protected], [email protected], [email protected]
Virtual synchronous generators: A survey and new perspectives
Hassan Bevrani a,b,⇑, Toshifumi Ise b, Yushi Miura b
a
b
Dept. of Electrical and Computer Eng., University of Kurdistan, PO Box 416, Sanandaj, Iran
Dept. of Electrical, Electronic and Information Eng., Osaka University, Osaka, Japan
a r t i c l e
i n f o
Article history:
Received 31 December 2012
Received in revised form 12 June 2013
Accepted 13 July 2013
Keywords:
Virtual inertia
Renewable energy
VSG
Frequency control
Voltage control
Microgrid
1
a b s t r a c t
In comparison of the conventional bulk power plants, in which the synchronous machines dominate, the
distributed generator (DG) units have either very small or no rotating mass and damping property. With
growing the penetration level of DGs, the impact of low inertia and damping effect on the grid stability
and dynamic performance increases. A solution towards stability improvement of such a grid is to provide virtual inertia by virtual synchronous generators (VSGs) that can be established by using short term
energy storage together with a power inverter and a proper control mechanism.
The present paper reviews the fundamentals and main concept of VSGs, and their role to support the
power grid control. Then, a VSG-based frequency control scheme is addressed, and the paper is focused
on the poetical role of VSGs in the grid frequency regulation task. The most important VSG topologies
with a survey on the recent works/achievements are presented. Finally, the relevant key issues, main
technical challenges, further research needs and new perspectives are emphasized.
! 2013 Elsevier Ltd. All rights reserved.
Abstract- The method to investigate the interaction between a
Virtual Synchronous Generator (VSG) and a power system is
presented here. A VSG is a power-electronics based device that
emulates the rotational inertia of synchronous generators. The
development of such a device started in a pure simulation
environment and extends to the practical realization of a VSG.
Investigating the interaction between a VSG and a power system
is a problem, as a power system cannot be manipulated without
disturbing customers. By replacing the power system with a real
time simulated one, this problem can be solved. The VSG then
interacts with the simulated power system through a power
interface. The advantages of such a laboratory test-setup are
numerous and should prove beneficial to the further
development of the VSG concept.
A solution towards stabilizing such a grid is to provide additional inertia, virtually. A virtual inertia can be established for
DGs/RESs by using short term energy storage together with a
power electronics inverter/converter and a proper control mechanism. This concept is known as virtual synchronous generator
(VSG) [3] or virtual synchronous machine (VISMA) [4]. The units will
then operate like a synchronous generator, exhibiting amount of
inertia and damping properties of conventional synchronous machines for short time intervals (in this work, the notation of
‘‘VSG’’ is used for the mentioned concept). As a result, the virtual
inertia concept may provide a basis for maintaining a large share
of DGs/RESs in future grids without compromising system stability.
The present paper contains the following topics: first the fundamentals and main concepts are introduced. Then, the role of VSGs
in microgrids control is explained. In continuation, the most
important VSG topologies with a review on the previous works
and achievements are presented. The application areas for the
VSGs, particularly in the grid frequency control, are mentioned. A
frequency control scheme is addressed, and finally, the main technical challenges and further research needs are addressed and the
paper is concluded.
Short term frequency stability in power systems is secured
mainly by the large rotational inertia of synchronous
machines which, due to its counteracting nature, smoothes out
the various disturbances. The increasing growth of dispersed
generation will cause the so-called inertia constant of the
power system to decrease. This may result in the power
system becoming instable [1]-[3]. A promising solution to
such a development is the Virtual Synchronous Generator
(VSG) [4]-[8], which replaces the lost inertia with virtual
inertia. The VSG consists of three distinctive components,
namely a power processor, an energy storage device and the
appropriate control algorithm [4] as shown in Fig. 1. This
system has been tested in a full Matlab/Simulink [21]
simulation environment with promising results.
2
accuracy in AC measurements (averaged over ≈ 5 cycles)
3
constraints on currents, voltages, power, etc.
The capacity of installed inverter-based distributed generators
(DGs) in power system is growing rapidly; and a high penetration
level is targeted for the next two decades. For example only in Japan, 14.3 GW photovoltaic (PV) electric energy is planned to be
connected to the grid by 2020, and it will be increased to 53 GW
by 2030. In European countries, USA, China, and India significant
targets are also considered for using the DGs and renewable energy
sources (RESs) in their power systems up to next two decades.
Compared to the conventional bulk power plants, in which the
synchronous machine dominate, the DG/RES units have either very
small or no rotating mass (which is the main source of inertia) and
damping property. The intrinsic kinetic energy (rotor inertia) and
damping property (due to mechanical friction and electrical losses
in stator, field and damper windings) of the bulk synchronous generators play a significant role in the grid stability.
With growing the penetration level of DGs/RESs, the impact of
low inertia and damping effect on the grid dynamic performance
and stability increases. Voltage rise due to reverse power from
PV generations [1], excessive supply of electricity in the grid due
to full generation by the DGs/RESs, power fluctuations due to variable nature of RESs, and degradation of frequency regulation
(especially in the islanded microgrids [2], can be considered as
some negative results of mentioned issue.
4
idc
To better study and witness the effects of virtual inertia, the
hardware of a real VSG should be tested within a power
system. Investigating the interaction between a real VSG and
a power system is not easy as a power system cannot be
manipulated without disturbing customers. Building a real
power system for testing purposes would be too costly. By
replacing the power system with a real time simulated one,
this problem can be solved. In this paper the testing of a real
hardware VSG in combination with a simulated power system
is described.
The power processor from Fig.1 is built from a Triphase®
[9], [10] inverter system. The Matlab/simulink VSG
algorithm is directly implemented on the inverter system
through a dedicated FPGA interface developed by Triphase®.
In order to test the hardware implemented VSG and to
study its effects within a power system, it is connected with a
real time digital simulator from RTDS® [17] through a power
interface (Fig 2).
R
Gdc
Cdc
L
+
+ ic
vx
vαβ
iload
C
−
−
DC cap & AC filter equations:
1
Cdc v̇dc = −Gdc vdc + idc − m> iαβ
2
C v̇αβ = −iload + iαβ
˙ = −Riαβ + 1 mvdc − vαβ
Liαβ
2
delays in measurement acquisition, signal processing, & actuation
I. INTRODUCTION
1. Introduction
iαβ
ix
guarantees on stability and robustness
⇑ Corresponding author at: Dept. of Electrical and Computer Eng., University of
Kurdistan, Sanandaj, PO Box 416, Iran. Tel.: +98 8716660073.
E-mail address: [email protected] (H. Bevrani).
2. Fundamentals and concepts
The idea of the VSG is initially based on reproducing the dynamic
properties of a real synchronous generator (SG) for the power
electronics-based DG/RES units, in order to inherit the advantages
of a SG in stability enhancement. The principle of the VSG can be
applied either to a single DG, or to a group of DGs. The first
Fig. 1. The VSG Concept.
0142-0615/$ - see front matter ! 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijepes.2013.07.009
modulation: ix = 12 m> iαβ , vx = 12 mvdc
✓
Fig. 2. RTDS and Power Interface and VSG in a closed loop.
The RTDS® simulates power systems in real time and is
often used in closed loop testing with real external hardware.
Keeping in mind that the ADCs and DACs, which are the
inputs and outputs of the RTDS, have a dynamic range of
±10V max rated at 5mA max and the Triphase® inverter
system is rated at 16kVA, it is clear that a power interface has
to come in between to make this union possible as it is shown
in Fig. 2.
The main function of the power interface is to replicate the
voltage waveform of a bus in a network model to a level of
400VLL at terminal 1 in Fig. 2. This terminal is loaded by the
VSG and the current flowing from/to the VSG is fed back to
the RTDS, to load the bus in the network model with that
current.
The simulated power system is a transfer from the
Matlab/Simulink environment, in which the system was
developed initially, to RSCAD [18] format.
In section II the requirements for testing a VSG and the
principle of a VSG are discussed and in section III the test set
model of a
synchronous
generator
θ̇ = ω
M ω̇ = −Dω + τm +
>
iαβ
Lm if
C v̇αβ = −Gload vαβ + iαβ
− sin(θ)
cos(θ)
if
˙ = −Riαβ − vαβ − ωLm if − sin(θ)
Ls iαβ
cos(θ)
today: use DC measurement, exploit analog storage, & passive control
This work is a part of the VSYNC project funded by the European
Commission under the FP6 framework with contract No:FP6 – 038584
(www.vsync.eu).
passive: (idc , iload ) → (vdc , vαβ )
11 / 35
12 / 35
See the similarities & the differences ?
iαβ
ix
idc
standard power electronics control would continue by
Gdc
Cdc
2
constructing voltage/current/power references
(e..g, droop, synchronous machine emulation, etc.)
L
+
+ ic
vx
vαβ
−
1
R
iload
C
−
DC cap & AC filter equations:
1
Cdc v̇dc = −Gdc vdc + idc − m> iαβ
2
C v̇αβ = −iload + iαβ
˙ = −Riαβ + 1 mvdc − vαβ
Liαβ
2
modulation: ix = 12 m> iαβ , vx = 12 mvdc
tracking these references at the converter terminals
typically by means of cascaded PI controllers
let’s do something different (smarter?) today . . .
passive: (idc , iload ) → (vdc , vαβ )
✓
model of a
synchronous
generator
θ̇ = ω
>
M ω̇ = −Dω + τm + iαβ
Lm if
C v̇αβ = −Gload vαβ + iαβ
− sin(θ)
cos(θ)
if
− sin(θ)
˙
Ls iαβ = −Riαβ − vαβ − ωLm if
cos(θ)
13 / 35
14 / 35
iαβ
R
L
ix
idc
Gdc
Cdc
1
iload
+
+ ic
vx
vαβ
DC cap & AC filter equations:
Cdc v̇dc = −Gdc vdc + idc
C
1
− m> iαβ
2
quadratic curves for
stationary P vs. (|V |, ω)
2 /4G
⇒ P ≤ Pmax = idc
dc
C v̇αβ = −iload + iαβ
˙ = −Riαβ + 1 mvdc − vαβ
Liαβ
2
−
−
Some properties & different viewpoints
200
Amplitude (V)
Model matching (6= emulation) as inner control loop
150
100
50
0
⇒ reactive power not
directly affected
0
0.5
1
1.5
2
×10 4
Active power P
matching control: θ̇ = η · vdc
⇒ (P, ω)-droop ≈ 1/η
− sin(θ)
with η, µ > 0
, m =µ·
cos(θ)
⇒ (P, |V |)-droop ≈ 1/µ
Frequency (Hz)
50
inverter
40
(idc30, iload )
20
10
_
0
0
2
⇒ pros: is balanced, uses natural storage, & based on DC measurement
⇒ virtual machine with M =
Cdc
,
η2
D=
Gdc
,
η2
τm =
idc
η , if
=
µ
ηLm
3
⇒ base for outer controls via idc & µ, e.g., virtual torque, PSS, & inertia
15 / 35
reformulation as virtual
& adaptive oscillator
remains passive:
(idc , iload ) → (vdc , vαβ )
m
1
(vdc , vαβ )
Cdc v̇dc = −Gdc vdc + i∗dc − m⊤ iαβ
2
C v̇αβ = −iload + iαβ
1
˙ = −Riαβ
L0.5
iαβ
+ mv1.5
dc − vαβ
1
2
2.5
2
Active power P
×10 4
µ
modulation
!
"
0 1
ξ˙ = vdc η ·
ξ
−1 0
Eye candy: response to a load step
0.2
iαβ
R
L
ix
+
+ ic
iload
vx
vαβ
C
g load(Ω-1)
0.15
idc
0.1
Gdc
Cdc
Gload
−
−
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
170
1
Amplitude (V)
time(s)
200
100
165
160
155
150
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
50
Frequency (Hz)
Vx
time(s)
0
-100
-200
0
0.1
0.2
0.3
0.4
0.5
time(s)
0.6
0.7
0.8
0.9
1
48
46
44
42
40
0
0.1
0.2
0.3
0.4
0.5
time(s)
17 / 35
optimal placement
of virtual inertia
η
vdc
16 / 35
Linearized & Kron-reduced swing equation model
mi θ̈i + di θ̇i = pin,i − pe,i
pe,i ≈
j∈N
f
Pgeneration
+η
Pgeneration
generator swing equations
P
Performance metric for emulation of rotational inertia
restoration time
nominal frequency
ω
bij (θi − θj )
effort
linearized power flows
max deviation
Pdemand
demand
likelihood of disturbance at #i: ti ≥ 0
ROCOF
f
state space representation:
System norm:
0
I
θ
0
θ̇
=
+
T 1/2 η
−M −1 L −M −1 D
ω
M −1
ω̇
{z
}
|
{z
}
|
A
nominal frequency
amplification of
disturbances: impulse (fault), step (loss of unit), white noise (renewables)
B
to
where M = diag(mi ), D = diag(di ), T = diag(ti ), & L = LT (Laplacian)
performance outputs: integral, peak, ROCOF, restoration time, . . .
18 / 35
Coherency performance metric & H2 norm
Algebraic characterization of the H2 norm
Energy expended by the system to return to synchronous operation:
Z
0
∞
X
{i, j}∈ E
aij (θi (t) − θj (t))2 +
H2 norm interpretation:
1
2
3
associated performance output:
impulses (faults) −→ output energy
y=
Xn
si ωi2 (t) dt
"
# 0
θ
1/2
ω
Q
i=1
R∞
0
1/2
Q1
0
19 / 35
Lemma: via observability Gramian
kG k22 = Trace(B T PB)
R∞ T
where P is the observability Gramian P = 0 e A t C T Ce At dt
I
P solves a Lyapunov equation: P A + AT P + Q = 0
I
A has a zero eigenvalue → restricts choice of Q
"
# 1/2
Q1
0
θ
1/2
y=
Q1 1 = 0
1/2
ω
0 Q2
I
P is unique for P [1 0] = [0 0]
2
y (t)T y (t) dt
white noise (renewables) −→ output variance lim E
t→∞
y (t)T y (t)
20 / 35
21 / 35
Problem formulation
P , mi
kG k22 = Trace(B T PB)
Xn
subject to
i=1
mi ≤ mbdg
mi ≤ mi ≤ mi ,
T
Fundamental learnings
→ performance metric
→ budget constraint
i ∈ {1, . . . , n} → capacity constraint
1
explicit closed-form solution is rational function
2
sufficiently uniform (t/d)i → strongly convex & fairly flat cost
3
non trivial in the presence of capacity constraints
→ observability Gramian
PA+A P +Q =0
→ uniqueness
P [1 0] = [0 0]
Dissimilar and Identical t/d ratios
Budget, Sum, Inertia1, Inertia2
6
25
dissimilar t/d
identical t/d
5
Optimal inertia allocation
minimize
Building the intuition: results for two-area networks
f(m1)
4
Insights
1
2
3

m appears as m−1 in system matrices A , B 


large-scale &
⇒
product of B(m) & P in the objective

non-convex


product of A(m) & P in the constraint
3
2
1
0
0
2
4
6
0
0.1
0.2
0.3
0.4
0.5
t1 =1-t 2
0.6
0.7
0.8
0.9
1
optimal inertia allocation
Primary Control . . . cont’d
Theorem: the primary control effort optimization reads equivalently as
minimize
mi
𝜔sync
subject to
P1
I
T
θ̇(t) D θ̇(t) dt
I
0
1/2
1/2
= 0 and Q2
i=1
Xn
ti
mi
i=1
mi ≤ mbdg
i ∈ {1, . . . , n}
Key take-aways:
∞
which is the H2 performance for the penalties Q1
Xn
mi ≤ mi ≤ mi ,
P
P2
Primary control effort → accounted for by integral quadratic cost
Z
0
23 / 35
𝜔*
∝ (Pi − Pi (θ))
m
Di θ̇i = Pi ∗ − Pi (θ)
5
22 / 35
P/θ̇ primary droop control
(ωi −
10
performance metric
𝜔
∗
10
m1 ∗ + m2 ∗
15
m1
Closed-form results for cost of primary control
ω∗)
8
m1 ∗
m2 ∗
mbdg
20
=D
optimal solution independent of network topology
√
allocation ∝ ti or mi = min{mbdg , mi }
Location & strength of disturbance are crucial solution ingredients
24 / 35
25 / 35
Taylor & power series expansions
Key idea: expand the performance metric as a power series in m
kG k22 = Trace(B(m)T P(m)B(m))
Motivation: scalar series expansion at mi in direction µi :
1
1
δµi
=
− 2 + O(δ 2 )
(mi + δµi )
mi
mi
numerical method for
the general case
Expand system matrices as Taylor series in direction µ:
(0)
(1)
(0)
(1)
A(m + δµ) = A(m,µ) + A(m,µ) δ + O(δ 2 )
B(m + δµ) = B(m,µ) + B(m,µ) δ + O(δ 2 )
Expand the observability Gramian as a power series in direction µ:
(0)
(1)
P(m + δµ) = P(m,µ) + P(m,µ) δ + O(δ 2 )
Explicit gradient computation
Expansion of system matrices & Gramian ⇒ match coefficients . . .
Formula for gradient at m in direction µ
1
nominal Lyapunov equation for O(δ 0 ):
P(0) = Lyap(A(0) , Q)
2
(1)
P
3
results
perturbed Lyapunov equation for O(δ 1 ) terms:
= Lyap(A
(0)
,P
(0)
A
(1)
+A
(1) T
P
(0)
)
expand objective in direction µ:
kG k22 = Trace(B(m)T P(m)B(m)) = Trace((. . .) + δ(. . .)) + O(δ 2 )
4
T
T
gradient: Trace(2 ∗ B (1) P (0) B (0) + B (0) P (1) B (0) )
⇒ use favorite method for reduced optimization problem
27 / 35
26 / 35
Heuristics outperformed by H2 - optimal allocation
Modified Kundur case study: 3 regions & 12 buses
transformer reactance 0.15 p.u., line impedance (0.0001+0.001i) p.u./km
Original, Optimal, and Capacity allocations
160
0.25
Scenario: disturbance at #4
m
m∗
m
Cost
0.2
120
I
1
5
4
3
8
7
2
I
6
optimal allocation ≈
matches disturbance
0.1
40
0.05
0
0
1
2
4
5
6
8
9
10
12
node
allocation subject to capacity constraints
Original, Optimal, and Uniform allocations
inertia emulation at all
undisturbed nodes is
actually detrimental
⇒ location of disturbance &
inertia emulation matters
trace
0.15
80
0.25
m
m∗
muni
160
Cost
0.2
120
0.15
80
trace
12
locally optimal solution
outperforms heuristic
max/uniform allocation
inertia
9
11
inertia
I
10
40
0.05
0.1
0
1
2
4
5
6
8
9
10
12
0
node
allocation subject to the budget constraint
28 / 35
29 / 35
Eye candy: time-domain plots of post fault behavior
Original, Optimal, and Uniform allocations
Angle Diff.
Freq #4
(a)
0.05
(b)
0.15
5
×10 -3
Freq #5
Control Effort
(c)
(d)
2
0.04
m
muni
m∗
1.5
4
0.03
0.1
1
3
0.02
0.05
0
-0.01
∆ω5
2
∆ω4
∆θ 1 -∆θ 4
0.01
Control effort
0.5
1
0
0
-0.5
conclusions
-1
-0.02
0
-0.03
-1.5
-0.05
-1
-2
-0.04
-0.05
0
50
100
150
-0.1
0
Time(s)
50
100
150
Time(s)
-2
0
50
100
Time(s)
150
-2.5
0
50
100
150
Time(s)
Take-home messages:
best oscillation
performance
smallest peak
frequency at #4
undisturbed sites minimal control
are irrelevant
effort mi · θ̈i
30 / 35
Conclusions on virtual inertia emulation
Recall: operation centered around (virtual) sync generators
Where to do it?
1
50.02
f [Hz]
50.01
H2 -optimal (non-convex) allocation
Primary Control
Tertiary Control
50.00
2
closed-form results for cost of primary control
3
numerical approach via gradient computation
f - Setpoint
49.99
49.98
Secondary Control
49.97
49.96
PP - Outage
How to do it?
Oscillation/Control
49.95
PS Oscillation
1
down-sides of naive inertia emulation
2
novel machine matching control
49.94
49.93
&
49.89
n
, decentralized, plug-and-play (passive), grid-friendly, user-friendly, . . .
/ suboptimal, wasteful in control effort, & need for new actuators
io
at
ul
Em
49.90
What else to do? Inertia emulation is . . .
tia
49.91
er
In
49.92
49.88
16:45:00
16:50:00
Frequency Mettlen, Switzerland
16:55:00
17:00:00
17:05:00
17:10:00
Frequency Athens
Source: W. Sattinger, Swissgrid
32 / 35
31 / 35
A control perspective of power system operation
This “ what else? ” has been broadly recognized
by TSOs, device manufacturers, academia, etc.
Conventional strategy: emulate generator physics & control
M ω̇(t) =
| {z }
Pmech − Dω(t) −
| {z }
| {z }
(virtual) inertia tertiary control primary control
Z
|
t
0
ω(τ ) d τ − Pelec
{z
}
Essentially all PID + setpoint control (simple, robust, & scalable)
M ω̇(t) =
| {z }
D
P
|{z}
set-point
− Dω(t) −
| {z }
P
|
t
0
Control engineers should be able to do better . . .
ω(τ ) d τ − Pelec
{z
I
Massive InteGRATion of power Electronic devices
“The question that has to be
examined is: how much power
electronics can the grid cope
with?” (European Commission)
secondary control
Z
17:15:00
8. Dezember 2004
current controls
what else?
}
33 / 35
all options are on the table and keep us busy . . .
34 / 35
Acknowledgements
Taouba Jouini
Catalin Arghir
Bala K. Poolla
Saverio Bolognani
appendix
Dominic Gross
35 / 35
Spectral perspective on different inertia allocations
The planning problem
sparse allocation of limited resources
Cone, Original, Optimal, and Uniform allocations
`1 -regularized inertia allocation (promoting a sparse solution):
3
cone
m
muni
m∗
2
minimize
P , mi
Imaginary Axis
1
subject to
0
Jγ (m, P) = kG k22 + γkm − mk1
Xn
i=1
mi ≤ mbdg
mi ≤ mi ≤ mi
i ∈ {1, . . . , n}
-1
P A + AT P + Q = 0
-2
P[1 0] = [0 0]
-3
-0.18
-0.16
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
Real Axis
0
0.02
where γ ≥ 0 trades off sparsity penalty and the original objective
Highlights:
• m = m → best damping asymptote & best damping ratio
• Spectrum holds only partial information !!
1
regularization term is linear & differentiable
2
gradient computation algorithm can be used with some tweaking
Relative performance loss (%) as a function of γ
Uniform disturbance to damping ratio
0% → optimal allocation, 100% → no additional allocation
power sharing → d ∝ P ∗ , assuming t ∝ source rating P ∗
100
8
80
Card
Perf. %
Cardinality-Loc
Cardinality-Uni
Performance-Loc
Performance-Uni
6
4
40
2
20
0
0
0
0.5
1
1.5
2
2.5
γ
1
2
60
Theorem: for ti /di = tj /dj the allocation problem reads equivalently as
Relative Performance Loss (%)
Cardinality
Localized and Uniform disturbances
10
minimize
mi
subject to
localized disturbance ⇒ (2 → 1) without affecting performance
i=1
Xn
si
mi
i=1
mi ≤ mbdg
mi ≤ mi ≤ mi , i ∈ {1, . . . , n}
Key takeaways:
optimal solution independent of network topology
√
allocation ∝ si or mi = min{mbdg , mi }
3
×10 -4
uniform disturbance ⇒ ∃ γ 1.3% loss ≡ (9 → 7)
Xn
What if freq. penalty ∝ inertia? → norm independent of inertia
Taylor & power series expansions
Taylor & power series expansions cont’d
Key idea: expand the performance metric as a power series in m
Expand the observability Gramian as a power series in direction µ
kG k22 = Trace(B(m)T P(m)B(m))
Motivation: scalar series expansion at mi in direction µi :
1
1
δµi
=
− 2 + O(δ 2 )
(mi + δµi )
mi
mi
Expand system matrices in direction µ, where Φ = diag(µ):
0
I
0
0
(0)
(1)
A(m,µ) =
, A(m,µ) =
−M −1 L −M −1 D
ΦM −2 L ΦM −2 D
0
0
(0)
(1)
B(m,µ) =
, B(m,µ) =
M −1 T 1/2
−ΦM −2 T 1/2
(0)
(1)
P(m) = P(m + δµ) = P(m,µ) + P(m,µ) δ + O(δ 2 )
Formula for gradient in direction µ
1
2
nominal Lyapunov equation for O(δ 0 ): P(0) = Lyap(A(0) , Q)
perturbed Lyapunov equation for O(δ 1 ) terms:
T
P(1) = Lyap(A(0) , P(0) A(1) + A(1) P(0) )
3
expand objective in direction µ:
kG k22 = Trace(B(m)T P(m)B(m)) = Trace((. . .) + δ(. . .)) + O(δ 2 )
4
T
T
gradient: Trace(2 ∗ B(1) P(0) B(0) + B(0) P(1) B(0) )
Gradient computation
Heuristics outperformed also for uniform disturbance
Original, Optimal, and Capacity allocations
Algorithm: Gradient computation & perturbation analysis
m
m∗
m
Scenario: uniform disturbance
0.15
100
0.1
50
Evaluate the system matrices A(0) , B(0) based on current inertia
Heuristics for placement:
trace
inertia
Input → current values of the decision variables mi
Output → numerically evaluated gradient ∇f of the cost function
1
Cost
150
0.05
0
1
max allocation in case of
capacity constraints
2
uniform allocation in case
of budget constraint
0
1
2
4
5
6
8
9
10
12
node
allocation subject to capacity constraints
2
Solve for P(0) =Lyap(A(0) , Q) using a Lyapunov routine
Original, Optimal, and Uniform allocations
Cost
90
3
For each node- obtain the perturbed system matrices
A(1) , B(1)
m
m∗
muni
0.15
Results & insights:
1
locally optimal solution
outperforms heuristics
2
optimal solution 6= max
inertia at each bus
60
+
0.1
trace
Compute
T
A(1) P(0) )
inertia
4
P(1) =Lyap(A(0) , P(0) A(1)
30
5
Gradient ⇒ Trace(2 ∗
T
B(1) P(0) B(0)
+
0.05
T
B(0) P(1) B(0) )
0
0
1
2
4
5
6
8
9
10
12
node
allocation subject to the budget constraint