Simulation of Power System Dynamics and
Transient Stabilization Using Flywheel Energy
Storage Systems
Kevin Bachovchin
kbachovc@andrew.cmu.edu
10th CMU Electricity Conference
Pre-conference Workshop
March 30, 2015
Joint work with
Prof. Marija Ilic
milic@ece.cmu.edu
Outline
 Introduce and motivate flywheels
 Methods for modeling power system dynamics and designing
control using flywheels
 Model nonlinear power system dynamics using the Lagrangian
formulation
 Variable speed drive controller for flywheels using time‐scale
separation and nonlinear passivity‐based control logic
 Transient stabilization of interconnected power systems using
flywheels
 Smart Grid in a Room Simulator (SGRS)
 Implementation of simulating power system dynamics in a distributed
manner
 Show demo of flywheel controller
2
Motivation for Flywheels
 Transactive energy control does not guarantee dynamic
stability
 Instabilities can happen on a fast time‐scale
 Interest in implementing more wind power (and other
renewable energy sources) into future power grids
 Wind power is difficult to predict and control
 Large sudden deviations in wind power can cause
 high deviations in frequency and voltage
 transient instabilities
 One possible solution is to add fast energy storage, such as
flywheel energy storage systems
3
Flywheel Energy Storage System
 Stores mechanical energy by
accelerating a rotor to a very high
speed
 Not appropriate for large scale
applications
 Low energy capacity
 Many advantages for small‐scale
transient applications
 Very efficient
 Small time constants
 Not limited to a certain number of charge‐
discharge cycles
4
Objective
 Use flywheels for transient stabilization of power grids in
response to large sudden wind disturbances
 Design nonlinear power electronic control so that the
flywheel absorbs the disturbance and the rest of the system
is minimally affected
P
P
P
P
5
Modeling of Power System Dynamics
 To design and test control for flywheels, it is necessary to first
derive the dynamic model for the interconnected power system
 Large interconnected power systems contain many coupled
electrical and mechanical components
 Conventional modeling of dynamics:
 Electrical systems: Kirchhoff’s voltage and current law equations
 Mechanical systems: Conservation of force
 Difficulty is determining the effect of subsystems on each other for
mixed energy systems
Source: H. H. Woodson, J. R. Melcher, “Electromechanical Dynamics, Part I: Discrete
Systems,” New York: Wiley, 1968.
6
Lagrangian Formulation
 Therefore, for mixed energy systems, often advantageous to
derive the dynamics using the Lagrangian formulation from
classical mechanics
 Reformulation of Newtonian mechanics
Newtonian mechanics: model in terms of forces
Lagrangian mechanics: model in terms of kinetic energy and
potential energy of the system
 Can be applied to other types of systems, such as electric systems, as
well as to mixed energy systems
Source: D. Jeltsema, J.M.A. Scherpen, "Multidomain modeling of nonlinear networks
and systems," IEEE Control Systems, vol.29, no.4, pp.28-59, Aug. 2009
7
Motivation for the Lagrangian Formulation
 Unified framework for the analyzing systems with multiple
types of energy
Displacement Flow
Qgen
Fgen
Mechanical
(Rotational)
θ
ω
Rayleigh
dissipation
R
1
Rmech   B 2
2
1
2
Electrical
Electromechanical
Lagrangian
L
Relec   Ri 2
θ,
ω,
L  Lelec  L mech R  Relec  Rmech
Forcing
Function
F
F
V
F,V
 Passivity‐based control
 Choose closed‐loop energy functions
 Need to derive error dynamics from those closed‐loop energy functions
in order to derive the control law
8
Example Using Lagrangian Formulation
Electrical Subsystem:
Mechanical Subsystem:
1
1
1
1
2
2
2
2
 M cos  iSa iR  M cos   2 / 3 iSb iR  M cos   2 / 3 iSc iR
L elec  LR iR 2  LS iSa 2  LS iSb 2  LS iSc 2  LSS iSa iSb  LSS iSa iSc  LSS iSbiSc
Relec
1
1
1
1
 RR iR 2  RS iSa 2  RS iSb 2  RS iSc 2
2
2
2
2
L mech 
1
J 2
2
1
2
Rmech  B 2
Felec   vR vSa vSb vSc 
Fmech   m 
 Compute dynamic equations using the Lagrangian equations:
L
R
d  L 

 F (k )  0


dt  Fgen (k )  Q gen (k ) Fgen (k )
Source: R. Ortega, A. Loria, P. Nicklasson, H. Sira-Ramirez, Passivity-based Control of
Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications,
Springer Verlag, New York 1998
.
9
Automated Modeling of Power System Dynamics
 Implemented an automated procedure for symbolically
deriving the dynamic equations using the Lagrangian
formulation
 User specifies the power system topology
 Code symbolically solves for the energies of the system
 Code computes dynamic equations by evaluating the Lagrangian
equations and re‐expresses in standard state space form
Power System
Topology
L , R, F
x  f ( x , u )
x (0)  x 0
Source: K. D. Bachovchin, M. D. Ilić, "Automated Modeling of Power System
Dynamics Using the Lagrangian Formulation," International Transactions on
Electrical Energy Systems [to appear, 2014]
10
Variable Speed Drives for Flywheels
 Use AC/DC/AC converter to regulate the speed of the flywheel (and hence the
energy stored) to a different frequency than the grid frequency
 Controllable inputs are the duty ratios of the switch positions in the power
electronics
Source: K. D. Bachovchin, M. D. Ilic, "Transient Stabilization of Power Grids Using
Passivity-Based Control with Flywheel Energy Storage Systems," IEEE Power & Energy
Society General Meeting, Denver, USA, July 2015..
11
Controller Implementation
 Time‐scale separation to simplify the control design
 Regulate both the flywheel speed and the currents into the power
electronics using passivity‐based control
Source: K. D. Bachovchin, M. D. Ilic, "Transient Stabilization of Power Grids Using
Passivity-Based Control with Flywheel Energy Storage Systems," IEEE Power & Energy
Society General Meeting, Denver, USA, July 2015..
12
Nonlinear Passivity‐Based Control
 Nonlinear control method
 Exploits the intrinsic energy properties of the system dynamics
 Robust due to the avoidance of exact cancellation of
nonlinearities
 Relies on Lyapunov stability argument
 For a dynamic system
, if there exists a Lyapunov function
such that

is positive definite

0
0 and
0∀
0
is negative definite

0
0 and
0∀
0

then
0 is an asymptotically stable equilibrium
Source: R. Ortega, A. Loria, P. Nicklasson, H. Sira-Ramirez, Passivity-based Control of
Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications,
Springer Verlag, New York 1998
.
13
Automated Passivity‐Based Control Law Derivation
State space model:
Closed-loop
energy functions:
Lyapunov function:
x  f ( x, u )
V  x   Wm '  x   We  x 
Wm '  x  , We  x 
where x  x  x
Closed-loop
dissipation function:
R  x 
Set point equations:
fr ( x D )  r
can derive control law in
an automated manner
u  g1 ( x, x Dn , r  )
Passivity-Based
Control Law: x Dn  g ( x, x Dn , r  )
2
dV  x 
D
dt
dV  x  dx

dx dt
is positive definite
If
is negative definite
Then
x  0, x  x D
Non-directly controlled desired
state variable dynamics must
be stable for control to be
physically realizable.
Source: K. D. Bachovchin, M. D. Ilić, "Automated Passivity-Based Control Law
Derivation for Electrical Euler-Lagrange Systems and Demonstration on Three-Phase
AC/DC/AC Converter," EESG Working Paper No. R-WP-5-2014, August 2014.
14
Power Electronics Passivity‐Based Control
Dynamic Model:
(Power electronic
time-scale)
x pe  f pe  x pe , u pe 
x pe  i1d i1q qC 1  T
T
u pe  u1d u1q 
Closed-loop energy functions:

1
Wm '  L1 i1d 2  i1q 2
2

Set Point Equations:
i1d D  i1d *
2

q
1
1
C
We 
2 C
Closed-loop dissipation function:
1
1
R  R1 i1d 2  i1q 2  RC1iR12
2
2

i1q D  i1q *

Lyapunov function:
2

q
1
1
2
2
1
C
V  Wm ' We  L1  i1d  i1q  
2
2 C1
2

q
dV dx
2
2
C
1

V
  R1 i1d  i1q  2
dx dt
C1 RC1


Positive definite function
Negative definite function
Source: K. D. Bachovchin, M. D. Ilić, "Automated Passivity-Based Control Law
Derivation for Electrical Euler-Lagrange Systems and Demonstration on Three-Phase
AC/DC/AC Converter," EESG Working Paper No. R-WP-5-2014, August 2014.
15
Passivity‐Based Control for Power Electronics
Controller
Automated Control Law:
u1d 
2  C1V1d  C1 R1i1d ref  C1 L1i1q ref 1 
u1q 
qC1D
D
dqC1

dt

2  C1V1q  C1 R1i1q ref  C1 L1i1d ref 1 
qC1D
C1 V1d i1d ref  V1q i1q ref  R1  i1d ref   R1  i1q ref   vS 2 d ref i2 d  vS 2 q ref i2 q
2
2
qC D
 A stable equilibrium for
D
 q
D
C
CRC
only exists when
V1d i1d  V1q i1q  R1  i1d   R1  i1q   vS 2 d ref i2 d  vS 2 q ref i2 q

 
ref
ref
ref
2
power input to power electronics
ref
2
power output of power electronics
Source: K. D. Bachovchin, M. D. Ilić, "Automated Passivity-Based Control Law
Derivation for Electrical Euler-Lagrange Systems and Demonstration on Three-Phase
AC/DC/AC Converter," EESG Working Paper No. R-WP-5-2014, August 2014.
16
Transient Stabilization Using Flywheels
P
P
P
P
 Want to choose set points so that the wind disturbance power goes
to the flywheel and rest of the system is minimally affected
17
Simulation Results: Flywheel
 Since the power output of the wind generator decreases during the
disturbance, the flywheel set point decreases
Wind Generator Mechanical Torque
Flywheel Speed
angular velocity(rad/sec)
0.26
torque(N*m)
0.255
0.25
0.245
0.24
0.235
0.23
0
0.2
0.4
time(sec)
0.6
0.8
1290

1280

ref
1270
1260
1250
1240
1230
0
0.2
0.4
time(sec)
0.6
0.8
18
Simulation Results: Power Electronics
 The set points for the power electronic currents are chosen so that
the total current out of Bus 2 remains constant during the
disturbance
Power Electronic and Wind Currents
Power Electronic Currents
2.4
current(A)
current(A)
-251
-252
i1d
-253
-254
iref
1d
0
0.2
0.4
0.6
i1q
36
current(A)
current(A)
1.8
0
0.2
0.4
0.6
0.8
20.02
37
iref
1q
35
34
33
2
1.6
0.8
i1d+iSdW
2.2
0
0.2
0.4
t(sec)
0.6
0.8
i1q+iSqW
20.01
20
19.99
19.98
0
0.2
0.4
t(sec)
0.6
0.8
19
Simulation Results: Rest of System
 With the control, the effect on the rest of the system is very
minimal and lasts only a short time
0.0745
voltage(V)
0.066
voltage(V)
Bus 2 Voltage: Direct Component
Bus 1 Voltage: Direct Component
Vd (Control)
Vd (No Control)
0.065
0.064
0.2
0.4
0.6
Vq (Control)
Vq (No Control)
0.2
0.4
t(sec)
0.6
0.2
0.4
0.6
0.8
2
0.8
Vq (Control)
1.995
1.99
0
0
2.005
voltage(V)
voltage(V)
0.073
Bus 2 Voltage: Quadrature Component
2
1.995
Vd (No Control)
0.0735
0.8
Bus 1 Voltage: Quadrature Component
1.99
0.074
0.0725
0
Vd (Control)
Vq (No Control)
0
0.2
0.4
t(sec)
0.6
0.8
20
SGRS: Modular Modeling of Open‐Loop Dynamics
 For the Smart Grid in a Room Simulator, a modular object‐
oriented approach is used for modeling power system
dynamics
 lends itself to distributed computing
 scalable for large systems
 Open‐loop dynamics of each object
depend on




State variables
Controllable inputs
Exogenous inputs
Port inputs
x k  f k  xk , pk , uk , mk 
Source: K. D. Bachovchin, M. D. Ilić, "Automated and Distributed Modular Modeling of
Large-Scale Power System Dynamics," EESG Working Paper No. R-WP-8-2014,
October 2014
21
SGRS: Modular Modeling of Closed‐Loop Dynamics
 Controllable inputs depend on
 State variables
 Outputs of connecting modules
 Internal set points ref
u k  g k  xk , yck 1 , yk ref
Variable Speed Drive Controller:
T
1

yk
 Internal set points depend on
 Outputs of connecting modules
 Set points from market ref
y k ref  h k yck 2 , r ref

u k  G k  xk , yck , r ref

u k  u1d u1q u2 d u2 q 
T


x k  i1d i1q qC1 iS 2 d iS 2 q iR 2   
T
y ck 1   vd vq 
2
ref
 2 ref i1d ref i1q ref 
T
y ck 2  iWd iWq 
ref
ref
ref T
r  iTotd iTotq 
Explicit
Equations

y ck   y ck 1 y ck 2 
T
i1d ref  iTotd ref   iWd
i1q ref  iTotq ref   iWq
T
22
SGRS: Modeling of Interconnected Power System
 Dynamics of the interconnected power system can be symbolically solved for
in a distributed manner
x i  fi ( xi , pi , ui , mi )
x j  f j ( x j , p j1 , p j 2 , u j , m j )
Bus 1 solves for
pi  g1 ( y j ), p j1  g 2 ( yi )
x k  f k ( xk , pk , uk , mk )
Bus 2 solves for
pk  h1 ( y j ), p j 2  h2 ( yk )
 Dynamics of each module depend only on its own state variables
and the outputs of connecting modules
x k  Fk ( xk , yck , uk , mk )
Source: K. D. Bachovchin, M. D. Ilić, "Automated and Distributed Modular Modeling of
Large-Scale Power System Dynamics," EESG Working Paper No. R-WP-8-2014,
October 2014
23
SGRS: Communication Structure for Distributed
Simulation of Dynamics and Control
Processor A
Initial Conditions
xi 0
yi0
y j0
Compute
for time step t
Take next
iteration step
Processor B
Initial Conditions
xj
yk0
0
y j0
Compute
for time step t
yin
n  n 1
Take next
iteration step
Initial Conditions
xk 0
Compute
for time step t
ykn
n  n 1
y jn
Processor C
Take next
iteration step
n  n 1
y jn
24
Conclusions
 Designed a novel variable speed drive for flywheels using three
time‐scale separations and passivity‐based control logic
 Demonstrated the effectiveness of this controller in the SGRS for
transiently stabilizing an interconnected power system against a
wind generator disturbance
Future Work
 Larger power systems with multiple wind generator disturbances
and multiple flywheels
 More general flywheel control logic for systems where the source
of the disturbance is not known
25