Power System Model Reduction Fall 2014 CURENT...
Transcript of Power System Model Reduction Fall 2014 CURENT...
Power System Model Reduction
Fall 2014 CURENT Course
October 29, 2014
Joe H. Chow
Rensselaer Polytechnic Institute, Troy, New York, USA
www.ecse.rpi.edu/homepages/chowj
Rensselaer Polytechnic Institute October 2014 JHC 2
Model Reduction of Large Power Systems
1. Simulation of power system dynamics for stability analysis on a digital computer: needs the most comprehensive power system model so that
i. the relevant dynamics can be accurately simulated given the computing resources
ii. the simulation can be completed in a reasonable amount of time.
2. One of the decisions is the geographic extent of the power system data set: disturbance always in the study region, but external system(s) also needed.
Rensselaer Polytechnic Institute October 2014 JHC 3
Model Reduction of Large Power Systems
1I
Study SystemExternal System
2I
kI
1bV
2bV
bkV
Gen + Controls
Gen + Controls
Gen + Controls
Gen + Controls
1P
Study SystemExternal System
1bV
2bV
Gen + Controls
Gen + Controls
Gen + Controls
Gen + Controls
2P
A
B
Internal
and
external
systems
Disturbance
Travelling
through the
external
system
Rensselaer Polytechnic Institute October 2014 JHC 4
Model Reduction Approaches
1. Coherency – some machines swinging together after a disturbancei. Time simulation (R. Podmore)ii. Modal coherency (R. Schleuter)iii. Slow coherency (J. Chow, et. al) – coherency with respect to
the slow interarea modesiv. Weak-link method – slow coherent areas are weakly
connected (N. Rao, J. Zaborzsky)
2. Aggregation – obtaining an equivalent power system modeli. Generator aggregation (R. Podmore)ii. Singular perturbations – provides improvements to
equivalent generator reactances (J. Chow, et. al)
Rensselaer Polytechnic Institute October 2014 JHC 5
Model Reduction Approaches
3. Linear model analysis of external systems
i. Modal truncation (J. Undrill, W. W. Price)
ii. Selective modal analysis (SMA) (G. Verghese, I. Peres-Arriaga, L. Rouco)
iii. Krylov method – moment matching (D. Chaniotis, M.A. Pai)
iv. Balanced truncaton – “best” reachibility (controllability) -observability (Shanshan Liu)
4. Artificial Neural Networks (F. Ma, V. Vittal)
5. Synchrophasor data (A. Chakrabortty, Chow)
1I
External System
2I
kI
1bV
2bV
bkV
Gen + Controls
Gen + Controls
Rensselaer Polytechnic Institute October 2014 JHC 6
Monographs
1982 2013
Rensselaer Polytechnic Institute October 2014 JHC 7
Topics
1. Slow coherency
2. Reduced order modeling of dominant transfer paths in large power systems using synchrophasor data (work with Dr. Aranya Chakrabortty)
Slow Coherency
A large power system usually consists of tightly connected controlregions with few interarea ties for power exchange and reservesharing
The oscillations between these groups of strongly connectedmachines are the interarea modes
These interarea modes are lower in frequency than local modesand intra-plant modes
Singular perturbations method can be used to show this time-scaleseparation
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 2 / 29
Klein-Rogers-Kundur 2-Area System
Gen 1
Gen 2
Load 3
1
2
3Gen 11
Gen 12
1310 20
101120 110 11
12
Load 13
Interarea mode
Local
mode
Local
mode
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 3 / 29
Coherency in 2-Area SystemDisturbance: 3-phase fault at Bus 3, cleared by removal of 1 linebetween Buses 3 and 101
0 1 2 3 4 51
1.001
1.002
1.003
1.004
1.005
1.006
1.007
Time (sec)
Mac
hin
e S
pee
d (
pu
)
Gen 1Gen 2Gen 11Gen 12
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 4 / 29
Power System Model
An n-machine, N -bus power system with classical electromechanicalmodel and constant impedance loads:
miδi = Pmi − Pei = Pmi −EiVj sin(δi − θj)
x′di= fi(δ, V ) (1)
where
machine i modeled as a constant voltage Ei behind a transientreactance x′dimi = 2Hi/Ω, Hi = inertia of machine i
Ω = 2πfo = nominal system frequency in rad/s
damping D = 0
δ = n-vector of machine angles
Pmi = input mechanical power, Pei = output electrical power
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 5 / 29
Power System Model
the bus voltage
Vj =√
V 2jre + V 2
jim, θj = tan−1
(
Vjim
Vjre
)
(2)
Vjre and Vjim are the real and imaginary parts of the bus voltagephasor at Bus j, the terminal bus of Machine i
V = 2N -vector of real and imaginary parts of load bus voltages
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 6 / 29
Power Flow EquationsFor each load bus j, the active power flow balance
Pej − Real
N∑
k=1,k 6=j
(Vjre + jVjim − Vkre − jVkim)
(
Vjre + jVjim
RLjk + jXLjk
)∗
− V2
j Gj = g2j−1 = 0 (3)
and the reactive power flow balance
Qej −Imag
N∑
k=1,k 6=j
(Vjre + jVjim − Vkre − jVkim)
(
Vjre + jVjim
RLjk + jXLjk
)∗
−V2
j Bj +V2
j
BLjk
2= g2j = 0
(4)
RLjk, XLjk, and BLjk are the resistance, reactance, and linecharging, respectively, of the line j-k
Pej and Qej are generator active and reactive electrical outputpower, respectively, if bus j is a generator bus
Gj and Bj are the load conductance and susceptance at bus j
Note that j denotes the imaginary number if it is not used as an index.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 7 / 29
Electromechanical Model
Mδ = f(δ, V ), 0 = g(δ, V ) (5)
M = diagonal machine inertia matrix
f = vector of acceleration torques
g = power flow equation
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 8 / 29
Linearized ModelLinearize (5) about a nominal power flow equilibrium (δo, Vo) to obtain
M∆δ =∂f(δ, V )
∂δ
∣
∣
∣
∣
δo,Vo
+∂f(δ, V )
∂V
∣
∣
∣
∣
δo,Vo
= K1∆δ +K2∆V (6)
0 =∂g(δ, V )
∂δ
∣
∣
∣
∣
δo,Vo
+∂g(δ, V )
∂V
∣
∣
∣
∣
δo,Vo
= K3∆δ +K4∆V (7)
∆δ = n-vector of machine angle deviations from δo
∆V = 2N -vector of the real and imaginary parts of the load busvoltage deviations from Vo
K1, K2, and K3 are partial derivatives of the power transferbetween machines and terminal buses, K1 is diagonal
K4 = network admittance matrix and nonsingular.
the sensitivity matrices Ki can be derived analytically or fromnumerical perturbations using the Power System Toolbox
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 9 / 29
Linearized Model
Solve (6) for∆V = −K−1
4 K3∆δ (8)
to obtainM∆δ = K1 −K2K
−14 K3∆δ = K∆δ (9)
where
Kij = EiEj (Bij cos(δi − δj)−Gij sin(δi − δj))|δo,Vo, i 6= j (10)
and Gij + jBij is the equivalent admittance between machine i and j.Furthermore,
Kii = −n∑
j=1,j 6=i
Kij (11)
Thus the row sum of K equal to zero. The entries Kij are known asthe synchronizing torque coefficients.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 10 / 29
Slow Coherent Areas
Assume a power system has r slow coherent areas of machines and theload buses that interconnect these machines
Define
∆δαi = deviation of rotor angle of machine i in area α from itsequilibrium value
mαi = inertia of machine i in area α
Order the machines such that ∆δαi from the same coherent areasappears consecutively in ∆δ.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 11 / 29
Weakly Coupled Areas
We attribute the slow coherency phenomenon to be primarily due tothe connections between the machines in the same coherent areas beingstiffer than those between different areas, which can be due to tworeasons:
The admittances of the external connections BEij much smaller
than the admittances of the internal connections BIpq
ε1 =BE
ij
BIpq
(12)
where E denotes external, I denotes internal, and i, j, p, q are busindices. This situation also includes heavily loaded high-voltage,long transmission lines between two coherent areas.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 12 / 29
The number of external connections is much less than the numberof internal connections
ε2 =γE
γI(13)
where
γE = maxα
γEα , γI = minα
γIα, α = 1, ..., r (14)
γEα = (the number of external connections of area α)/Nα
γIα = (the number of internal connections of area α)/Nα
where Nα is the number of buses in area α.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 13 / 29
Internal and External ConnectionsFor a large power system, the weak connections between coherent areascan be represented by the small parameter
ε = ε1ε2 (15)
Separate the network admittance matrix into
K4 = KI4 + ǫKE
4 (16)
where KI4 = internal connections and KE
4 = external connections.The synchronizing torque or connection matrix K is
K = K1 −K2(KI4 + ε(KI
4 ))−1K3
= K1 −K2(KI4 (I + ε(KI
4 )−1KE
4 ))−1K3 = KI + ǫKE (17)
whereKI = K1 −K2(K
I4 )
−1K3, KE = −K2KE4εK3 (18)
In the separation (17), the property that each row of KI sums to zerois preserved.c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 14 / 29
Slow Variables
Define for each area an inertia weighted aggregate variable
yα =
nα∑
i=1
mαi ∆δαi /m
α, α = 1, 2, . . . , r (19)
where mαi = inertia of machine i in area α, nα = number of machines
in area α, and
mα =
nα∑
i=1
mαi , α = 1, 2, . . . , r (20)
is the aggregate inertia of area α.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 15 / 29
Denote by y = the r-vector whose αth entry is yα.The matrix form of (19) is
y = C∆δ = M−1a UTM∆δ (21)
whereU = blockdiag(u1, u2, . . . , ur) (22)
is the grouping matrix with nα × 1 column vectors
uα =[
1 1 . . . 1]T
, α = 1, 2, . . . , r (23)
Ma = diag(m1,m2, . . . ,mr) = UTMU (24)
is the r × r diagonal aggregate inertia matrix.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 16 / 29
Fast VariablesSelect in each area a reference machine, say the first machine, anddefine the motions of the other machines in the same area relative tothis reference machine by the local variables
zαi−1 = ∆δαi −∆δα1 , i = 2, 3, . . . , nα, α = 1, 2, . . . , r (25)
Denote by zα the (nα − 1)-vector of zαi and conside zα as the αthsubvector of the (n− r)-vector z. Eqn. (25) in matrix form is
z = G∆δ = blockdiag(G1, G2, . . . , Gr)∆δ (26)
where Gα is the (nα − 1)× nα matrix
Gα =
−1 1 0 . 0
−1 0 1 . 0
. . . . .
−1 0 0 . 1
(27)
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 17 / 29
Slow and Fast Variable Transformation
A transformation of the original state ∆δ into the aggregate variable yand the local variable z
[
y
z
]
=
[
C
G
]
∆δ (28)
The inverse of this transformation is
∆δ =(
U G+)
[
y
z
]
(29)
whereG+ = GT (GGT )−1 (30)
is block-diagonal.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 18 / 29
Slow SubsystemApply the transformation (28) to the model (9), (17)
May = ǫKay + ǫKadz
Mdz = ǫKday + (Kd + ǫKdd)z (31)
where
Md = (GM−1GT )−1, Ka = UTKEU
Kda = UTKEM−1GTMd, Kda = MdGM−1KEU
Kd = MdGM−1KIM−1GTMd,Kdd = MdGM−1KEM−1GTMd (32)
Ka, Kad, and Kda are independent of KI
System (31) is in the standard singularly perturbed form
ε is both the weak connection parameter and the singularperturbation parameter
Neglecting the fast dynamics, the slow subsystem is
May = εKay (33)
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 19 / 29
Formulation including Power NetworkApply (28) to the model (6)-(7)
May = K11y +K12z +K13∆V
Mdz = K21y +K22z +K23∆V
0 = K31y +K32z + (KI4 + εKE
4 )∆V (34)
where
K11 = UTK1U, K12 = UTK1G+, K13 = UTK2, K21 = (G+)TK1U
K22 = (G+)TK1G+, K23 = (G+)TK2, K31 = K3U, K32 = K3G
+(35)
Eliminating the fast variables, the slow subsystem is
May = K11y +K13∆V
0 = K31y +K4∆V (36)
This is the inertial aggregate model which is equivalent to linking theinternal nodes of the coherent machines by infinite admittances.c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 20 / 29
2-Area System ExampleConnection matrix
K =
−9.4574 8.0159 0.5063 0.9351
8.7238 −11.3978 0.9268 1.7472
0.6739 0.9520 −9.6175 7.9917
1.3644 1.9325 8.1747 −11.4716
(37)
Decompose K into internal and external connections
KI =
−8.0159 8.0159 0 0
8.7238 −8.7238 0 0
0 0 −7.9917 7.9917
0 0 8.1747 −8.1747
(38)
εKE =
−1.4414 0 0.5063 0.9351
0 −2.6739 0.9268 1.7472
0.6739 0.9520 −1.6258 0
1.3644 1.9325 0 −3.2969
(39)
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 21 / 29
EM Modes in 2-Area System
λ(M−1K) = 0,−14.2787,−60.7554,−62.2531 (40)
with the corresponding eigenvector vectors
v1 =
0.5
0.5
0.5
0.5
, v2 =
0.4878
0.4031
−0.5672
−0.5271
, v3 =
0.6333
−0.7446
0.1924
−0.0863
, v4 =
0.1102
−0.1494
−0.8098
0.5566
(41)
Interarea mode:√−14.279 = ±j3.779 rad/s
Local modes:√−60.755 = ±j7.795 rad/s and√
−62.253 = ±j7.890 rad/s
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 22 / 29
Slow and Fast Subsystems of 2-Area System
Slow dynamics:
Ma =1
2π × 60
[
234 0
0 234
]
, εKa =
[
−4.1154 4.1154
4.9227 −4.9227
]
(42)
The eigenvalues of M−1a Ka are 0 and −14.561 ⇒ an interarea mode
frequency of√−14.561 = ±j3.816 rad/s.
Fast local dynamics:
Md =1
2π × 60
[
58.500 0
0 55.611
]
, Kd =
[
−8.3699 0
0 −8.0628
]
(43)The eigenvalues of M−1
d Kd are −53.939 and −54.660 ⇒ local modes of±j7.3443 and ±j7.3932 rad/s.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 23 / 29
Finding Coherent Groups of Machines
1 Compute the electromechanical modes of an N -machine power
2 Select the (interarea) modes with frequencies less than 1 Hz
3 Compute the eigenvectors (mode shapes) of these slower modes
4 Group the machines with similar mode shapes into slow coherentgroups
5 These slow coherent groups have weak or sparse connectionsbetween them
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 24 / 29
Grouping AlgorithmPlot rows of the slow eigenvector Vs of the interarea modes
1
2
0
1
2
0 1
1
Use Gaussian elimination to select the r most separated rows asreference vectors and group them into Vs1 and reorder Vs as
Vs =
[
Vs1
Vs2
]
⇒[
Vs1
Vs2
]
V −1s1 =
[
I
L
]
=
[
I
Lg
]
+
[
0
O(ε)
]
(44)
Use Vs1 to form a new coordinate systemc©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 25 / 29
Coherent Groups in 2-Area System
Vs =
0.5 0.4878
0.5 0.4031
0.5 −0.5672
0.5 −0.5271
Gen 1
Gen 2
Gen 11
Gen 12
, Vs1 =
[
0.5 0.4878
0.5 −0.5672
]
Gen 1
Gen 11
(45)Then
V ′sV
−1s1 =
1 0
0 1
0.9198 0.0802
0.0380 0.9620
Gen 1
Gen 11
Gen 2
Gen 12
(46)
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 26 / 29
Coherency for Load Buses
Vθ =
0.5 0.4283
0.5 0.3535
0.5 0.2556
0.5 0.3844
0.5 −0.5018
0.5 −0.4667
0.5 −0.3556
0.5 0.3128
0.5 −0.0523
0.5 −0.4671
0.5 −0.4125
Bus 1
Bus 2
Bus 3
Bus 10
Bus 11
Bus 12
Bus 13
Bus 20
Bus 101
Bus 110
Bus 120
VθV−1
s1 =
0.9436 0.0564
0.8727 0.1273
0.7800 0.2200
0.9020 0.0980
0.0620 0.9380
0.0953 0.9047
0.2006 0.7994
0.8342 0.1658
0.4880 0.5120
0.0949 0.9051
0.1466 0.8534
(47)
Bus 101
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 27 / 29
17-Area Partition of NPCC 48-Machine System
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 28 / 29
EQUIV and AGGREG Functions for Power System
Toolbox
L group: grouping algorithm
coh-map, ex group: tolerance-based grouping algorithm
Podmore: R. Pormore’s algorithm of aggregating generators atterminal buses
i agg: inertial aggregation at generator internal buses
slow coh: slow-coherency aggregation, with additional impedancecorrections
These functions use the same power system loadflow input data files asthe Power System Toolbox.
c©J. H. Chow (RPI-ECSE) Coherency October 29, 2014 29 / 29
Rensselaer Polytechnic Institute October 2014 JHC 8
Power Transfer Paths/Interfaces
Rensselaer Polytechnic Institute October 2014 JHC 9
Power Transfer Paths/Interfaces
1. The integrity of the power transfer interfaces is crucial for a large interconnected system to function properly. A disruption of a major transfer path may have a “spillover” effect into parallel power transfer paths in neighboring regions.
2. Questions:a. How can PMU data be used to monitor the integrity of
the power transfer interfaces, for both internal and external visibility?
b. How many PMUs do we need and where should they be located?
c. Are there analytical concepts and approaches to extract stability margin information from PMU data?
Rensselaer Polytechnic Institute October 2014 JHC 10
Power Transfer Paths/Interfaces
P1
Zone A
Exports Power
Zone B
Imports Power
PMU 1 PMU 2
PMU 3
P2
P3
Power transfer between two areas with multiple lines and PMU monitoring
Some US Interconnection interfaces may consist of
• a limited number of transmission lines (CA, NY, NE)
• many transmission lines
Rensselaer Polytechnic Institute October 2014 JHC 11
Inter-area Model Estimation
2-Area Power System 2-Machine Equivalent
• Problem: Given synchronized voltage phasors (magnitude and phase) at Buses 3 and 13 and current phasor between the two buses in the 2-area power system, develop an equivalent 2-machine model to represent the power transfer between the two areas.
• Variations: • Voltage phasors at Bus 101 also available.• Voltage phasor at Bus 3, the current phasor going from Bus 3 to Bus 101, and the
impedance between Buses 3 and 13. • Difficulty is how to look outward into the importing and exporting areas. • Need dynamic data, i.e., can’t do it with State Estimator solution. In particular, need
interarea mode oscillations in the voltage and current phasors.
3 13
3V 13V
I
2
e er jx
1jx 2jx
2 2,E 1 1,E
~ ~
2
e er jx101V~
101
1aG 2aG1
1G
2G
3G
4G
10
2 1220 120
3 13101
4 14
11011
4L 14L
Rensselaer Polytechnic Institute October 2014 JHC 12
Inter-area Model Estimation
• Measurement-based model reduction approach• Use bus voltage and line current phasors• Develop new concepts and calculation methods:
Exact extrapolation equations can be developed for an ideal 2-machine system – Dynamic Model Estimation (DME) algorithm
Extend DME to 2-area system – Interarea Model Estimation (IME) algorithm: verify with disturbances
• Additional research and customization needed: transfer paths with intermediate voltage support, transfer interfaces with multiple transmission lines, impact of disturbances, …
Rensselaer Polytechnic Institute October 2014 JHC 13
Dynamic Model Estimation
Two machine power system
Classical model representation
zTi = transformer impedance
zi = transformer impedance + direct-axis
transient reactance
lossem PPPH
2(i =1, 2)
sm
e
sm
loss
s
mm
m
H
HH
z
EEP
H
EHEH
z
rP
H
PHPHP
)cos()cos( 1221
2
21
2
12
2112
where
• Two generators G1 and G2 with
inertias H1 and H2 are connected
to Buses 1 and 2
• G1 supplies power to G2 (load)
• Dynamic model – swing equations
DME Problem Formulation for a 2-machine system
1 2
1V 2V
2 2r jx
~ ~
2z
1 1 1E E 2 2 2E E
1 1r jx
1z I
ez
e er jx~ ~
1G1 2
1V 2VI
ez
1 1T Tr jx
1E
2 2T Tr jx
2G
2E~ ~
~ ~
1Tz
e er jx
2Tz
Rensselaer Polytechnic Institute October 2014 JHC 14
Dynamic Model Estimation
• Assume PMUs are located at Buses 1 and 2
• Available Phasor Variables –
DME Problem : Given the available phasor variables, that exhibit a few cycles of
oscillations, and assuming E1 and E2 to be constant, compute
to completely characterize the dynamic behavior of the 2-machine system.
• Because and ,
the problem reduces to the estimation of
, , , , 1 1 2 2 , IV V I
, , , , , , , 1 1 2 2 1 2 1 2, eE E z z z H H
IVVze
~/)
~~( 21 IjzVE
~~~111 IjzVE
~~~222
, , 1 2 1 2, z z H H
DME Problem Formulation for a 2-machine system (Contd.)
1 2
1V 2V
2 2r jx
~ ~
2z
1 1 1E E 2 2 2E E
1 1r jx
1z I
ez
e er jx~ ~
Rensselaer Polytechnic Institute October 2014 JHC 15
Dynamic Model Estimation
• Key idea : Amplitude of voltage oscillation at any point on the transfer path is a
function of its electrical distance from the two fixed voltage sources.
]))sin()cos(([ ]))sin()cos(()1([),(~
2112 bEabEjbaEaErxV
where
22 ''
''
ee
ee
xr
xxrra
22 ''
''
ee
ee
xr
rxxrb
• Voltage Magnitude at B
)sin()cos()(2|),(~
|),(| 22
21 bbaaEEcrxVrxV
)())1(( 222
1
222
2 abEabEc where
Reactance Extrapolation
1 1,E I V x~
2 2, 0E
A B Cz
Rensselaer Polytechnic Institute October 2014 JHC 16
Dynamic Model Estimation
Reactance Extrapolation
• Linearize model about (δ0, ω0 = 0, Vss) :
),,()( 0baJxV
where the Jacobian is
)]cos()sin()[(),,(
2),,(:),,( 00
22
0
210
0
0
bbaabaV
EEbaVbaJ
• Assume uniform impedance along transfer path → and ex
xa 0b
)1( ),,(
)sin(2),0,(
0
0210 aa
baV
EEaJ
(*)
• Note : J(x,δ0) in (*) has a numerator varying
with x, and a denominator equal to steady
state voltage magnitude at B.
(*)
Rensselaer Polytechnic Institute October 2014 JHC 17
Dynamic Model Estimation
• J(x,δ0) in (*) can be used to estimate x1 and x2 if voltage oscillations are measured
or calculated at an additional intermediate bus between Bus 1 and 2
• Following a perturbation, let
iV
iV
iss
im
Busat Voltage state-Steady
Busat Swing Voltage of Amplitude
• From (*) , with321 , )1( , , iaaAVVV iiissimin
Computed from
PMU measurements
1 22 sin( oA E E
• The reactances x1 and x2, and the unknown constant A can be solved
numerically (3 nonlinear equations with 3 unknowns)
Reactance Extrapolation
1 2
1V 2V
1ez1z
2 2,E 1 1,E
~ ~3V
~
3
2ez2z
Rensselaer Polytechnic Institute October 2014 JHC 18
Dynamic Model Estimation
Inertia Estimation
• From linearized model
21
21
HH
HHH
)0( er
where f s is the measured swing frequency and is the aggregate
Inertia
• For a second equation in H1 and H2, use conservation of angular momentum
0 )()(222 221122112211 dtPPPPdtHHHH emem
1
2
2
1
H
H
• However, ω1 and ω2 are not available from PMU data,
• Estimate ω1 and ω2 from the measured frequencies ξ1 and ξ2 at Buses 1 and 2
)(2
)cos(
2
1
21
021
xxxH
EEf
e
s
Rensselaer Polytechnic Institute October 2014 JHC 19
Dynamic Model Estimation
• Again use linearization to derive bus frequency and machine speed expressions
12111
2121211111
)cos(2
)cos()(
cba
cba
22122
2221212122
)cos(2
)cos()(
cba
cba
where 22
221
22
1 ),1( ,)1( iiiiiii rEcrrEEbrEa
exxx
xr
21
11
e
e
xxx
xxr
21
1
2
(i=1, 2)
• ξ1 and ξ2 are measured, and ai, bi, ci are
known from reactance extrapolation
• Hence, we can calculate ω1/ω2 and solve
for H1 and H2
and ,
0 0.2 0.4 0.6 0.8 1
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
1
2
1
2
Normalized Reactance r
Fre
qu
en
cy O
scill
atio
n M
ag
nitu
de
(r/
s)
Inertia Estimation
Rensselaer Polytechnic Institute October 2014 JHC 20
Dynamic Model Estimation
Example
• Illustrate DME on classical 2-machine model (re = 0)
• Disturbance is applied to the system and the response simulated in MATLAB
Voltage Oscillations at 3 buses
0376.0 0326.0 0301.0
0136.1 0317.1 0320.1
0371.0 0316.0 0292.0
321
321
321
nnn
ssssss
mmm
VVV
VVV
VVV
DME Algorithm
x1 = 0.3382 pu
x2 = 0.3880 pu
• Exact values: x1 = 0.34 pu, x2 = 0.39 pu
Rensselaer Polytechnic Institute October 2014 JHC 21
Dynamic Model Estimation
Example (cont.)
• Jacobian curve fit for reactance extrapolation
Rensselaer Polytechnic Institute October 2014 JHC 22
Dynamic Model Estimation
Example (cont.)
• Inertia estimation
• Find the bus frequencies by passing the bus voltage angles through a derivative
filter with bandwidth higher than the inter-area mode frequency (0.912 Hz)
102.0)(
s
ssG
DME Algorithm
H1 = 6.48 pu, H2 = 9.49 pu
• Exact values: H1 = 6.5 pu, H2 = 9.5 pu
Rensselaer Polytechnic Institute October 2014 JHC 23
Dynamic Model Estimation
Example (cont.)
• Frequency curve fit for inertia estimation
Rensselaer Polytechnic Institute October 2014 JHC 24
Inter-area Model Estimation
• Consider inter-area dynamics across the transfer interface of a multi-
machine power system
• Extend the DME algorithm to estimate an equivalent two-machine model
• Assumption: Interface exhibits a single dominant mode of inter-area
oscillation
• A new approach to power system model reduction based on PMU
measurements
Rensselaer Polytechnic Institute October 2014 JHC 25
IME Problem : Given the phasor measurements that exhibit a
few cycles of inter-area oscillations, and assuming E1 and E2 to be constant, compute
to completely characterize the dynamic behavior of the reduced 2-machine system.
, , , , 3 3 13 13 , IV V I
, , , , , , , 1 1 2 2 1 2 1 2, e a aE E z z z H H
• Similar derivations as DME – but in this case local modes are present in system
response
• Apply a modal estimation method such as ERA to isolate inter-area mode
Inter-area Model Estimation
2-Area Power System 2-Machine Equivalent
1
1G
2G
3G
4G
10
2 1220 120
3 13101
4 14
11011
4L 14L
3 13
3V 13V
I
2
e er jx
1jx 2jx
2 2,E 1 1,E
~ ~
2
e er jx101V~
101
1aG 2aG
Rensselaer Polytechnic Institute October 2014 JHC 26
• Disturbance applied to a 2-area system
Bus 3 Voltage Bus 13 Voltage
Bus 101 Voltage
Inter-area Model Estimation
• With 300 MW power transfer the inter-area oscillation
frequency is 0.574 Hz, and the local mode
frequencies are 1.293 Hz and 1.308 Hz.
• IME estimates: x1 = 0.5069 pu, x2 = 0.0618 pu
Ha1 = 18.98 pu, Ha2 = 12.55 pu
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• Disturbance applied to a 2-area system
Inter-area Model Estimation
Frequency at Bus 3 Frequency at Bus 13
Rensselaer Polytechnic Institute October 2014 JHC 28
Inter-area Model Estimation
0 0.2 0.4 0.6 0.8 1-3
-2
-1
0
1
2
Normalized Reactance rF
req
ue
nc
y O
sc
illa
tio
n M
ag
nit
ud
e (
r/s
)
x1
x2
xe
Reactance Curve Fit Frequency Curve Fit
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Inter-area Model Estimation• Disturbance plots comparing full model to reduced models
Voltage Oscillations at Bus 3 Voltage Oscillations at Bus 13
Angular Difference Speed Difference
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Inter-area Model Estimation
Some important observations• Inter-area model estimation is independent of the severity of
the disturbance event resulting in the same post-fault condition (robustness)
• Inter-area model is dependent on the amount of inter-area power transfer. In the 2-area system example, larger amount of power transfers results in:
i. Higher angular separation between the sending end and the receiving end
ii. Slower frequency of oscillationiii. Smaller reactance at the sending end and larger reactance
at the receiving area iv. Equivalent inertias of the two areas also change
Rensselaer Polytechnic Institute October 2014 JHC 31
Conclusions and Future Research
• New techniques in using synchronized phasor data to identify power transfer paths/interfaces and hence for PMU siting.
• Establish power-angle curves and energy function analysis on each power transfer path. Monitor the power-angle curves as the system evolves through a sequence of disturbances.
• Monitor energy levels in the transfer paths to determine stability (loss of synchronism or negative damping) and to determine which transfer path is under the most stress.
• Improve system transmission planning and operations planning studies Different types of studies Different ways to see and evaluate results
• Extend calculations to dynamic voltage stability.
Rensselaer Polytechnic Institute October 2014 JHC 32
This work was supported primarily by the ERC Program of the
National Science Foundation and DOE under NSF Award Number
EEC-1041877.
Other US federal and state government agencies and industrial
sponsors of CURENT research are also gratefully acknowledged.
Acknowledgements