Upload
dokhuong
View
299
Download
3
Embed Size (px)
Citation preview
Synchronized Phasor Measurements and State Estimation
Ali AburDepartment of Electrical and Computer Engineering
Northeastern University, Bostonhttp://www.ece.neu.edu/~abur
University of VermontSeptember 22, 2017
Outline
Synchronized Phasor MeasurementsSCADA/PMU Hybrid State EstimationPhasor-Only Linear State Estimation
WLS formulationLAV formulation
Dynamic State Estimation and ObservabilityConcluding Remarks
2
Historical Timeline
3
StateEstimation
1970
SCADA-OnlyMonitoring
PMUMeasurements
HybridState Estimation
DynamicState Estimation
Present Future 1960 1980
Phasor-OnlyState Estimation
Historical Timeline
• SCADA only monitoring– Vulnerable to errors in measurements, network
model parameters and topology– No direct measurement of phase angles
• Introduction of SE using SCADA– Phase angles can be estimated– Errors can be detected and removed
4
[*] Schweppe, F.C., Wildes, J., and Rom, D., “Power system static state estimation: Parts I, II, III”, Power Industry Computer Applications (PICA), Denver, CO, June 1969.
Static state estimation
– State variables: voltage phasors at all system buses – Measurements:
• Power injection measurements • Power flow measurements• Voltage/Current magnitude measurements• Synchronized phasor measurements
SCAD
AM
easu
rem
ents
5
State estimation: data/info flow diagram
Analog MeasurementsPi, Qi, Pf, Qf, V, I
Topology Processor
State Estimator
Network Observability Check
V, θ Bad Data Processor
Network Parameters, Branch Status, Substation Configuration
Parameter and Topology Errors Detection, Identification,
Correction
Load ForecastsGeneration Schedules
6
PM +
– Available every 1/30 seconds– Both voltage and current phasors– More accurate than SCADA but not error free
US DoE Office of Electricity Delivery and Energy Reliability “Advancement of Synchrophasor Technology in projects funded by the American Recovery and Reinvestment Act of 2009”, March 2016.
7
Under the Recovery Act SGIG and SGDP programs, their numbers rapidly increased : 166 1700
8
Phasor Measurement Units (PMU)Phasor Data Concentrators (PDC)[*]
GPS CLOCK
AntennaPMU 1
PT
CT
PDC
[*] IEEE PSRC Working Group C37 Report
CORPORATEPDC
REGIONALPDC
DATA STORAGE& APPLICATIONS
DATA STORAGE& APPLICATIONS
PMU 2
PMU 3
PMU 4PMU K
Reference phasor
+ϴ1
--ϴ2
--ϴ3 +ϴ1
--ϴ2
--ϴ3
Reference
Reference
18
Measurements provided by PMUs
PMUV phasor
I phasors
10
ALL 3-PHASES ARE TYPICALLY MEASUREDBUTONLY POSITIVE SEQUENCE COMPONENTS ARE REPORTED
𝑉𝑉𝐴𝐴𝑉𝑉𝐵𝐵𝑉𝑉𝐶𝐶
= [𝑇𝑇]𝑉𝑉0𝑉𝑉+𝑉𝑉−
𝑉𝑉+ = 13[𝑉𝑉𝐴𝐴+∝ 𝑉𝑉𝐵𝐵+∝ 2 𝑉𝑉𝐶𝐶]
∝= 𝑒𝑒𝑗𝑗𝑗𝜋𝜋3
11
Tutorial Example: Placement of PMUs
: Power Injection: Power Flow
: Voltage Magnitude
: PMU
BUSES REACHED:
B1, B2, B3, B4, B5,B6, B11, B12, B13,B9, B10, B14, B7,B8
12
Tutorial Example: Placement of PMUs
: Power Injection: Power Flow
: Voltage Magnitude
: PMU
BUSES REACHED:
B1, B2, B3, B4, B5,B6, B11, B12, B13,B9, B10, B14, B7,B8
CONSIDERINGZERO INJECTIONAT BUS B7 !
13
Branch PMU Placement for Full Observability
Only 7 branch PMUs make the entire system observable.
Use of Synchrophasor Measurements
Given unlimited number of available channels per PMU, it is sufficient to place PMUs at roughly 1/3rd of the system buses to make the entire system observable just by PMUs.
Systems No. of zero injections
Number of PMUs
Ignoring zeroInjections
Using zero injections
14-bus 1 4 3
57-bus 15 17 12
118-bus 10 32 29
Incorporation of PMUs in State Estimators:Hybrid Estimation
Hybrid State Estimation:• Use of hybrid measurements• Use of hybrid estimation methods
Challenges:• Different scan rates of SCADA and PMUs
Every 2-3 seconds versus every 33 ms• Different accuracy classes• Lack of full observability by PMU measurements• Coordinating two different estimators running
together
Re: SEL Synchrophasors Brochure
16
SCADA and PMU Measurements
SCADA measure.
SCADA measure.
PMU measur. PMU measur. PMU measur. PMU measur. PMU measur.
t1 t2
Using conventional SCADA-based SE System collapses without warningUsing mixed measurement based SE Tracks state and takes control action
17
Hybrid SE Using Both SCADA and PMU Measurements
Challenge: Different scan rates of SCADA and PMU measurements.
18
Possible Implementation of a Hybrid Scheme
19
Linear Estimator
Non-linear Estimator
Test Case
• IEEE 57 bus system– 9 branch PMUs– 32 Power injection measurements– 32 Power flow measurements
• Voltage collapse at bus 22 – No PMU at the bus.
20
57-Bus System
Voltage CollapseBus Location
21© Ali Abur
Voltage at Bus 22 Tracked by the Two Estimators
0 10 20 30 40 50 60 70 80 90
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
Time (ms.)
Vol
tage
mag
nitu
de (p
u)
True valuesWLS based estimatesLAV based estimates
WLS-based
Method
LAV-based
MethodAverage
Run Time 4.5 ms 9.7 ms
22
Historical Timeline
• Phasor Only State Estimation–Requires phasor based observability–Needs to be faster than scan rate of
PMUs–Should handle bad data (detect and
remove)
23
Incorporation of PMUs in State Estimators:Using Only Synchronized Phasor Measurements
PMU-Only State Estimator:• Use of only phasor measurements• Use of robust estimation methods
Challenges:• Requires a large number of PMUs for full observability• Estimation should be faster than PMU scan rate • Robustness should preferably be built-in
Measurement equations
onlyparametersnetworkofFunctionHModelLinearXHZ
tsMeasuremenPhasorXhH
ModellinearNonXhZtsMeasuremenSCADA
x
:
)(:)(
υ
υ
+⋅=
∇−+=
25
A.G. Phadke, J.S. Thorp, and K.J. Karimi, “State Estimation with Phasor Measurements”, IEEE Transactions on Power Systems, vol. 1, no.1, pp. 233-241, February 1986.
Phasor-only WLS state estimation
varianceerroriiRERHRHG
solutionDirectZRHGX
sidualrXHZrtoSubject
rMinimize
problemestimationstateWLSModelLinearXHZ
i
TT
T
m
i i
i
),(:)cov(}{;
ˆ
eˆ
:
2
1
11
2
2
σ
υυυ
σ
υ
=⋅==
=
⋅−=
+⋅=
−
−−
∑
26
Phasor-only WLS state estimation
[ ]
matrix incidence bus-branch :matrix admittancebranch :
matrixidentity :
AYU
VAY
U
currentsBranchvoltagesBus
IV
Z
b
b
m
mm
υ+⋅
⋅
=
⇒⇒
=
27
Consider a fully measured system:
Note: Shunt branches are neglected initially, they will be introduced later.
Phasor-only WLS state estimation
[ ]
[ ] [ ] υ
υ
++⋅=
++⋅
⋅+
=
++
=
jfeH
jfeAjbg
U
jdcjfe
IV
Let
F
mm
mm
m
m
)(
28
Phasor-only WLS state estimation:Complex to real transformation
[ ] υ
υ
+
⋅=
+
⋅
−=
fe
HZ
fe
gAbAbAgA
UU
dcfe
m
m
m
m
][
29
Phasor-only WLS state estimation:Exact cancellations in off-diagonals of [G]
( )( )
++++
=
−⋅
−=⋅=
AggbbAUAbbggAU
gAbAbAgA
UU
gAbAbAgA
UU
HHG
TTT
TTT
T
T
00
30
[G] matrix:• Is block – diagonal• Has identical diagonal blocks• Is constant, independent of the state
R is assumed to be identity matrix without loss of generality
Phasor-only WLS state estimation:Correction for shunt terms
XHRHGX
XHZRHGX
ZRHGXXE
XEHuEXHu
uXHXHHZ
shT
shTcorr
T
sh
sh
sh
ˆˆ)ˆ(
ˆ}{
}{}{
)(][
11
11
11
⋅−=
⋅−=
==
⋅=+⋅=
+⋅=+⋅+=
−−
−−
−−
υυ
31
Very sparse
Fast Decoupled WLS Implementation Results
32
Test Systems Used
SystemLabel
Number of Buses
Number of Branches
Number of Phasor
Measurements
A 159 198 222
B 265 340 361
C 3625 4836 4982
Case-1: No bad measurement.Case-2: Single bad measurement.Case-3: Five bad measurements.
Cases simulated:
33
MEAN CPU TIMES OF 100 SIMULATIONS
System CaseCPU Times (ms)
WLS Decoupled WLS
A1 5 2.42 5.7 2.73 9.3 3.9
B1 7.5 3.52 8.7 3.93 14.8 5.8
C1 137.4 75.92 169.5 95.73 284.7 165.6
Fast Decoupled WLS Implementation Results
L1 (LAV) Estimator
]1,,1,1[
ˆ
||1
=
+⋅=
⋅∑=
T
m
i
T
crXHZtoSubject
rcMinimize
Robust against gross errors
L1 estimator
35
• Computationally efficient. Fast Linear Programming (LP) code exists to solve large scale systems.
• L1 estimator automatically rejects bad data given sufficient local redundancy, hence bad data processing is built-in.
Conversion to Equivalent LP Problem
36
zrHxtsrT
=+ ..||c min
0 ..
c min
≥=
yzMyts
yT
][][
]00[c
IIHHMVUXXy
ccTTTT
bTa
mmnnT
−−==
=
VUrXXx ba
−=
= -
Phasor-Only Robust State Estimation
Robust LAV Based Estimator Testbed
Objectives• Perform static state estimation using a redundant set of PMU measurements• Maintain robustness against bad data
140-Bus NPCC System
139
133
134
135
132
128
92
120
118
123
117
119
114
90
89
105
112
113
8384
85
86
87
115
116
122
88
97 96
95
93
94
111
91
13
2414
25
12
715
8
18
17
2620
19
23
1122
10
1627
2928
9
30
6
531
4
32
1
21
33
3235
34
73
39
3738
40
41
42
44
45
43
49
77 74
7675
80
81125
78 7982
110
108
109
107
10410
6
131
130
127
124
129
5251
6163
6258
59
60
68
66
13713
6
138
57
56
53
67
65
64
48
47
50
46
126
6970
71
72
98
121
55
54
101
102 10
3
100
99
36
ISO-NENYISO
PJMMISO
IESO
Hydro Quebec
140
3625 Bus + 4836 Branch Utility System
Case a: No bad measurement.Case b: Single bad measurement.Case c: Five bad measurements.
38
Case a Case b Case c
LAV 3.33 s. 3.36 s. 3.57 s.
WLS 2.32 s. 9.38 s. 50.2 s.
Phasor-only state estimation:WLS versus L1 (LAV)
39
WLS :– Linear solution (exact cancellations
in [G] leading to decoupled formulation)
– Requires bad-data analysis– Normalized residuals test
(CPU increases with BD)
L1 (LAV):– Linear programming (computationally
competitive with WLS)
– Built-in bad-data analysis (CPU is relatively insensitive to BD)
Historical Timeline
• Dynamic State Estimation–Load and generator dynamic models–Wide-area versus local estimation–Tool to facilitate dynamic security
assessment
40
Basic Formulation
Dynamic state vector for the generators is augmented by the vector of all bus voltage magnitudes and phase angles.
Considering a system with N buses, the augmented state vector will be:
𝑥𝑥𝑘𝑘𝑇𝑇 = [𝛿𝛿𝑘𝑘𝑇𝑇 𝜔𝜔𝑘𝑘𝑇𝑇 𝑉𝑉𝑘𝑘𝑇𝑇 𝜃𝜃𝑘𝑘𝑇𝑇] at time instant k
41
Modeling [ DSE ]
42
Gen1
Gen2
GenNG
Load 1 Load 2
Load NL
Transmission Network[Lines + Transformers]
Estimated Measurements
DSE: Single machine / zonal / wide-areaDetailed Gen and Load models Frequency and power angle estimation
Zone-2
Zone-1
Zone NZ
Tracking the network anddynamic gen/load statevariables in real-time
Basic Formulation• Use all available measurements to form z• Discretize the dynamic state and
measurement equations • Form the set of discrete time equations:
43
kkkxfkx υ+=+ ),(1
kkk ekxhz += ),(
Extended Kalman Filter• Prediction:
• Correction:
44
)1,ˆ(ˆ 11 −= +−−
− kxfx kkkTkkk
Tkkkk LQLFPFP 111111 −−−−
+−−
− +=
( ) 1−−− += Tkkk
Tkkk
Tkkk MRMHPHHPK
( )),ˆ(ˆˆ kxhzKxx kkkkk−−+ −+=
( ) −+ −= kkkk PHKIP
k-1
k
−kx̂
+−1ˆkx
+kx̂
−−1ˆkx
DSE: Direct and Two-Stage Implementations
45
11ˆ,ˆ θV
u
11ˆ,ˆ QP
zx̂
Robust Linear Phasor
Estimator
DSEZwide-area
Zwide-area DSE x̂
TWO-STAGE
DIRECT
Timeline: Centralized/Direct DSE
46
SSE /Wide-area
DSE /Wide-area
DSE
Timeline: Local/Two-Stage DSE
47
DSE / Local
SSE /Wide-area
DSE /Wide-area
SSE
DSE
Robust Dynamic State Estimation
GG
GG
GG
GG
G G
Zone #1 Zone #2
Zone #n
UKFEstimated inputs &measurements x̂LAV
Measurements associated with each zone
0 5 10 15 20 25 3030
35
40
45
50
55
60
65
70
time [sec]
Rot
or a
ngle
[Deg
]
δ
δ+
0 5 10 15 20 25 3045
50
55
60
65
70
time [sec]
Rot
or a
ngle
[D
eg]
δ
δ+
Bad data present starting at t=15 sec. until y=20 sec.
Robust DSE Using LAV estimates DSE Using Raw Measurements
Observability Analysis for Time Varying Systems
For a time-varying dynamic system, the outcome of observability analysis will also be time dependent.
Outcome of observability analysis will no longer be binary, but the degree (or strength) of observability for a given measurement set at a given time instant will be of interest.
One metric to quantify this strength is the smallest singular value of the approximated observability matrix.
Approach
Two Alternative Observability Analysis Methods:
1. Use of small signal approximation and computeobservability matrix for linear dynamic systems.
2. Use Lie derivatives to compute the observability matrixand its smallest singular value.
Linear Time-Invariant Discrete-Time System
Observability matrix:
Dynamic system will be observable if the row rank of Õis equal to n (dimension of the state vector).
X(k+1) = A X(k) + B U(k)Z(k) = C X(k) + D U(k)
Small Signal Approximation
First order approximation for matrices A and C can be calculated at discrete time step k:
First order approximation for matrices A and C can be calculated at discrete time step k, yielding the approximate observability matrix (Õk) :
Results
Observability Analysis for Time Varying Systems
Pro:The main advantage of the linear approximation based approach is its computational simplicity.
Con:Linear approximation based results may occasionally be highly inaccurate in particular under highly nonlinear operating conditions.
Observability Analysis: NL Systems
55
In case of nonlinear systems, observability will no longer be a global property but will be determined locally around a given operating state or equilibrium point.
This can be done via the use of Lie derivatives of the nonlinear measurement function h with respect to the nonlinear function describing system dynamics [*].
[*] K. Muske and T. Edgar, Nonlinear State Estimation, Prentice-Hall, 1997.
Observability Analysis: NL Systems
56
�̇�𝑿 = 𝒇𝒇 𝒙𝒙 𝒕𝒕 + 𝒖𝒖(𝒙𝒙 𝒕𝒕 )
𝒁𝒁 = 𝒉𝒉 𝒙𝒙 𝒕𝒕
Lie derivative of h with respect to f will be given by:𝑳𝑳𝒇𝒇𝒉𝒉 = 𝛁𝛁𝒉𝒉 � 𝒇𝒇
By definition:𝑳𝑳𝒇𝒇𝟎𝟎𝒉𝒉 = 𝒉𝒉
𝑳𝑳𝒇𝒇𝒌𝒌𝒉𝒉 =𝝏𝝏(𝑳𝑳𝒇𝒇𝒌𝒌−𝟏𝟏𝒉𝒉)
𝝏𝝏𝑿𝑿� 𝒇𝒇
Observability Analysis: NL Systems
57
Defining Ω as:
and a gradient operator as:
𝛀𝛀 =
𝐿𝐿𝑓𝑓0(ℎ1) ⋯ 𝐿𝐿𝑓𝑓0(ℎ𝑚𝑚)𝐿𝐿𝑓𝑓1(ℎ1) ⋯ 𝐿𝐿𝑓𝑓1(ℎ𝑚𝑚)
⋮ ⋯ ⋮𝐿𝐿𝑓𝑓𝑛𝑛−1(ℎ1) ⋯ 𝐿𝐿𝑓𝑓𝑛𝑛−1(ℎ𝑚𝑚)
Ο = 𝑑𝑑Ω =
𝑑𝑑𝐿𝐿𝑓𝑓0(ℎ1) ⋯ 𝑑𝑑𝐿𝐿𝑓𝑓0(ℎ𝑚𝑚)𝑑𝑑𝐿𝐿𝑓𝑓1(ℎ1) ⋯ 𝑑𝑑𝐿𝐿𝑓𝑓1(ℎ𝑚𝑚)
⋮ ⋯ ⋮𝑑𝑑𝐿𝐿𝑓𝑓𝑛𝑛−1(ℎ1) ⋯ 𝑑𝑑𝐿𝐿𝑓𝑓𝑛𝑛−1(ℎ𝑚𝑚)
The observability matrix “O” defined above must have full rank in order for the system to be observable.
58
Results:
Validation Via Simulations:
59
0.09
ω
Smallest Singular ValuesRotor Speed / Syn. speed
0.127
δ
Meas.
0 5 10 15 20 25
Time [sec]
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Smal
lest
sin
gula
r val
ue o
f the
O m
atrix
0 5 10 15 20 25
Time [sec]
0
0.05
0.1
0.15
0.2
0.25
Smal
lest
sin
gula
r val
ue o
f the
O m
atrix
0 5 10 15 20 25
Time [Sec]
50
60
70
80
90
100
110
Ro
tor
an
gle
[D
eg
]
+
0 5 10 15 20 25 30
Time [Sec]
50
60
70
80
90
100
110R
otor
ang
le [D
eg]
+
Validation Via Simulations:
60
0.03
Smallest Singular ValuesRotor Speed / Syn. speed
Qe
Meas.
0 5 10 15 20 25
Time [Sec]
52
54
56
58
60
62
64
66
68
Rot
or a
ngle
[Deg
]
+
Pe
0 5 10 15 20 25
Time [Sec]
40
60
80
100
120
140
160
Rot
or a
ngle
[Deg
]
+
0 5 10 15 20 25
Time [sec]
0
0.5
1
1.5
2
2.5
Small
est s
ingula
r valu
e of
O m
atrix
2.0
0 5 10 15 20 25
Time [sec]
0
0.05
0.1
0.15
0.2
0.25
Smal
lest
sin
gula
r val
ue o
f the
O m
atrix
0 5 10 15 20 25
Time [sec]
2
4
6
8
10
12
14
16
18
20Sm
alles
t sing
ular v
alue o
f O m
atrix
Using Pe and Qe
Results
61
0 5 10 15 20 25
Time [sec]
0
5
10
15
20
25
30
Small
est s
ingula
r valu
e of
O m
atrix
Using all state variables
Smallest Singular Value Plots
Remarks and Conclusions
• Use of only phasor measurements simplifies the problemformulation and enables direct (non-iterative) solution.
• Hybrid SE can be beneficial in tracking system states duringslow moving emergencies.
• LAV-estimator becomes a computationally competitive androbust alternative to WLS when using PMUs.
• Strength of observability for different measurementconfigurations appears to be consistent with the ability of theDSE to track the true trajectory of the dynamic states.
• Observability analysis can facilitate sensor selection foroptimal tracking of dynamic states.
62
Acknowledgements
• National Science Foundation• NSF/PSERC• DOE/Entergy Smart Grid Initiative Grants• NSF/ERC CURENT
63
Jun Zhu, Ph.D. 2009
Murat Gol , Ph.D. 2014
Liuxi Zhang, Ph.D. 2014
A. Rouhani, Ph.D. 2017
Thank You
Any Questions?
64