Synchronized Phasor Measurements and State Estimation

Ali AburDepartment of Electrical and Computer Engineering

Northeastern University, Bostonhttp://www.ece.neu.edu/~abur

abur@ece.neu.edu

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