Bayesian Linear State Estimation using Smart Meters and PMUs Measurements

in Distribution Grids

Luca Schenato University of Padova

Joint work with

Grazia Barchi, Univ. of Trento, Italy

David Macii, Univ. of Trento, Italy

Alexandra von Meier, U.C. Berkeley

Kameshwar Poolla, U.C. Berkeley

Reza Arghandeh, CIEE

Calif. Inst. for Ener. and Envir.

Can we bring PMU to distribution grids ?

!  Develop a ultra-high-resolution micro-PMU for measuring voltage angles

!  develop a wireless network optimized for power distribution systems

!  deploy a hundred of these PMUs at participating utilities

!  Investigate diagnostic and control applications

Micro-synchrophasors for distribution systems

Traditional Distribution Grid

Medium Voltage Grid (MV) Low

Voltage Grid (LV)

Only passive loads (households, small businesses and factories)

Transformer

!  Passive loads !  Oversized for peak loads !  No need for (real-time) monitoring

Future Distribution Grid

Active loads (DRES) are being added !!

!  DRES might create reverse power flow !  Safety breakers status might not be known !  Rapid power-flow changes NEED FOR REAL-TIME MONITORING (ESTIMATION)

What is state estimation ?

1/60 Hz

V0

Voltage profile at every node at every point in time: vi(t) At steady-state and neglecting harmonics: v0=sin(ωt) vh(t)≈Vhsin(ωt+θh)

k h

0

θ

V

Im

Re 1

V θ

Why voltage phasors?

k

h

ξ Line currents

Load currents

Voltage phasor + grid impendences provide all necessary information

Voltage & line current" line power flows

Voltage & load current" load power flows

Grid state estimation

!  Global picture of the grid status !  Enabling Technology :

!  voltage regulation, stability monitoring, contingency analysis, dispatching, fault detection, topology identification

!  Hard problem !  Well studied in transmission grids (Schweppe and

Wildes, 1970) !  Non-linear stochastic problem !  Lines are mostly resistive in distribution grids !  Distributiuon grids are often 3-phase unbalanced

Can we measure voltage phasors?

Phasor Measurement Unit (PMU) •  GPS timestamps •  Measure magnitude and

phase w.r.t. global time •  Few PMU: costly (5K$) •  Voltage phasor only •  Real-time •  Total Vector Error (TVE):

|vtrue-vmeas|<1%-0.1%

Smart meter •  Measures active (p) and reactive power (q) •  Measures average power over 5-15min •  Data available at “control center” at midnight •  Historical data available for each household

Available information: smart meters

1 R. Sevlian and R. Rajagopal, “Short term electricity load forecasting on varying levels of aggregation,” submitted to IEEE Transactions on Power Systems

24h 0h

Available information: PMUs

From polar to cartesian b=c=σPMU

Objectives of this work

!  What is the value of PMU in distribution grids ?

!  What is the value of Smart Meters ?

!  PMU measurement error of PMUs vs P-Q prediction error of smart meters ?

!  What is the optimal positioning if only few PMU available ?

Contribution

!  Simple Power-Flow solver with P-Q loads

!  Linear approximation of PF with P-Q loads

!  Bayesian Linear State Estimation

!  Same performance but faster then WLS

!  Off-line optimal placement of PMUs

Power Grid modeling (single phase)

0 0.5 1 1.5 2 2.5 3

−1

−0.5

0

0.5

1

Smart Grid Topology

1 2

3

4 5

6

1

2

3

4

5

6

busno busfeederPMUvPMUiPMU

ξ

switch onswitch off

L: admittance matrix

6

4 6

Node models

0 0.5 1 1.5 2 2.5 3

−1

−0.5

0

0.5

1

Smart Grid Topology

1 2

3

4 5

6

1

2

3

4

5

6

busno busfeederPMUvPMUiPMU

ξ

switch onswitch off

LINEAR

NO

N LIN

EAR

Power Flow with PQ loads

0 0.5 1 1.5 2 2.5 3

−1

−0.5

0

0.5

1

Smart Grid Topology

1 2

3

4 5

6

1

2

3

4

5

6

busno busfeederPMUvPMUiPMU

ξ

switch onswitch off

Power Flow

Solver

Power Flow equations: A linear approximation

V0 V0

V0

V0

V0

V0

V0 V0

V0

V0 V0

V0

Bayesian Linear State Estimation (BLSE)

BLSE: Prior distribution

BLSE: posterior distribution

Weighted-Least-Squares State Estimator (WLS)

z = h(x)+ eNon-linear measurement model

The problem is usually solved by Weighted-Least-Squares method:

minx= J(x) =

i=1

N+M

∑ zi − hi (x)[ ]2 / Rii = z− h(x)[ ]T R−1 z− h(x)[ ]

A. Abur, A. Gomez Exposito, “Power System State Estimation: Theory and Implementation,” CRC Press, 2004

BLSE and WLS: block diagram

BLSE

WLSE

Performance Analysis Example - 15-nodes radial distribution network

D.Das, D. Kothari, and A. Kalam, “Simple and efficient method for load flow solution of radial distribution networks,” International Journal of Electrical Power & Energy Systems, vol. 17, no. 5, pp. 335 – 346, 1995.

Performance metric:

Theoretical (off-line)

Empirical (Monte Carlo)

Simulation results: linear vs nonlinear model

NO PMU

Load prediction uncertainty

Simulation results: BLSE vs WSE

Simulation result: Greedy vs Optimal placement

ARMSE as a function of number of PMUs with σL = 0.1% and σPMU=0.1% for BLSE

Simulation result: performance vs PMU accuracy

Simulation result: the value of phase measurements

0 5 10 150

1

2

3

4

5

6

7

8 x 10−3

Volta

ge A

RM

SE

Number of PMUs0 5 10 15

0

1

2

3

4

5

6

7

8 x 10−3

Volta

ge A

RM

SE

Number of PMUs

Magnitude only Magnitude and phase (PMU)

IEEE 123 node Test Feeder

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3 x 10−3

Number of PMUs

ARM

SE

load uncertanty=50%, PMU TVE=0.1%

WLSRMSE TheoreticalRMSE Monte Carlo

1

3

45 6

2

7 8

12

11 14

10

2019

2221

1835

37

40

135

3332

31

2726

2528

2930

250

48 4749 50

51

4445

46

42

43

41

3638 39

66

6564

63

62

60160 67

575859

54535255 56

13

34

15

16

1796

95

94

93

152

92 90 88

91 89 87 86

80

81

8283

84

78

8572

7374

75

77

79

300111 110

108

109 107

112 113 114

105

106

101

102103

104

450100

97

99

6869

7071

197

151

150

61 610

9

24

23

251

195

451

149

350

76

98

76

On-going work: 3-phase

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

MUSCAS et al.: IMPACT OF DIFFERENT UNCERTAINTY SOURCES 3

Fig. 1. PI model of three phase branch.

node. Voltages are expressed in rectangular form (i.e., thereal and imaginary parts of node voltages). All the branchesare represented through a three-phase PI model, where bothseries impedance and shunt admittance terms are taken intoconsideration. Mutual impedances among the phases havebeen considered, thus considering coupling among the dif-ferent phases (which are here referred to as a, b, and c,respectively). Furthermore, measurement devices are supposedto provide three-phase measurements [25], [27]. In Fig. 1,the model of the line between the buses 1 and 2 is shown:the phases of buses 1 and 2 are, respectively, a, b, c anda′, b′, c′ the subscript l indicates the current on the branchesand the subscript y identifies the current flowing through theground.

B. Measurements

The voltage at the substation and the powers supplied todownstream feeders are usually the only real-time measure-ments available in traditional distribution systems. However,it is widely accepted that the estimation obtained starting fromsuch a small measurement set is not sufficiently accurate. Forthis reason, in this paper, the possibility to assume the avail-ability of other measurements in the field has been applied.

Such measurements are described in the following.1) Smart meters (SMs): three-phase active and reactive

power injection at the node [25].2) Nonsynchronized measurements (NSMs): three-phase

node voltage magnitude, branch current magnitude, andalso the phase angle between the node voltage andbranch current of each phase. The cluster of voltagemagnitude, branch current magnitude, and phase anglebetween the node voltage and branch current is handledby the algorithm as active and reactive power flowmeasurement [26].

3) Synchronized measurements provided by PMUs:three-phase synchronized voltage phasors at the nodesand current phasors of the connected branches [27].

The following available prior knowledge of the network hasbeen included in the following estimation.

1) Zero injection buses: the lack of loads or generators insome nodes is considered as a three-phase measurementof zero active and reactive power injection with highaccuracy.

2) Pseudomeasurements: at the buses where no measure-ments devices are present, the three-phase active and

reactive power absorption is inferred based on historicaldata or statistical assumption.

C. Measurement Model

The presence of power measurements, both flow and injec-tion, having a nonlinear relation with the voltage state, isaddressed by converting them to equivalent rectangular currentmeasurements, using, in the iterative procedure, the voltagesestimated at the previous iteration [7]

˙Iin ji =

( ˙Sin j

V i−1

)∗= Re

(˙Iin j

i)

+ j · Im(

˙Iin ji)

(3)

Ili =

(Sl

V i−1

)∗= Re

(Il

i)

+ j · Im(

Ili)

(4)

where ´Iin ji

and Ili

are, respectively, the phasors in rectan-gular coordinates of injection current and the branch currentcalculated at the i th iteration of the Newton method, ´Sin j

and Sl are, respectively, the complex power injection andpower flow measurements, and V i−1 is the bus voltage phasorestimated at the previous iteration of the Newton method. Thisapproach permits the SE to be drastically simplified, becausethe Jacobian matrix (i.e., the matrix of partial derivatives ofthe estimated quantities) is constant and composed only byelements from the admittance matrix [7]. Thus, it is calculatedonly once, with a relatively low computational effort, andexploited for all the following iterations. Therefore, whena complex power measurement is given, the actual inputsat the algorithm are the real and imaginary parts of thecurrent.

The equations of the h(x) vector, for the three-phase case,are listed below for the four types of measurements thatare provided to the estimator, in order: bus voltage phasor,bus voltage magnitude, branch current phasor, and injectioncurrent measurements.

1) Bus voltage phasor measurements: the h(x) vector con-tains 6 · Nx elements, where Nx is the number of busvoltage phasor units (e.g., number of PMUs)

h(x) =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Re(V1_a)

Re(V1_b)

Re(V1_c)

Im(V1_a)

Im(V1_b)

Im(V1_c)...

Im(VNx _c)

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

where V is the voltage phasor, the subscript indicates thenumber of the corresponding node with voltage phasormeasurements (from 1 to Nx ), and the correspondingphase (a, b, c).

Summary !  BLSE better than WLS for PQ uncertainty >30% and PMU

Total Vector Error <1%

!  BLSE numerically much superior than WLS (non-iterative method and provides unique solution)

!  BLSE allows for off-line computation of estimation confidence bars. Allows off-line: !  Optimal PMU placement !  Evaluation of trade-offs between # of PMUs, their accuracy and performance

!  PMU with total error vector of 1% might not be sufficient for distribution networks

Questions ?

URL: http://automatica.dei.unipd.it/people/schenato.html