Bayesian linear state estimation using smart meteres and PMU ...
Transcript of Bayesian linear state estimation using smart meteres and PMU ...
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
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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
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4 5
6
1
2
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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
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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