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SYNCHRONIZED PHASOR MEASUREMENTTECHNIQUES
A.G. Phadke
Lecture outline:
Development of Phasor Measurement Units
Power system state estimation
Control with feed-back
Adaptive relaying
Remedial Action Schemes
Phasor Estimation
Off-nominal frequency phasors
Evolution of PMUs
Standards
Applications of PMUs
Comtrade Synchrophasor
History of Wide-Area Measurements
Wide-area measurements in power systems have been used in EMS functions for a long time. Economic Dispatch, tie line bias control etc. all require wide area measurements.
EMC
However, the birth of modern wide-area measurement systemscan be traced back to a very significant event which took placein 1965.
The state estimators as we know them today were developedfollowing the technical assessments of the causes of the failuresin 1965.
The Birth of the PMUs
Computer Relaying developments in 1960-70s.
Symmetrical Component Distance Relay Development.
Significance of positive sequence measurements.
Importance of synchronized measurements.
Development of first PMUs at Virginia Tech ~ 1982-1992
Development funded by AEP, DOE, BPA, and later NYPA
First prototype units assembled at Va Tech and installedon the BPA, AEP, NYPA systems.
(b)
GPSreceiver
PMU
Signalconditioning
unit
UserInterface
PHASOR ESTIMATION
Introduction to phasors
Real
Imag
inar
y
The starting time defines the phase angle of the phasor.
This is arbitrary. However, differences between phase angles are
independent of the starting time.
t=0
Sampling process, Fourier filter for phasors
sin and cosfunctions
t
Input signal
xnxn-1.
.x1
Dat
asa
mpl
es
cosi
nes
sine
s
Phasor X = -- xk(cosk - j sink)2N
Sampling process, Fourier filter for phasors
Fourier filters can also be described as:
Least-squares on a period Cross-correlation with sine and cosine Kalman filters (under many circumstances)
t
Xc - jXs = -- xk(cosk - j sink)Phasor X = (AXc+BXs)+j(CXc+DXs)
2N
Phasors from fractional cycle:
High speed relaying
Non-recursive phasor calculations
t
1
1
2
2 = 1 + k
The non-recursive phasor rotates in the forwarddirection, one sample angle per sample.
Recursive phasor calculations
t
1
1The recursive phasor remains fixed if the inputwaveform is constant.
2= 1
2 = 1
Effect of noise on phasor calculations
Harmonics eliminated correctly if Nyquistcriterion is satisfied.
Non-harmonic components
Random Noise
True Phasor
Circle ofuncertainty
Size
of c
ircle
of
unce
rtai
nty
Measurement data window
Motivation for synchronization
By synchronizing the sampling processes fordifferent signals - which may be hundreds of milesapart, it is possible to put their phasors on the samephasor diagram.
Substation A Substation B
At different locations
Sources for Synchronization
Pulses Radio GOES GPS
Anti-aliasingfilters
16-bitA/D conv
GPSreceiver
Phase-lockedoscillator
AnalogInputs
Phasormicro-processor
Modems
A phasor measurement unit
Except for synchronization, the hardware is the sameas that of a digital fault recorder or a digital relay.
Sampling process, Fourier filter for phasors
sin and cosfunctions
t
Input signal
xnxn-1.
.x1
Dat
asa
mpl
es
cosi
nes
sine
s
Phasor X = -- xk(cosk - j sink)2N
Sampling clock based on nominal frequency
Fixed clocks, DFT at off-nominal frequency
Consider frequency excursions of 5 Hz The definition of phasor is independent of frequency
t=0
off-nominal sineoff-nominal cosine
nominal sine nominal cosine
x
off-nominal signal
Phasor:
X = (Xm /2) jXm
tDat
asa
mpl
es
cosi
nes
sine
s
X = -- xk(cosk - j sink)2N
Sampling clock based on nominal frequency
Input signal at off-nominal frequency:
^
Fixed clocks, DFT at off-nominal frequency Using the normal phasor estimation formula withxr being the first sample, the estimated phasor is:
Xr = PX jr()t + QX* jr(+)t^where t is the sampling interval, is the actualsignal frequency, and 0 is the nominal frequency.P and Q are independent of r, and are given below:
N(0)t2P =
sin
(0)tN sin2
N(0)t2 j(N-1)
N(+0)t2Q =
sin
(+0)tN sin2
N(0)t2 -j(N-1)
Fixed clocks, DFT at off-nominal frequency
At off-nominal frequency constant input, the phasor estimate is no longer constant, butdepends upon sample number r.
The principal effect is summarized in the Pterm. It shows that the estimated phasorturns at the difference frequency.
The Q term is a minor effect, and has arotation at the sum frequency.
For normal frequency excursions, P is almostequal to 1, and Q is almost equal to 0.
Fixed clocks, DFT at off-nominal frequency
For small deviations in frequency, P is almost 1 and Q is almost 0.
-5 -4 -2 0 2 4 50.988
0.992
0.996
1
Frequency deviation
Mag
nitu
de
0
5
10
15
Phas
e Sh
ift
degr
ees
(dot
ted)
-5
-10
-15
The function P
Fixed clocks, DFT at off-nominal frequency
For small deviations in frequency, P is almost 1 and Q is almost 0.
-0.05-0.04
-0.02
0.02
0.040.05
Frequency deviation
Mag
nitu
de
Phas
e Sh
ift
degr
ees
(dot
ted)
-5 -4 -2 0 2 4 50
5
10
15
20
25
30
0.00
The function Q
Fixed clocks, DFT at off-nominal frequency
A graphical representation of Xr:^
(0)
(+0)
PX
Xr^
QX*
Errors havebeen exaggeratedfor illustration.
In reality, Q is verysmall.
Fixed clocks, DFT at off-nominal frequency
If a cycle by cycle phasor is estimated at off-nominalfrequency, the magnitude and angle will show a ripple at (+0), and the average angle will show a constant slope corresponding to (0)
Phasor index r
Mag
nitu
de
Ang
le
Ripples are at (+0)
Slope of angle is (0)
Fixed clocks, Symmetrical Components atoff-nominal frequency
If the off-nominal frequency input is unbalanced, andhas symmetrical components of X0, X1, and X2, theestimated symmetrical components are given by
= P jr()t + Q jr(+)tXr0
Xr1
Xr2
^
^
^
X0
X1
X2
X0
X2
X1
*
*
*
Note that positive sequence creates a ripple in thenegative sequence estimate, and vice versa. The zerosequence is not affected by the other components.
Also, more importantly, if the input has no negativesequence then the positive sequence estimate iswithout the corrupting ripple.
Balanced 3-phase voltages at
Positive sequencevoltage at
Fixed clocks, Symmetrical Components atoff-nominal frequency
The ripple components of the three phase voltagesare equal and 120 apart, and thus cancel in the positive sequence estimate.
Frequency De
viation Df
Per unit negative sequence
Com
pone
nt a
t (w
+w0)
Error in positive sequence estimate as a function of per unit negative sequence and frequency deviation.
Summary of fixed clock DFT estimation of phasors:
For small frequency deviations, a single phaseinput with constant magnitude and phase will lead to an estimate having minor error terms.
The principal effect is the rotation of the phasorestimate at difference frequency (0), and a small ripple component at the sum frequency (+0).
A pure positive sequence input at off-nominalfrequency produces a pure positive sequenceestimate without the ripple. The positive sequenceestimate rotates at the difference frequency.
APPLICATIONS FOR MONITORING, PROTECTIONAND CONTROL
Frequency measurement with phasors
3-phase voltagesat
d/dt
Positive sequencevoltage at
0
timefreq
uenc
y
Present practice
ControlCenter
Measurementsare primarilyP, Q, |E| = [z]
State is the vectorof positive sequencevoltages at all network buses [E]
Measurementsare scannedand are NOTsimultaneous
Phasor measurement based state estimation offersmany advantages as will be seen later.
Power system state estimation
ControlCenter
Estimation with phasors Since the currents andvoltages arelinearly relatedto the state vector,The estimatorequations arelinear, and noiterations arerequired.
[Z] = [A] [E] , and once again the weighted leastsquare solution is obtained with a constantgain matrix.
Power system state estimation
Incomplete observability estimators:
One of the disadvantages of traditional state estimatorsis that at the very minimum complete tree of the network must be monitored in order to obtain a state estimate.The phasor based estimators have the advantage thateach measurement can stand on its own, and a relativelysmall number of measurements can be used directlyif the application requirements could be met.
Monitoringor controlsite
For example considerthe problem of controllingoscillations between twosystems separated bygreat distance.
In this case, only twomeasurements would be sufficient to provide a usefulfeed-back signal.
Incomplete observability estimators:
How many PMUs must be installed?
For complete observability, about 1/3 the numberof buses (along with the currents in all the connectedlines) in the system need to be monitored.
PMU Indirect
Incomplete observability estimators:
PMU placement for incomplete observability andinterpolation of unobserved states:
PMU
Indirectlyobserved
Unobserved
The unobserved set can be approximated by interpolation from the observed set
[Eun-observed ] = [B][Eneighbors]
State estimation with phasor measurements:
Summary:
Linear estimator True simultaneous measurements Dynamic monitoring possible
Complete observability requires PMUs at 1/3buses
Incomplete observability possible Few measurements become useful for control
ADVANCED CONTROL FUNCTIONSPresent system: model based controls
Controller
Measurements ControlledDevice
Control with feed-back
Phasor based: Feedback based controlADVANCED CONTROL FUNCTIONS
ControlledDevice
ControllerMeasurements
Control with feed-back
Example of control with phasor feed-back
System A System B
Power demand Controller
A- B
time
Performance withconstant power control law
Desired performance
Example 1: HVDC Controller
1640 MW
820 MW
(200+j20) MVA
590 MVAR
680 MVAR3 phase faultcleared in 3 cycles
G1
G2
Example 1: HVDC Controller
Control law 1: Constant current, constant voltage on HVDCControl law 2: Optimal controller
1
2
t (seconds) 5
DefinitionAdaptive protection is a protectionphilosophy which permits and seeksto make adjustments in various protection functions automatically in order to make them moreattuned to prevailing power systemconditions.
Adaptive Relaying
Adaptive out-of-step relaying
Conventionalout-of-steprelaying
stable
unstable
Adaptive out-of-step relaying
pmupmu
P
tob
serv
e
predict
Controlled Security & Dependability
A
r
b
i
t
r
a
t
i
o
n
L
o
g
i
c
System State
And
Vote
ProtectionNo
ProtectionNo
ProtectionNo
1
2
3
Or
T
o
C
i
r
c
u
i
t
B
r
e
a
k
e
r
s
Adaptive Relaying
FUTURE PROSPECTS
Applications a very active area of investigation
Intense industry interest in installations of PMUs New revised standard a step forward System post-mortem analysis the first application State estimation is an ideal application Control and adaptive relaying applications will follow