Identifying System-Wide Early Warning Signs of Instability in Stochastic Power Systems

0.27 MB·

by  Samuel Chevalier

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Identifying System-Wide Early Warning Signs of
Instability in Stochastic Power Systems
Samuel C. Chevalier and Paul D. H. Hines
College of Engineering and Mathematical Sciences
University of Vermont, Burlington, VT
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1 Abstract—Prior research has shown that spectral decompo- term voltage stability, as defined by a joint IEEE PES/CIGRE
0 sition of the reduced power flow Jacobian (RPFJ) can yield task force [3].
2 participationfactorsthatdescribetheextenttowhichparticular There is increasing evidence that as a dynamical system
l bsyussteesmc.oRnetsreibaurctehthoaspaalrstoicsuhloawrnsptehcattrablotchomvaproiannecnetsanodf aauptoocwoerr- approaches a bifurcation, early warning signs (EWSs) of the
u
relation of time series voltage data tend to increase as a power looming transition appear in the statistical properties of time
J
system with stochastically fluctuating loads approaches certain series data from that system. This fact has been evidenced
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critical transitions. This paper presents evidence suggesting that in many complex systems, including ecological networks,
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a system’s participation factors predict the relative bus voltage
financial markets, the human brain, and power systems [4],
variance values for all nodes in a system. As a result, these
] [5].Researchershaveevenfoundthathumandepressiononset
h participation factors can be used to filter, weight, and combine
p realtimePMUdatafromvariouslocationsdispersedthroughout can be predicted by these same statistical properties [6]. In
- a power network in order to develop coherent measures of the statistical physics literature this phenomenon is known as
c global voltage stability. This paper first describes the method Critical Slowing Down (CSD) [7]. When stressed, systems
o of computing the participation factors. Next, two potential uses
experiencing CSD require longer periods to recover from
s of the participation factors are given: (1) predicting the relative
. stochastic perturbations. Two of the most well-documented
s bus voltage variance magnitudes, and (2) locating generators
c at which the autocorrelation of voltage measurements clearly signs of CSD are increased variance and autocorrelation [5].
i indicateproximitytocriticaltransitions.Themethodsaretested Real power systems, particularly as renewable energy pro-
s
y using both analytical and numerical results from a dynamic ductionincreases,areconstantlysubjecttostochasticallyfluc-
h model of a 2383-bus test case. tuating supply and demand. The presence of stochastic power
p
Index Terms—Power system stability, phasor measurement injections has motivated research to quantify the presence of
[
units,timeseriesanalysis,autocorrelation,criticalslowingdown, CSD in bulk power networks, particularly as early warning
3 spectral analysis. signs of bifurcations such as voltage collapse (a type of
v Saddle-nodebifurcation[8])oroscillatoryinstability(atypeof
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6 I. INTRODUCTION Hopf bifurcation). Through simulations, reference [9] demon-
strated that both variance and autocorrelation in bus voltages
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OnaparticularlyhotdayinJulyof1987,thepowersystem
0 increase substantially as several power systems approached
infrastructure in Tokyo Japan saw a dramatic increase in de-
0 saddlenodebifurcation.Similarly,reference[10]computesan
1. mandasmillionsofairconditioningunitswereturnedonline. auto-correlation function for a power system model to gauge
This demand spike occurred very rapidly and caused system
0 collapse probability. Finally, variance and autocorrelation are
wide voltages to sag. Voltage collapse soon followed, leaving
6 measured in an unstable power system in [4] across many
1 almost3millionpeoplewithoutelectricalpower.Accordingto
state variables. These results indicate that variance of bus
: [1]and[2],inadequateoperationalplanningcoupledwithpoor
v voltages and autocorrelation of line currents show the most
situationalawarenesswereprimarycausesoftheblackout.Un-
i usefulsignalsofCSD,whereascurrentangles,voltageangles,
X fortunately,thisisnotanisolatedvoltagecollapseincident:in
generator rotor angles, and generator speeds did not generally
r order to optimize limited infrastructure, many power systems
a yield sufficiently strong signs of CSD to provide actionable
arefrequentlyoperatedclosetocritical(orbifurcation)points,
early warning of bifurcation.
leaving them vulnerable to the devastating effects of voltage
Although CSD does consistently appear in these systems
collapse. This paper seeks to improve situational awareness
before bifurcations, not all variables in a complex system
by providing a method that can use information gleaned from
exhibit CSD sufficiently early to be useful EWSs [11]. For
network models to combine streams of synchrophasor data
instance, reference [4] destabilized a simulated power system
in ways that provide useful information about a particular
by over stressing all load buses. Signals were then collected
system’s proximity to stability limits. Ultimately, the goal
from many nodes in this system, and certain nodes did not
of this work is to provide operators with enhanced tools
conclusively show measurable early CSD warning signs. In
for gauging a system’s long-term stability. In this paper, we
order to mitigate this problem, we propose a method that
particularly focus on identifying early warning signs of long
combines measurements from a variety of locations using
spectral analysis of the power flow Jacobian, as introduced in
ThisworkwassupportedbytheUSDOE,award#de-oe0000447,andby
theUSNSF,award#ECCS-1254549. reference [12]. By understanding which variables are the best
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indicators of long-term voltage stability, we aim to develop is defined:
measuresthatareusefulforassessingthestabilityofanentire ∆Q=(cid:2)J −J J−1J (cid:3)∆V=[J ]∆V (2)
system. QV Qθ Pθ PV R
It is well known that voltage collapse can cause non-
Assuming that Newton-Raphson converges to a power flow
convergence in AC power flow solvers[8]. Additionally, ref-
solution for the system being studied, the matrix J is
R
erence [13] shows that only under very strict conditions will
non-singular and can be written as the product of its right
the load flow Jacobian show unambiguous signs of dynamic
eigenvector matrix R, its left eigenvector matrix L, and its
instability.Sinceourexperimentsdonotmeettheseconditions,
diagonal eigenvalue matrix Λ, such that:
spectral analysis of the load flow Jacobian, as proposed in
this paper, does not provide direct warning signs of dynamic JR =RΛL (3)
instability. However, since spectral analysis is used in our
Theleftandrighteigenvectorscanthenbeorthonormalized
approach merely as a means of deciding how to combine
such that, for the right eigenvector r (column vector) and the
PMU data from diverse locations, our method provides a i
left eigenvector l (row vector), the Konecker delta function
complement to conventional approaches that focus only on j
defines their relationship:
spectral analysis of the load flow Jacobian. It is also well
knownthatthesaddle-nodebifurcationisthemaximumupper (cid:40)
1 i=j
limit on system loadability. But this upper limit is seldom ljri =δj,i = (4)
0 i(cid:54)=j
reached, because system dynamics generally become unstable
well before the saddle-node bifurcation occurs [13]. Indeed, We begin our method by decomposing J using a simple
R
reference [14] shows that a Hopf bifurcation will frequently similarity transform. The transform is substituted into (2):
precede a saddle-node bifurcation. Therefore, monitoring for
∆Q=
voltage instability is only one important aspect of overall
 λ 0 ··· 0 
power system stability;there is a strong link between the 1 .  l1 
 .  l
oncocduerrbeinfucrecoaftioanH.opfbifurcationandtheproximityofasaddle- (cid:2) r1 r2 ··· rn (cid:3) 0... λ2 ... 0.  ...2 ∆V
By combining spectral analysis and CSD theory, this paper l
0 ··· 0 λ n
n
shows that the RPFJ contains valuable information about
 r r ··· r  λ (l ·∆V) 
1,1 2,1 n,1 1 1
voltage stability. We show that participation factors resulting r r r λ (l ·∆V)
 1,2 2,2 n,2  2 2 
from a spectral analysis of this matrix can be used to weight =  ... ... ...  ...  (5)
and filter real-time PMU data, thus suggesting a method for
r r ··· r λ (l ·∆V)
combiningthedataintolow-dimensionalmetricsoflong-term 1,n 2,n n,n n n
voltage stability. This paper does not seek to define these The effect of this transform can be made more clear by
metrics in detail; instead, we present a tool (spectral analysis investigating how changing voltage affects the change in
oftheRPFJ)thatprovidesafoundationforthedevelopmentof injected reactive power of a single bus ((cid:52)Q1 for example).
such metrics in future work. Section II of this paper outlines This is shown in (6).
the mathematical methods and motivation for forming and
(cid:52)Q = r λ (l ·∆V)+r λ (l ·∆V)+ (6)
1 1,1 1 1 2,1 2 2
decomposingtheRPFJ.SectionIIIpresentsthe2383testcase
···+r λ (l ·∆V)
and shows the potential usefulness of the participation factors n,1 n n
Finally, our conclusions are presented in Section IV.
In order to determine how the reactive power at bus i is
affected by the voltage at only bus i, we simply hold all other
II. SPECTRALANALYSISOFTHEPOWERFLOWJACOBIAN voltage magnitudes constant. If we choose i=1, the voltage
This section presents a method for using spectral decompo- differentialvectorbecomes∆V=[ (cid:52)V 0 ··· 0 ].The
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sition of the RPFJ to identify and weight variables that will reactive power differential equation changes accordingly.
most clearly show evidence of CSD. Further information on
(cid:52)Q =(λ r l +λ r l +···+λ r l )(cid:52)V
this spectral decomposition approach can be found in [12]. 1 1 1,1 1,1 2 2,1 2,1 n n,1 n,1 1
(7)
The standard power flow Jacobian matrix, which is a
linearizationofthesteadystatepowerflowequations,isgiven At this point, we can define and incorporate the participation
by (1). factors. The indices in the following equation refer to the jth
row and the ith column of the right eigenvector matrix R and
(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
∆P = JPθ JPV ∆θ (1) the ith row and the jth column of the left eigenvector matrix
∆Q J J ∆V
Qθ QV L.
ρ =R L (8)
i,j j,i i,j
In order to perform V-Q sensitivity analysis (an important
aspect of voltage stability analysis), we assume that the Therefore, ρ defines how the jth state is affected by the
i,j
incrementalchangeinrealpower∆Pisequalto0.Inthisway, ith eigenvalue.Clearly,individualreactivepowerstatescanbe
we can study how incremental changes in injected reactive expressedasasuperpositionofeigenvaluesofvaryingdegrees
poweraffectsystemvoltages.Setting∆P=0andrearranging of participation. If we compute the reactive power changes at
terms to remove ∆θ, the expression for the reduced Jacobian each bus based on the voltage changes at each corresponding
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local bus, we obtain the following set of equations. which nodes will show the strongest EWSs [4]. Therefore,
thenovelapproachoutlinedinthispaperusesresultsfromthe
(cid:52)Q =(λ ρ +λ ρ +···+λ ρ )(cid:52)V
1 1 1,1 2 2,1 n n,1 1 static decomposition above to weight and interpret incoming
(cid:52)Q =(λ ρ +λ ρ +···+λ ρ )(cid:52)V dynamic data.
2 1 1,2 2 2,2 n n,2 2
.
.
. III. EXPERIMENTALRESULTS:USINGPARTICIPATION
FACTORSTOCOMBINEPMUDATA
(cid:52)Q =(λ ρ +λ ρ +···+λ ρ )(cid:52)V
n 1 1,n 2 2,n n n,n n
A. Polish Test Case Overview
In these equations, a reactive power state is expressed
as a superposition of eigenvalues. Conversely, we can also Inordertotestourmethods,weusedatafromthe2383-bus
express each eigenvalue as a superposition of different state dynamic Polish test system. This network contains 327 four-
contributions. The reason why such an expression is useful is variable synchronous generators. Each generator is equipped
shownthrough(12).RecognizingthatR=L−1,thefollowing
with a three-variable turbine governor model for frequency
manipulations may be made.
control and a four-variable exciter model (AVR) for volt-
∆Q= age regulation. There are 322 shunt loads (all connected to
 λ 0 ··· 0  generator buses) and 1503 active and reactive loads spread
1  l 
 ..  l1 throughout the system. In order to push the system towards
(cid:2) r1 r2 ··· rn (cid:3) 0... λ2 ... 0.  ...2 ∆V valolltlaogaedsco(ellxacpespet,fworetehmospeloaytetadchaedsimtopgleenuenriaftoorrmbuloseasd)i.nTghoisf
l
0 ··· 0 λn n methodisjustifiedin[3].HalfofthePQbusloadsaremodeled
asvoltagecontrolledloads,whiletheotherhalfaremodeledas
 λ 0 ··· 0 
 l  1  l  frequency controlled loads. Parameters controlling the voltage
1 . 1
 l...2 ∆Q= 0... λ2 ... 0..  l...2 ∆V cinon[1tr5o]l,lewdhlioleadpsaraarmeetmerosdceolendtroalfltienrgtthhee NfreoqrduiecncTyesctonStyrostlelemd
l l loads are modeled after the 39 bus test system described in
n 0 ··· 0 λ n
n [4].
Now, we can isolate a single eigenvalue (λ1, for example). As in [4], we model this larger network using a set of
stochastically forced differential algebraic equations, which
l ∆Q=λ l ∆V (9)
1 1 1
can be written as:
x˙ =f(x,y) (13)
l (cid:52)Q +l (cid:52)Q +···+l (cid:52)Q = (10)
1,1 1 1,2 2 1,n n
λ (l (cid:52)V +l (cid:52)V +···+l (cid:52)V ) 0=g(x,y,u) (14)
1 1,1 1 1,2 2 1,n n
Clearly, the relationship between (cid:52)Q and (cid:52)V for the jth where f and g represent the differential and algebraic equa-
isolated state (holding all else constant) is given by the tions governing the system, x and y are the differential
following expression. and algebraic variables of these equations, and u represents
stochastic power (load or supply) fluctuations. u follows a
q (cid:52)V (cid:52)V 1
1,j j = j = (11) mean-reverting Ornstein-Uhlenbeck process:
q (cid:52)Q (cid:52)Q λ
1,j j j j
u˙ =−Eu+ξ (15)
This is true for all states of a given eigenvalue. Therefore, the
spectral componentthat will havethe largest voltagevariation where E is a diagonal matrix whose diagonal entries equal
for a given reactive power change will have the smallest the inverse correlation times t−1 of load fluctuations and
corr
eigenvalue. For this reason, the participation factors of this ξ is a vector of zero-mean independent Gaussian random
eigenvaluewillbeofgreatinteresttostudy.Thejtheigenvalue variables. A further description of our noise model can be
canbewrittenasasummationofnuniquestates.Inthisway, found in Sec. II A of [4]. Also given in [4] is a method for
(12) shows how each state participates in the jth eigenvalue analyticallycomputingthecovarianceandcorrelationmatrices
of a system. for all state and algebraic variables. We used this method to
pre-compute the variance of voltages in the 2383-bus Polish
λ =λ ρ +λ ρ +···+λ ρ (12)
j j j,1 j j,2 j j,n system. After thorough testing, we found the analytically-
There are many different ways to use the eigenvalues and calculated covariance and correlation matrices to be just as
eigenvectorsofJ .Forinstance,[12]suggestusingthesmall- accurate on the large Polish system as they were on the small
R
est eigenvalue of J to gauge proximity to bifurcation. Such 39 bus system. Thus, the data presented in the following
R
stabilityanalysis,though,isbasedsolelyonthedecomposition two sections use the analytically calculated results rather than
of a model based static matrix and is highly limited in nature, averaged dynamic simulation results.
as outlined by Pal in the discussion section of [12]. Instead, In order to push the system towards a critical transition
we propose that J can be leveraged as tool to combine (voltage collapse), we increase all loads and generator set
R
and thus interpret streams of PMU data. Detecting Critical pointsbyaconstantloadingfactorb,whichrangesfromb=1
Slowing Down in time series data is a purely data driven up to b = 1.92. We empirically found that voltage collapse
stability assessment, but it can be difficult to understand occurs when the load factor increases past b=1.923.
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0.2 0.18
ors a
ct0.15 0.16
a
F
n
articipatio0.00.51 Factors00..1124 Bus 466
P 0 on 0.1
200 250 300 350 400 450 500 ati
p
2x 10−6 artici0.08 Bus 240
b P0.06
2σ 1.5 us
ge B0.04
a
olt 1 Bus 218
V 0.02
us 0.5
B
0
0 1 1.2 1.4 1.6 1.8 2
200 250 300 350 400 450 500 Load Level
Bus Number
Figure 2. Participation factors for three different buses at different load
Figure 1. Participation factors and voltage variance values for buses levels.Asthesystemisincreasinglyloaded(rightuptobifurcation),bus466
200 through 500 from the loaded 2383 bus system. Bus voltage variances (themostunstablebus)seesasharpincreaseinparticipationtotheinstability.
(below)correlatealmostperfectlywiththeparticipationfactorsofthesmallest Bus 240 (the 5th most unstable bus) sees a very slight increase, while bus
eigenvalueoftheRPFJ(above). 218(the10thmostunstablebus)beginstoseeadecrease.
The concept of a limit-induced bifurcation is an important uniform loading condition). This is equivalent to saying that
topic discussed in [16]. Power system limits, such as reactive the spectral components do not change significantly. This is a
power generation limits, are an important aspect of stability useful result, since state-estimator derived power flow models
analysis. However, in order to focus our analysis on voltage areonlytypicallycomputedperiodicallyduringpowersystems
collapse without the possibility of additional bifurcations due operations.
to limits, we increased limits in our test case so that the As indicted previously, participation factors of the most
system can run up to b=1.92 without hitting a limit-induced unstable nodes provide a very clear indication of the relative
bifurcation.Thissimplificationallowsustofocusourstudyon bus voltage variance strengths. Therefore, as the system is
the effects of pure voltage collapse. Future work will extend increasingly loaded, the most unstable nodes will begin to
our method to the case of other types of bifurcations. have larger and larger participation factors as their relative
variance strengths grow relative to other, more stable nodes.
Fig. 2 shows an example of this for the 2383 bus system.
B. Evidence for Bus Voltage Variance Prediction
As the system is loaded, the relative strength of the most
As indicated by (11), the smallest eigenvalue of JR corre- unstable bus’ participation increases almost linearly, but when
sponds to the spectral component that will yield the largest the critical transition approaches, the participation begins to
voltage variation for a given variation in reactive power. climb more steeply.
When the participation factors corresponding to the smallest
eigenvalue are plotted, they are shown to directly predict the
C. Evidence for Locating Generators with Elevated Voltage
relativebusvoltagevariancestrengths.Fig.1showstwoplots.
Autocorrelation
The top plot corresponds to the participation factors for buses
200 through 500, and the bottom plot shows the true bus CSD theory predicts that signals from a system approach-
voltage variance, derived analytically, for buses 200 through ing a critical transition will show increasing levels of auto-
500. The remaining system buses are left out for the sake correlation, R(∆t). This can be due to the system’s reduced
of clarity. Despite the fact that the participation factors are ability to respond to high frequency fluctuations [18], but the
completely blind to the dynamics of the system, they are still system also begins to return to an equilibrium state more
quite successful at predicting the relative variance strengths. slowly after perturbations [17]. In a power system, system-
wide increases inautocorrelation are typically indicators of
As shown in reference [17], increasing voltage variance increasinglyunstablegeneratordynamics.Thesedynamicsare
is due to buses that are operating closer to the limit along driven by the load variations, since this is where the noise is
the PV curve. Therefore, participation factors of the smallest being injected.
eigenvaluealsoidentifythenodevoltageswhich,asthesystem Fig. 1, clearly shows that a small number of buses in our
is overloaded, begin to diverge from their nominal values. test case have particularly high voltage variances measure-
These tend to be the nodes that are primarily responsible for ments. Looking at the topology of this network, we find that
non-convergence in the power flow equations. Interestingly, these nodes are in fact separated by only a small number
as PQ buses in the system are increasingly loaded, the recal- of transmission lines suggesting that these buses represent a
culated participation factors do not change drastically (for a weakloadpocket.Theparticipationfactorsarethususefulfor
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stability metrics. The detailed development of these metrics is
n 0.9 a topic for future research.
o
ati
el
orr REFERENCES
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AUTHORBIOGRAPHIES
IV. CONCLUSIONS
Paul D. H. Hines (SM‘14) received the Ph.D. in Engineering and
This paper presents evidence that participation factors from Public Policy from Carnegie Mellon U. in 2007 and M.S. (2001)
a spectral decomposition of the reduced power flow Jacobian and B.S. (1997) degrees in Electrical Engineering from the U. of
Washington and Seattle Pacific U., respectively. He is currently
can be used to design methods for combining sychrophasor
Associate Professor in Electrical Engineering at the U. of Vermont.
measurementstoproducesystem-wideindicatorsofinstability
in power systems. Our approach uses spectral information
SamuelC.Chevalier(S‘13)receivedaB.S.inElectricalEngineering
from the power flow Jacobian, which can be updated every
fromtheUniversityofVermontin2015.Heiscurrentlypursuingan
fewminutesthroughtheSCADAnetworkincombinationwith
M.S. degree in Electrical Engineering from UVM, and his research
highsample-rate voltagemagnitudemeasurements, whichcan
interests include stochastic power system stability and Smart Grid.
be collected from synchronized phasor measurement systems
deployed throughout the system. The results suggest that that
a combination of a power flow model and streaming PMU
data analysis can be used to be used to develop system wide