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Understanding early indicators of critical transitionsin power systems from autocorrelation functionsGoodarz Ghanavati, Student Member, IEEE, Paul D. H. Hines, Member, IEEE, Taras I. Lakoba, Eduardo
Cotilla-Sanchez, Member, IEEE
Abstract—Many dynamical systems, including power systems,recover from perturbations more slowly as they approach criticaltransitions—a phenomenon known as critical slowing down. Ifthe system is stochastically forced, autocorrelation and variancein time-series data from the system often increase before thetransition, potentially providing an early warning of comingdanger. In some cases, these statistical patterns are sufficientlystrong, and occur sufficiently far from the transition, that theycan be used to predict the distance between the current operatingstate and the critical point. In other cases CSD comes too late tobe a good indicator. In order to better understand the extent towhich CSD can be used as an indicator of proximity to bifurcationin power systems, this paper derives autocorrelation functions forthree small power system models, using the stochastic differentialalgebraic equations (SDAE) associated with each. The analyticalresults, along with numerical results from a larger system,show that, although CSD does occur in power systems, itssigns sometimes appear only when the system is very close totransition. On the other hand, the variance in voltage magnitudesconsistently shows up as a good early warning of voltage collapse.Finally, analytical results illustrate the importance of nonlinearityto the occurrence of CSD.
Index Terms—Autocorrelation function, bifurcation, criticalslowing down, phasor measurement units, power system stability,stochastic differential equations.
I. INTRODUCTION
There is increasing evidence that time-series data taken fromstochastically forced dynamical systems show statistical pat-terns that can be useful in predicting the proximity of a systemto critical transitions [1], [2]. Collectively this phenomenon isknown as Critical Slowing Down, and is most easily observedby testing for autocorrelation and variance in time-series data.Increases in autocorrelation and variance have been shown togive early warning of critical transitions in climate models[3], ecosystems [4], the human brain [5] and electric powersystems [6], [7], [8].
Scheffer et al. [1] provide some explanation for why in-creasing variance and autocorrelation can indicate proximityto a critical transition. They illustrate that increasing autocor-relation results from the system returning to equilibrium moreslowly after perturbations, and that increased variance results
This work was supported in part by the US Dept. of Energy, award#de-oe0000447, and in part by the US National Science Foundation, award#ECCS-1254549 .
G. Ghanavati, P. Hines and T.Lakoba are with the College of Engineeringand Mathematical Sciences, University of Vermont, Burlington, VT (e-mail:[email protected]; [email protected]; [email protected]).
E. Cotilla-Sanchez is with the School of Electrical Engineeringand Computer Science at Oregon State University. (e-mail: [email protected]).
from state variables spending more time further away fromequilibrium. Some further explanation of CSD in stochasticsystems can be found by looking at the theory of fast-slowsystems [9]. In many stochastic systems with critical transi-tions there are two time scales; slow trends gradually movethe “equilibrium” operating state toward, or away from pointsof instability, and random perturbations cause fast changesin the state variables. In power systems, loads have slowpredictable trends, such as load ramps in the morning hours,and fast stochastic ones, such as random load switching orrapid changes in renewable generation. Reference [9] usesthe mathematical theory of the stochastic fast-slow dynamicalsystems and the Fokker–Planck equation to explain the use ofautocorrelation and variance as indicators of CSD.
While CSD is a general property of critical transitions [10],its signs do not always appear early enough to be usefulas an early warning, and do not universally appear in allvariables [10], [11]. References [10] and [11] both show, usingecological models, that the signs of CSD appear only in a fewof the variables, or even not at all.
Several types of critical transitions in deterministic powersystem models have been explained using bifurcation theory.Reference [12] explains voltage collapse as a saddle-nodebifurcation. Reference [13] describes voltage instability causedby the violation of equipment limits using limit-induced bi-furcation theory. Some types of oscillatory instability can beexplained as a Hopf bifurcation [14], [15]. Reference [16]describes an optimization method that can find saddle-nodeor limit-induced bifurcation points. Reference [17] shows thatboth Hopf and saddle-node bifurcations can be identified ina multi-machine power system, and that their locations canbe affected by a power system stabilizer. In [18], authorscomputed the singular points of the differential and algebraicequations that model the power system.
Substantial research has focused on estimating the proximityof a power system to a particular critical transition. Refer-ences [13],[19]–[21] describe methods to measure the distancebetween an operating state and voltage collapse with respectto slow-moving state variables, such as load. Although thesemethods provide valuable information about system stability,they are based on the assumption that the current networkmodel is accurate. However, all power system models includeerror, both in state variable estimates and network parameters,particularly for areas of the network that are outside of anoperator’s immediate control.
An alternate approach to estimating proximity to bifurcationis to study the response of a system to stochastic forcing, such
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as fluctuations in load, or variable production from renewableenergy sources. To this end, a growing number of papersstudy power system stability using stochastic models [22]–[26]. Reference [22] models power systems using StochasticDifferential Equations (SDEs) in order to develop a measure ofvoltage security. In [25], numerical methods are used to assesstransient stability in power systems, given fluctuating loads andrandom faults. Reference [26] uses the Fokker-Planck equationto calculate the probability density function (PDF) for statevariables in a single machine infinite bus system (SMIB), anduses the time evolution of this PDF to show how random loadfluctuations affect system stability.
The results above clearly show that power system stabilityis affected by stochastic forcing. However, they provide littleinformation about the extent to which CSD can be usedas an early warning of critical transitions given fluctuatingmeasurement data. Given the increasing availability of high-sample-rate synchronized phasor measurement unit (PMU)data, and the fact that insufficient situational awareness hasbeen identified as a critical contributor to recent large powersystem failures (e.g., [27], [28]) there is a need to betterunderstand how statistical phenomena, such as CSD, mightbe used to design good indicators of stress in power systems.
Results from the literature on CSD suggest that autocorre-lation and variance in time-series data increase before criticaltransitions. Empirical evidence for increasing autocorrelationand variance is provided for an SMIB and a 9-bus testcase in [6]. Reference [29] shows that voltage variance atthe end of a distribution feeder increases as it approachesvoltage collapse. However, the results do not provide insightinto autocorrelation. To our knowledge, only [7], [8] deriveapproximate analytical autocorrelation functions (from whicheither autocorrelation or variance can be found) for statevariables in a power system model, which is applied to theNew England 39 bus test case. However, the autocorrelationfunction in [7], [8] is limited to the operating regime veryclose to the threshold of system instability. Furthermore, thereis, to our knowledge, no existing research regarding whichvariables show the signs of CSD most clearly in power system,and thus which variables are better indicators of proximity tocritical transitions. In [30], the authors derived the generalautocorrelation function for the stochastic SMIB system. Thispaper extends the SMIB results in [30], and studies twoadditional power system models using the same analyticalapproach. Also, this paper includes new numerical simulationresults for two multi-machine systems, which illustrate insightsgained from the analytical work.
Motivated by the need to better understand CSD in powersystems, the goal of this paper is to describe and explainchanges in the autocorrelation and variance of state variables inseveral power system models, as they approach bifurcation. Tothis end, we derive autocorrelation functions of state variablesfor three small models. We use the results to show that CSDdoes occur in power systems, explain why it occurs, anddescribe conditions under which autocorrelation and variancesignal proximity to critical transitions. The remainder ofthis paper is organized as follows. Section II describes thegeneral mathematical model and the method used to derive
autocorrelation functions in this paper. Analytical solutionsand illustrative numerical results for three small power sys-tems are presented in Secs. III, IV and V. In Sec. VI, theresults of numerical simulations on two multi-machine powersystem models including the New England 39 bus test caseare presented. Finally, Sec. VII summarizes the results andcontributions of this paper.
II. SOLUTION METHOD FOR AUTOCORRELATIONFUNCTIONS
In this section, we present the general form of the StochasticDifferential Algebraic Equations (SDAEs) used to model thethree systems studied in this paper. Then, the solution of theSDAEs and the expressions for autocorrelations and variancesof both algebraic and differential variables of the systems arepresented. Finally, the method used for simulating the SDAEsnumerically is described.A. The Model
All three models studied analytically in this paper includea single second-order synchronous generator. These systemscan be described by the following SDAEs:
δ + 2γδ + F1
(δ, y, η
)= 0 (1)
F2
(δ, y, η
)= 0 (2)
where δ is angle of the synchronous generator’s rotor relativeto a synchronously rotating reference axis, y is the vector ofalgebraic variables, γ is the damping coefficient, F1, F2 forma set of nonlinear algebraic equations of the systems, and η isa Gaussian random variable. η has the following properties:
E [η (t)] = 0 (3)E [η (t) η (s)] = σ2
η · δI (t− s) (4)
where t, s are two arbitrary times, σ2η is the intensity of noise,
and δI represents the unit impulse (delta) function (whichshould not be confused with the rotor angle δ). There area variety of sources of noise, such as random load switchingor variable renewable generation, in power systems. To ourknowledge, no existing studies have quantified the correlationtime of noise in power systems. Thus, in this paper, we assumethat the correlation time of noise is negligible relative to theresponse-time of the system, which means that E[η(t)η(s)] =0 for all s significantly greater than t. It is important to notethat the variance of η is infinite according to (4), because thedelta function is infinite at t = s, which means that particularcare is needed when simulating (1) and (2) numerically (seeSec. II-C).
In order to solve (1) and (2) analytically, we linearized F1
and F2 around the stable equilibrium point using first-orderTaylor expansion. Then (1) and (2) were combined into asingle damped harmonic oscillator equation with stochasticforcing:
∆δ + 2γ∆δ + ω20∆δ = −fη (5)
where ω0 is the undamped angular frequency of the oscillator,f is a constant, and ∆δ = δ− δ0 is the deviation of the rotorangle from its equilibrium value. Both ω0 and f change with
3
the system’s equilibrium operating state. Equation (5) can bewritten as a multivariate Ornstein–Uhlenbeck process [31]:
z (t) = Az (t) +B
[0
η (t)
](6)
where z =[
∆δ ∆δ]T
is the vector of differential vari-ables, ∆δ is the deviation of the generator speed from itsequilibrium value, and A and B are constant matrices asfollows:
A =
[0 1−ω2
0 −2γ
](7)
B =
[0 00 −f
](8)
Given (7), the eigenvalues of A are −γ±√γ2 − ω2
0 . At ω0 =0, one of the eigenvalues of matrix A becomes zero, and thesystem experiences a saddle-node bifurcation.
Equation (5) can be interpreted in two different ways:using either Ito SDE and Stratonovich SDEs. In the Itointerpretation [32], noise is considered to be uncorrelated.However, in the Stratonovich interpretation [33], which is amore natural choice physically, noise has finite, albeit verysmall, correlation time [31]. Ito calculus is often used indiscrete systems, such as finance, though a few papers haveapplied the Ito approach to power systems [22], [25]. On theother hand, the Stratonovich method is often used in continu-ous physical systems or systems with band-limited noise [34].The Stratonovich interpretation also allows the use of ordinarycalculus, which is not possible with the Ito interpretation.Because B is a constant matrix in this paper, the Ito andStratonovich interpretations result in the same solution [34].This paper follows the Stratonovich interpretation because itallows one to use ordinary calculus.
Following the method in [35], if γ < ω0 (which holdsuntil very close to the bifurcation in two of our systems), thesolution of (6) is as follows:
∆δ(t) = f ·ˆ t
−∞exp (γ (t′ − t)) η (t′) · (9)
sin (ω′(t′ − t))ω′
dt′
∆δ (t) = − f ·ˆ t
−∞exp (γ (t′ − t)) η (t′) · (10)
sin (ω′(t′ − t) + φ)ω0
ω′dt′
where t′ is the variable of integration, ω′ =√ω2
0 − γ2 isthe frequency of the underdamped harmonic oscillator, andφ = arctan(ω′/γ).
In the system considered in Sec. IV, ω0 in (5) is equalto zero for all system parameters, so the condition γ < ω0
does not hold. Therefore, the solution of (5) in that system isdifferent from (9), (10) as follows:
∆δ = −fˆ t
−∞exp (−2γ (t− t′)) η (t′) dt′ (11)
B. Autocorrelation and Variance of Differential Variables
Given that the eigenvalues of A have negative real partbefore the bifurcation (because γ > 0), one can calculate thestationary variances and autocorrelations of ∆δ and ∆δ using(3), (4), (9) and (10). The variances of the differential variablesare as follows:
σ2∆δ =
f2σ2η
4γω20
(12)
σ2∆δ
=f2σ2
η
4γ(13)
If γ < ω0, the normalized autocorrelation functions for ∆δand ∆δ are as follows:
E [∆δ (t) ∆δ (s)]
σ2∆δ
= exp (−γ∆t)ω0
ω′· (14)
sin (ω′∆t+ φ)
E[∆δ (t) ∆δ (s)
]σ2
∆δ
= exp (−γ∆t)−ω0
ω′· (15)
sin (ω′∆t− φ)
where ∆t = t− s.If ω0 = 0, the variance of ∆δ can be calculated from (13)
and the autocorrelation of ∆δ is as follows:
E[∆δ (t) ∆δ (s)
]σ2
∆δ
= exp (−2γ∆t) (16)
C. Autocorrelation and Variance of Algebraic Variables
In order to compute the autocorrelation functions of thealgebraic variables, we calculated the algebraic variables aslinear functions of the differential variable ∆δ and the noiseη, by linearizing F2 in (2):
∆yi (t) = Ci,1∆δ (t) + Ci,2η (17)
where yi is an algebraic variable, and Ci,1, Ci,2 are constantvalues. Then, the autocorrelation of ∆yi is as follows for t ≥s:
E [∆yi (t) ∆yi (s)] = C2i,1 · E [∆δ (t) ∆δ (s)] + (18)
Ci,1Ci,2·E [∆δ (t) η (s)] +
C2i,2 · E [η (t) η (s)]
In deriving (18), we used the fact that E [∆δ (s) η (t)] = 0since the system is causal. Equation (18) shows that, in orderto calculate the autocorrelation of ∆yi (t), it is necessaryto calculate E [∆δ (t) η (s)]. Using (9), E [∆δ (t) η (s)] is asfollows:
E [∆δ (t) η (s)] = − exp (−γ∆t) · fω′· (19)
sin (ω′∆t)σ2η
which indicates that cov (∆δ, η) = 0.In order to use (18) to compute the variance of ∆yi, we
need to carefully consider our model of noise in numericalcomputations. According to (4), the variance of η is infinite,because the delta function is infinite at t = s, which would
4
mean that the variance of ∆yi could be infinite. However, thenoise in numerical simulations must have a finite variance. Todetermine it, we rewrite (6) as follows:
dz (t) = Az (t) dt+BdW (t) (20)
where dW (t) = ηdt is the Wiener process. It is well-known that the variance of dW (t) is σ2
ηdt [31]. In numericalsimulations, dt = τint, where τint is the integration timestep. Thus, E
[dW 2
num
]= E
[(ηnumτint)
2]
= σ2ητint. Hence,
E[η2
num
]= σ2
η/τint. With this definition of noise, (17) meansthat the variance of ∆yi is:
σ2∆yi = C2
i,1σ2
∆δ + C2i,2
σ2η
τint(21)
where τint is the integration time step in numerical simulations.In order to match analytical results with numerical simulations,we divided the noise intensity by the integration step size inthe second term of the right-hand side of (21). Combining (12)and (21) results in the following:
σ2∆yi =
(C2i,1f2
4γω20
+C2i,2
τint
)σ2η (22)
Combining (12), (14), (18) , (19) and (22), we calculated thenormalized autocorrelation function of ∆yi:
E [∆yi (t) ∆yi (s)]
σ2∆yi
= exp (−γ∆t) sin (ω′∆t+ φ∆yi) ·
Ci,1fω0
√λ
ω′(C2i,1f2 + 4C2
i,2γω2
0
) (23)
where λ =
√Ci,1f
(Ci,1f − 8Ci,2γ
2)
+(4Ci,2ω0γ
)2, φ∆yi =
arctan(Ci,1fω
′/(Ci,1fγ−4Ci,2γω
20)).
D. Numerical Simulation
In order to calculate numerical results that can be comparedto the analytical ones, (1) and (2) were solved using atrapezoidal ordinary differential equation solver, with a fixedtime step of integration, τint. We chose the integration stepsize τint to be much shorter than the the smallest period ofoscillation T = 2π/ω′, between the periods for all bifurcationparameter values.
In order to determine numerical mean values in this paper,each set of SDEs was simulated 100 times. In each case theresulting averages were compared with analytical means.
III. SINGLE MACHINE INFINITE BUS SYSTEM
Analysis of small power system models can be very helpfulfor understanding the concepts of power system stability.The single machine infinite bus system has long been usedto understand the behavior of a relatively small generatorconnected to a larger system through a long transmission line.This SMIB system has been used, for example, to explorethe small signal stability of synchronous machines [36] andto evaluate control techniques to improve transient stability
and voltage regulation [37]. In the recent literature, there isincreasing interest in stochastic behavior of power systems, inpart due to the increasing integration of variable renewableenergy sources. A few of these papers use stochastic SMIBmodels. In [38], it is suggested that increasing noise in thestochastic SMIB system can make the system unstable andinduce chaotic behavior. Reference [26] (mentioned in Sec. I)also studied stability in a stochastic SMIB system.
In this section, we use the autocorrelation functions derivedin Sec. II to calculate the variances and autocorrelations ofthe state variables of a stochastic SMIB system. Analysis ofthese functions provides analytical evidence for, and insightinto, CSD in a small power system.
A. Stochastic SMIB System Model
Fig. 1 shows the stochastic SMIB system. Equation (24),which combines the mechanical swing equation and the elec-trical power produced by the generator, fully describes thedynamics of this system:
Mδ +Dδ +(1 + η)E′a
Xsin (δ) = Pm (24)
where (η ∼ N (0, 0.01)) is a white Gaussian random variableadded to the voltage magnitude of the infinite bus to accountfor the noise in the system, M and D are the combined inertiaconstant and damping coefficient of the generator and turbine,and E
′
a is the transient emf. The reactance X is the sum ofthe generator transient reactance (X
′
d) and the line reactance(Xl), and Pm is the input mechanical power. The value ofparameters used in this section are given below:
D = 0.03 purad/s , H = 4MW.s
MV A , X′d = 0.15pu,
Xl = 0.2pu, ωs = 2π · 60rad/s
Note that M = 2H/ωs, where H is the inertia constantin seconds, and ωs is the rated speed of the machine. Thegenerator and the system base voltage levels are 13.8kVand 115kV, and both the generator and system per unitbase are set to 100MVA. The generator transient reactanceX
′
d = 0.15 · (13.8/115)2 pu, on the system pu base. The thirdterm on the left-hand side of (24) is the generator’s electricalpower (Pg).
In order to test the system at various load levels, wesolved the system for different equilibria, with the generator’smechanical and electrical power equal at each equilibrium:
Pm = Pg0 =E
′
a
Xsin (δ0) (25)
where δ0 is the value of the generator rotor angle at equilib-rium.
5
(1+η)/0°)
E’a)/δ)
)))))X’d) )))))Xl)Vg)/θg)
Figure 1. Stochastic single machine infinite bus system used in Sec. III. Thenotation Vg θg represents Vg exp [jθg ].
B. Autocorrelation and Variance
In this section, we calculate the autocorrelations and vari-ances of the algebraic and differential variables of this systemusing the method in Sec. II. Equations (1) and (2) describethis system for which the following equalities hold:
γ =D
2M;ω0 =
√E′a cos δ0MX
; y =[Vg θg
]T(26)
f =Pg0M
;F1
(z, y, η
)=
((1 + η)E
′
a
Xsin δ − Pm
)/M (27)
where ∆Vg = Vg−Vg0,∆θg = θg−θg0 are the deviations of,respectively the generator terminal busbar’s voltage magnitudeand angle from their equilibrium values. Equations (26) and(27) show that f increases with δ0 while ω0 decreases withδ0.
In order to calculate the algebraic equations, which formF2 (δ, y, η) in (2), we wrote Kirchhoff’s current law at thegenerator’s terminal:
E′aejδ − VgejθgjX ′d
+1 + η − Vgejθg
jXl= 0 (28)
Separating the real and imaginary parts in (28) gives thefollowing:
Vg sin (θg) = αE′
a sin (δ) (29)
Vg cos (θg) = αE′
a cos (δ) (30)+ (1 + η) (1− α)
where α = Xl/(Xl +X′
d). Equations (29) and (30) combineto make F2 (δ, y, η) in (2).
Linearizing (29) and (30) yields the coefficients in (17),which are necessary for calculating the autocorrelations andvariances of the algebraic variables (here y1 = ∆Vg, y2 =∆θg):
C1,1 = αE′
a sin (θg0 − δ0) (31)C1,2 = (1− α) cos (θg0) (32)
C2,1 = αE′
a cos (θg0 − δ0) (33)C2,2 = − (1− α) sin (θg0) (34)
Fig. 2 shows the decrease of ω′, which is the absolutevalue of the imaginary part of the eigenvalues of A in (7),with Pm. Note that the bifurcation occurs at Pm = 5pu. Thisfigure illustrates how it can be difficult to accurately foresee a
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 b 0
2
4
6
8
10
12
14
16
Pm
(pu)
ω’
Figure 2. The decrease of ω′ with Pm in the SMIB system. Near thebifurcation, ω′ is very sensitive to changes in Pm. In this figure, and mostthat follow, b is the value of the bifurcation parameter (Pm in this system)at the bifurcation.
bifurcation by computing the eigenvalues of a system (as in,e.g., [19]), if there is noise in the measurements feeding thecalculation. The value of ω′ ∼ (Pm − b)
1/4 does not decreaseby a factor of two (compared to its value at Pm = 1.0pu)until Pm = 4.83pu (only < 3.4% away from the bifurcation).It decreases by another factor of two at Pm = 4.99pu (0.2%away from the bifurcation). Also, note that the real part of theeigenvalues are equal to −γ until very close to the bifurcation(0.1% away from the bifurcation), so they do not provide auseful indication of proximity to the bifurcation. Thus, onecan confidently predict from ω′ the imminent occurrence ofthe bifurcation only very near it, which may be too late toavert it. On the other hand, we will demonstrate below that forthis system, autocorrelation functions can provide substantiallymore advanced warning of the bifurcation.
Using autocorrelation as an early warning sign of potentialbifurcations requires that one carefully select a time lag,∆t = t−s, such that changes in autocorrelation are observable.To understand the impact of different time lags, we computedthe autocorrelation as function of ∆δ (see Fig. 3). From (14),the autocorrelation of δ(t) crosses zero at ∆t0 = 2π−φ
ω′ .The implication is that choosing ∆t close to ∆t0 allows oneto observe a monotonic increase of autocorrelation as Pmincreases. For ∆t > ∆t0, autocorrelation may not increasemonotonically, or the autocorrelation for some values of Pmmay be negative. For example, in Fig. 3 for ∆t = 0.3s, theautocorrelation decreases first and then increases with Pm.On the other hand, for ∆t considerably smaller than ∆t0, theincrease of the autocorrelation may not be large enough to bemeasurable. In Fig. 3, the curves converge as ∆t→ 0. Giventhat the smallest period of oscillation (T = 2π/ω′) in thissystem is 0.41s, we chose ∆t = 0.1s for the autocorrelationcalculations in this section.
Using (12)–(15), we calculated the variances and autocor-relations of ∆δ, ∆δ at different operating points. In Figs. 4and 5, these analytical results are compared with the numericalones. To initialize the numerical simulations, we assumed thatVg0 = 1pu and solved for E′a in (29), (30) to obtain Vg = Vg0(for η = 0). We chose the integration step size τint to be0.01s, which is much shorter than the the smallest period
6
0 0.1 0.2 0.3 0.4 0.5−1
−0.5
0
0.5
1A
uto
co
rre
latio
n o
f ∆
δ
∆t(sec)
Pm
= 0.1pu
Pm
= 2pu
Pm
= 4pu
Pm
= 4.75pu
Figure 3. Autocorrelation function of ∆δ. ∆t = 0.1s is close to 1/4 of thesmallest period of the function for all values of Pm.
1 2 3 4 b0
50
100
150
200
σ2 ∆
δ/σ2 η
Pm(pu)
NumericalAnalytical
1 2 3 4 b0
0.5
1
1.5
σ2 ∆
δ
Pm(pu)
NumericalAnalytical
q 8020
= 26.2 q 8020
= 15.9
Figure 4. Variances of ∆δ,∆δ versus mechanical power (Pm) values.
of oscillation (T = 0.41s). The numerical results are shownfor the range of bifurcation parameter values for which thenumerical solutions were stable.
In order to determine if variance and autocorrelation mea-surably increase as load approaches the bifurcation, we com-puted the ratio of each statistic when load is at 80% of thebifurcation value to the value when load is at 20% of b. Thisratio, q 80
20in Figs. 4 and 5, is defined as follows:
q 8020
=Autocorrelation of u or σ2
u|Pm=0.8b
Autocorrelation of u or σ2u|Pm=0.2b
(35)
where u is the plot’s variable. In subsequent figures, q 8020
isdefined similarly.
Fig. 4 shows that the variances of both ∆δ and ∆δ increasesubstantially with Pm, and thus appear to be good warningsigns of the bifurcation. However, the two variances growwith different rates. (This becomes clear when comparing theratios q 80
20for ∆δ and ∆δ.) The difference becomes even
more noticeable near the bifurcation where the variance of∆δ increases much faster than the variance of ∆δ. This iscaused by the term ω2
0 in the denominator of the expressionfor the variance of ∆δ in (12). In Fig. 5, the autocorrelationsof ∆δ and ∆δ increase with Pm. Similar to the variances, theautocorrelations are good early warning signs of the bifurca-tion as well. Comparing Figs. 4 and 5 with Fig. 2 (where anequivalent q 80
20would be 1.28) shows that the autocorrelations
and variances of ∆δ and ∆δ provide a substantially strongerearly warning sign, relative to using eigenvalues to estimatethe distance to bifurcation in this system.
The results for the algebraic variables are mainly similar.
1 2 3 4 b0
0.2
0.4
0.6
0.8
1
Auto
corr
elat
ion
of∆
δ
Pm(pu)
NumericalAnalytical
1 2 3 4 b0
0.2
0.4
0.6
0.8
1
Auto
corr
elat
ion
of∆
δ
Pm(pu)
NumericalAnalytical
q 8020
= 4.1 q 8020
= 22.5
Figure 5. Autocorrelations of ∆δ,∆δ versus mechanical power (Pm) values.The autocorrelation values are normalized by dividing by the variances of thevariables.
1 2 3 4 b0
0.05
0.1
0.15
0.2
σ2 ∆
Vg/σ
2 η
Pm(pu)
NumericalAnalyticalFirst termSecond term
1 2 3 4 b0
50
100
150
200
σ2 ∆
θg/σ
2 η
Pm(pu)
NumericalAnalyticalFirst termSecond term
q 8020
= 0.7 q 8020
= 26.4
Figure 6. Variances of ∆Vg and ∆θg versus mechanical power (Pm) levels.The two terms comprising the variances in (21) are also shown.
Figs. 6,7 show the variances and autocorrelations of ∆Vg,∆θgas a function of load. In Fig. 6, the variance of ∆Vg decreaseswith Pm until the system gets close to the bifurcation, whilethe variance of ∆θg increases with Pm even if the systemis far from the bifurcation. The autocorrelations of both ∆Vgand ∆θg in Fig. 7 increase with Pm. However, the ratio q 80
20
in (35) is much larger for ∆Vg than for ∆θg . This is causedby the autocorrelation of ∆Vg being very close to zero forsmall values of Pm.
C. Discussion
These results can be better understood by observing thetrajectory of the eigenvalues of the SMIB system (Fig. 8). Nearthe bifurcation, the eigenvalues are very sensitive to changesin the bifurcation parameter. As a result, the system is inthe overdamped regime (ω0 < γ) for much less than 0.1%distance in terms of Pm to the critical transition. This implies
1 2 3 4 b0
0.2
0.4
0.6
0.8
1
Auto
corr
elat
ion
of∆
Vg
Pm(pu)
NumericalAnalytical
1 2 3 4 b0
0.2
0.4
0.6
0.8
1
Auto
corr
elat
ion
of∆
θ g
Pm(pu)
NumericalAnalytical
q 8020
= 76.7 q 8020
= 4.0
Figure 7. Autocorrelations of ∆Vg and ∆θg versus Pm.
7
−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−15
−10
−5
0
5
10
15Im
Re
b0 = 69.32Pm = 4.66
b0 = 0.57Pm = 0.05
b0 = 0.57Pm = 0.05
b0 = 69.32Pm = 4.66
b0 = 89.78Pm = 5.00 b0 = 90
Pm = 5.00b0 = 89.95Pm = 5.00
b0 = 89.78Pm = 5.00
b0 = 90Pm = 5.00
b0 = 89.95Pm = 5.00
Figure 8. Eigenvalues of the first system as the bifurcation parameter(mechanical power) is increased. The arrows show the direction of theeigenvalues’ movement in the complex plane as Pm is increased. The valuesof Pm and δ0 are given for several eigenvalues.
that, at least for this system, the autocorrelation function in[7], [8], is valid only when the system is within 0.1% of thesaddle-node bifurcation. Because the method in [7], [8] canprovide a good estimate of the autocorrelations and variancesof state variables only for a very short range of the bifurcationparameter, it may not be particularly useful as an early warningsign of bifurcation.
From Figs. 4–7, we can observe that, except for the varianceof ∆Vg , the variances and autocorrelations of all state variablesincrease when the system is more loaded. This demonstratesthat CSD occurs in this system as it approaches bifurcation,as suggested both by general results [9], and prior work forpower systems [6], [7].
In addition to validating these prior results, several newobservations can be made. For example, the signs of CSDare more clearly observable in some variables than in others.While all of the variables show some increase in autocorrela-tion and variance, they are less clearly observable in ∆Vg .The variance of ∆Vg decreases with Pm slightly until thevicinity of the bifurcation. In comparison, the variance of ∆θgalways increases with Pm. Fig. 6 shows the two terms of theexpressions for the variances of ∆Vg and ∆θg in (21). Thesecond term of the variance of ∆θg is very small comparedto the first term, and the first term is always dominant andgrowing. On the other hand, the second term of the varianceof ∆Vg is more significant for small Pm. This term decreaseswith Pm, which can be observed from the expression for C1,2
in (32). Accordingly, decrease of C1,2 with Pm causes thethe variance of ∆Vg to decrease with Pm until the vicinity ofthe bifurcation. In conclusion, the variance of ∆θg is a betterindicator of proximity to the bifurcation. Because the variables∆δ and ∆δ are highly correlated with ∆θg , their variances arealso good indicators of proximity to the bifurcation.
The rate at which autocorrelation increases with Pm differssignificantly in Figs. 5 and 7. In Fig. 5, the ratio q 80
20in
(35) is 5.5 times larger for ∆δ than for ∆δ. The normalizedautocorrelation functions of ∆δ and ∆δ are as follows:
E [∆δ (t) ∆δ (s)] /σ2∆δ = exp (−γ∆t)
ω0
ω′(36)
· sin (ω′∆t+ φ)
E[∆δ (t) ∆δ (s)
]/σ2
∆δ= exp (−γ∆t)
ω0
ω′(37)
· sin (ω′∆t+ π − φ)
The difference between the two functions is in the phaseof the sine function which causes the values of the twoautocorrelations to be different. q 80
20is so much larger for ∆δ
than for ∆δ because of the time lag (∆t) used to computeautocorrelation. ∆t = 0.1s is close to the zero crossing of theautocorrelation function of ∆δ, causing the large q 80
20. This
difference illustrates the importance of choosing an appropriatetime lag.
It is important to note that although the growth ratio of theautocorrelation for ∆δ is not large compared to ∆δ, it can beincreased by subtracting a bias value from the autocorrelationvalues for Pm = 0.2b(pu) and Pm = 0.8b(pu). For example,if the value of 0.075 is subtracted from the autocorrelationvalues, the ratio q 80
20increases from 4.1 to 13.0. When using
this approach, it is recommended that the new base value (here,autocorrelation of ∆δ for Pm = 0.2b(pu)) is chosen to be atleast 25% of the original value, in order to reduce the impactof measurement noise.
The results also show that it is the nonlinearity of thissystem that causes CSD to occur. One of the elements of thestate matrix (−ω2
0) in (7) changes with Pm because of thenonlinear relationship between the electrical power (Pg) andthe rotor angle, causing the eigenvalues to change with Pm.If the relationship between Pg and δ were linear, the statematrix A would be constant. Indeed, in [31], it is shown thatthe stationary time correlation matrix of (6) can be calculatedusing the following equation:
E[Z (t)ZT (s)
]= exp [−A∆t]σ (38)
where σ is the covariance matrix of the state variables. Thus,the normalized autocorrelation matrix depends only on A andthe time lag. As a result, if the state matrix is constant, theautocorrelation for a specific time lag will also be constant.Thus, in this system, CSD is caused by the nonlinear relation-ship between Pg and the rotor angle.
IV. SINGLE MACHINE SINGLE LOAD SYSTEM
The first system illustrates how CSD can occur in a gen-erator connected to a large power grid, through a long line.In this section we use a generator to represent the bulk grid,and look for signs of CSD caused by a stochastically varyingload. Some form of the single machine single load (SMSL)model used in this section has been used extensively to studyvoltage collapse (e.g., [13], [39]).
A. Stochastic SMSL System Model
The second system (shown in Fig. 9) consists of onegenerator, one load and a transmission line between them. Therandom variable η defined in (3) and (4), is added to the loadto model its fluctuations. The load consists of both active andreactive components. In order to stress the system, the baselineload Sd is increased, while keeping the noise intensity (Sd0)and the load’s power factor constant.
A set of differential-algebraic equations comprising the swingequation and power flow equations describe this system. Theswing equation and the generator’s electrical power equation
8
Sd+ηSd0&
&&&&&Xl&Vg/θg& Vl/θl&
&&&&&rl&
E’a&/δ&
&&&&&X’d&
Figure 9. Single machine single load system.
are given below:
Mδ +Dδ = Pm − Pg (39)
Pg = E′
aVlGgl cos (δ − θl) (40)
+E′
aVlBgl sin (δ − θl) + E′2a Ggg
where Vl, θl are voltage magnitude and angle of the loadbusbar, Ggl, Ggg and Bgl are as follows:
Ggg = −Ggl = Re
(1
rl + jXl
)(41)
Bgl = −Im(
1
rl + jXl
)(42)
The power flow equations at the load bus are as follows:
− Pd − Pd0η = VlE′
aGgl cos (θl − δ) (43)
+VlE′
aBgl sin (θl − δ) + V 2l Gll
−Qd −Qd0η = VlE′
aGgl sin (θl − δ) (44)
−VlE′
aBgl cos (θl − δ)− V 2l Bll
where Gll = Ggg, Bll = −Bgl, and Pd0, Qd0 are con-stant values. The parameters of this system are similar tothe SMIB system, with the following additional parameters:rl = 0.025Ω, Pd0 = 1pu, pf = 0.95lead, where rl is the line’sresistance and pf is the load’s power factor.
In this system, Vl, θl− δ are the algebraic variables, and δ,δ are the differential variables. The algebraic equations (43)and (44) define Vl and θl−δ, which then drive δ through (39)and (40). By linearizing (40) and the power flow equationsaround the equilibrium, we simplified (39) to the following:
∆δ +D
M∆δ = −C5
Mη (45)
where C5 is a function of the system state at the equilibriumpoint. The derivation of (45) and the expression for C5 arepresented in Appendix A. Comparing (5) with (45) yields thefollowing:
γ =D
2M,ω0 = 0, f =
C5
M(46)
The expression for the autocorrelation of ∆δ is given in(16). Note that the normalized autocorrelation of ∆δ does notchange with the bifurcation parameter (Pd), as it did for theSMIB system. In Appendix A, it is shown that ∆Vl and ∆δ−∆θl are proportional to η (see (54) and (55)). As a result, theyare memoryless; the variables have zero autocorrelation.
Figs. 10 and 11 show the analytical and numerical solutionsof the variances of ∆δ, ∆Vl and ∆δ−∆θl. Unlike the SMIB
0.5 1 1.5 2 2.5 b 0.06
0.08
0.1
0.12
0.14
0.16
σ2 ∆
δ
Pd(pu)
NumericalAnalytical
q 8020
= 1.3
Figure 10. Variance of ∆δ for different load levels. The variance increasesmodestly with Pd as the system approaches the bifurcation.
0.5 1 1.5 2 2.5 b 0
5
10
15
20
25
30
35
40
σ2 ∆
δ−∆
θl/
σ2 η
Pd(pu)
NumericalAnalytical
0.5 1 1.5 2 2.5 b 0
5
10
15
σ2 ∆
Vl/σ2 η
Pd(pu)
NumericalAnalytical
q 8020
= 2.4 q 8020
= 50.3
Figure 11. Variances of ∆δ −∆θl and ∆Vl for different load levels. Bothvariances increase with Pd as the system approaches the bifurcation.
system, the variance of ∆Vl is a good early warning sign ofthe bifurcation. It is also much more sensitive to the increaseof Pd compared to ∆δ −∆θl and ∆δ.
B. Discussion
As was the case with the SMIB system, when the powerflowing on the transmission line in this system reaches itstransfer limit, the algebraic equations become singular. How-ever, unlike the previous system, the differential equations ofthis system do not become singular at the bifurcation point ofthe algebraic equations. Fig. 12 shows the sample trajectoriesof the two systems’ rotor angles. Both signals are Gaussianstochastic processes. The rotor angle in the SMIB system isan Ornstein–Uhlenbeck process while the rotor angle in theSMSL system varies like the position of the brownian particle[40]. The existence of the infinite bus in the former systemcauses this difference.
One difference between the SMSL system and the SMIBsystem is the absence of the term comprising ∆δ in (45)compared with (5). This causes the linearized state matrix tobe independent of the bifurcation parameter. From (38), onecan show that the normalized autocorrelation of ∆δ dependsonly on A and the time lag. Since A is constant in this system,the autocorrelation of ∆δ will be constant for a specific ∆t.
The increase of the variances of both differential andalgebraic variables is due to the non-linearity of the algebraicequations. Fig. 13 shows that as the load power increases,the perturbation of the load power causes a larger deviationin the load busbar voltage magnitude. Consequently, variance
9
0 50 1000.7
0.8
0.9
1
1.1
1.2
1.3
1.4δ(r
ad
)
t(s)0 50 100
−8
−6
−4
−2
0
2
δ(r
ad
)t(s)
a b
Figure 12. A sample trajectory of the rotor angle of (a) the SMIB system(b) the SMSL system.
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 330.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Pd(pu)
Vl(
pu)
0 5−0.04
0.04
!V
l(pu)
t(s)
0 5−0.04
0.04
t(s)
!V
l(pu)
Figure 13. The load bus voltage as a function of load power. The loadbus voltage magnitude becomes increasingly sensitive to power fluctuationsas the system approaches the bifurcation. This increased sensitivity raises thevoltage magnitude’s variance.
of this algebraic variable increases with Pd. Likewise, thisnonlinearity causes the coefficient C5 in (45) to increase asthe load power is increased, increasing the variance of ∆δ.One can show that if the line resistance (rl) is neglected inthis system, C5 in (45) will be replaced by Pd0. In this case, thevariance of ∆δ is constant, since the differential and algebraicequations are fully decoupled.
While voltage variance increases with load, this systemdoes not technically show CSD before the bifurcation, sinceincreases in both variance and autocorrelation are essential toconclude that CSD has occurred [9]. Also, the eigenvalues ofthe state matrix of this system do not vary with load. Thisconfirms that CSD does not occur in this system, since thepoles of the dynamical system do not move toward the right-half plane as the bifurcation parameter increases [1], [9].
V. THREE-BUS SYSTEM
Real power systems have properties that are common toboth the SMIB in Sec. III and the SMSL in Sec. IV. In orderto explore CSD for a system that has both an infinite bus, andthe potential for voltage collapse case, this section looks atthe three-bus system in Fig. 14.
A. Model and Results
The three-bus system consists of a generator connected toa load bus through a transmission line, which is connected toan infinite bus through another transmission line. In the SMIB
system, the bifurcation occurred in the differential equations.Increasing the load in the three-bus system causes a saddle-node bifurcation in the algebraic equations F1
(δ, y, 0
)=
0, F2
(δ, y, 0
)= 0 (in terms of (1), (2)), as in the SMSL
system. However, unlike in the SMSL system, the bifurcationin these algebraic equations also causes a bifurcation in thedifferential equation (5).
E’a$/δ$
$$$$$X’d$ $$$$$Xl2$Vg$/θg$
$$$$$Xl1$
Pd+Pd0η$
Vl/θl$ 1/0$
Figure 14. Three–bus system.
We studied this system for two different cases. Our goalfrom studying these two cases was to show that the CSDsigns for some variables can vary differently with changing thesystem parameters. In case A, the parameters of this system aresimilar to those in the SMIB system except for the following:
Xl1 = 0.1pu, Xl2 = 0.35pu, X ′d = 0.1pu
In Case B, the following parameters were used:
Xl1 = 0.3pu, D = 0.001 purad/s
The algebraic equations of the three-bus system are asfollows:(
E′aVlX
sin (δ − θl)−2
3Pd
)/M = 0 (47)
E′aVlX
sin (δ − θl)−VlXl2
sin (θl)− Pd0η = Pd (48)
E′aVlX
cos (δ − θl) +VlXl2
cos (θl) = V 2l · (49)(1
X+
1
Xl2
)where X = X
′
d +Xl1, Vl, θl are voltage magnitude and angleof the load busbar. Equation (47) is equivalent to F1
(δ, y, 0
)in
(1), and (48), (49), which are the simplified active and reactivepower flow equations at the load busbar, are equivalent toF2
(δ, y, 0
)in (2). We assumed that Pg0 = 2Pd/3, which is
reflected in (47).The following equalities relate this system to the general
model in (5):
γ =D
2M;ω2
0 =−C6
M; f =
−C7
M(50)
where C6 and C7 are functions of the system state at theequilibrium point. The derivation and expressions for C6, C7
are presented in Appendix B. Fig. 15 shows C6, C7 versus Pd.When the load increases, C6 approaches 0, and a bifurcationin the differential equation (5) and (50) occurs.
Using (50), the expressions in Sec. II-B, and (72), (73) inAppendix B, we calculated the variances and autocorrelationsof ∆δ,∆δ,∆Vl and ∆θl. We chose the autocorrelation time
10
0 1 2 b−3
−2
−1
0
1
Pd(pu)
C6,C
7&
C2 7/C
6
C6
C7
C7
2/C
6
2.8 2.9 b −3
−2
−1
0
1
C6,C
7&
C2 7/C
6
Pd(pu)
C6
C7
C7
2/C
6
Figure 15. Three variables C6, C7 and C27/C6 derived by linearizing the
Three-bus system model. The left panel shows the variables versus Pd forCase B. The right panel shows a close-up view of the variables near thebifurcation. Note that as Pd → Pd,cr , C6 → 0 while C7 approaches a finitevalue of ∼ 0.6. C2
7/C6 →∞, as Pd → Pd,cr .
lag ∆t of the variables to be equal to 0.14s taking a similarapproach as in Sec. III-B. Although the chosen ∆t may notbe optimal for all of the variables, it represents a reasonablecompromise between simplicity (choosing just one ∆t) andusefulness as early warning signs. Figs. 16–19 compare theanalytical solutions with the numerical solutions of the vari-ances and autocorrelations of ∆δ, ∆δ, ∆Vl and ∆θl.
Fig. 17 shows that although the growth rates of the autocor-relations of ∆δ,∆δ are not large, the autocorrelations increasemonotonically in both cases. As mentioned in Sec. III-C, it ispossible to have larger indicators (growth ratios) by subtractinga bias value from the autocorrelations. On the other hand,the variances of ∆δ,∆δ in Fig. 16, do not monotonicallyincrease for case B. We will explain this behavior in thenext subsection. As a result, they are not reliable indicatorsof proximity to the bifurcation.
Fig. 18 shows that although both variances of ∆Vl and∆θl increase with Pd, increase of the variance of ∆Vl ismore significant. Also, the variance of ∆θl does not increasemonotonically with Pd for case B. As a result, the varianceof ∆Vl seems to be a better indicator of the system stability.
In Fig. 19, the autocorrelation of ∆Vl until very near thebifurcation is small compared to those in Fig. 17. This iscaused by C26 being very small in (72), so ∆Vl is tied to thedifferential variables weakly. As a result, ∆Vl behaves in partlike η—the white random variable, and hence its autocorrela-tion is not a good indicator of proximity to the bifurcation. Inaddition, nonmonotonicity of the autocorrelations of ∆Vl, ∆θlfor case B in Fig. 19 shows that they are not good earlywarning signs of bifurcation.
B. Discussion
After studying this system with a range of different parame-ters, we found that autocorrelations of the differential variablesand variance of the voltage magnitude are consistently goodindicators of proximity to the bifurcation.
On the other hand, as shown in Fig. 16, variance inthe differential variables is not a reliable indicator. Namely,variances change non-monotonically (i.e., they do not alwaysincrease) and, importantly, may exhibit very abrupt changes.Fig. 15 provides some clues as to the reason for this latterphenomenon. In this figure, the absolute value of C7 decreases
0 1 2 3 4 5 b0
5
10
15
20
25
30
σ2 ∆
δ/σ2 η
Pd(pu)
Case A (N)Case A (A)Case B
0 1 2 3 4 5 b0
0.02
0.04
0.06
0.08
0.1
σ2 ∆
δ
Pd(pu)
Case A (N)Case A (A)Case B
q 8020
(1) = 1.6
q 8020
(2) = 0.9
q 8020
(1) = 1.2
q 8020
(2) = 0.8
Figure 16. Variances of ∆δ,∆δ versus load power (Pd). The ratiosq 8020
(1), q 8020
(2) are for case A, case B respectively. CaseA(N), CaseA(A)denote numerical and analytical solutions for case A.
0 1 2 3 4 5 b0
0.2
0.4
0.6
0.8
1
Auto
corr
ela
tion
of
∆δ
Pd(pu)
Case A (N)Case A (A)Case B
0 1 2 3 4 5 b0
0.2
0.4
0.6
0.8
1
Auto
corr
ela
tion
of
∆δ
Pd(pu)
Case A (N)Case A (A)Case B
q 8020
(1) = 1.9
q 8020
(2) = 1.3
q 8020
(1) = 3.3
q 8020
(2) = 1.3
Figure 17. Autocorrelations of ∆δ,∆δ versus Pd. Both of the autocorrela-tions increase with Pd.
0 1 2 3 4 5 b0
0.5
1
1.5
2
2.5
3
σ2 ∆
Vl/σ2 η
Pd(pu)
Case A (N)
Case A (A)
Case B
0 1 2 3 4 5 b0
5
10
15
20
σ2 ∆
θl/σ2 η
Pd(pu)
Case A (N)
Case A (A)
Case B
q 8020
(1) = 45.0
q 8020
(2) = 47.1
q 8020
(1) = 2.0
q 8020
(2) = 1.0
Figure 18. Variances of ∆Vl,∆θl versus Pd.
0 1 2 3 4 5 b0
0.05
0.1
0.15
0.2
0.25
Auto
corr
ela
tion
of
∆V
l
Pd(pu)
Case A (N)
Case A (A)
Case B
0 1 2 3 4 5 b0
0.1
0.2
0.3
0.4
0.5
Auto
corr
ela
tion
of
∆θ
l
Pd(pu)
Case A (N)
Case A (A)
Case B
q 8020
(1) = 1.7
q 8020
(2) = 1.2
q 8020
(1) = 1.6
q 8020
(2) = 0.9
Figure 19. Autocorrelations of ∆Vl,∆θl versus Pd.
11
with Pd and becomes zero very close to the bifurcation point,at Pd|C7=0. Therefore, the variances of ∆δ and ∆δ, whichare proportional to C2
7 , decrease and vanish at Pd|C7=0. Pastthis point, |C7| increases, while C6 continues to decreaseand vanishes at b. Therefore, the variances of ∆δ and ∆δ,which are proportional to C2
7/C6, increase to infinity in thevery narrow interval
(Pd|C7=0, b
); see Fig. 15. This explains
the sharp features in Fig. 16; a similar explanation can be givento such a feature in Fig. 19. Therefore, neither the variances of∆δ,∆δ or the autocorrelations of ∆Vl,∆θl are good indicatorsof proximity to bifurcation.
The results for this system clearly show that not all of thevariables in a power system will show CSD signs long beforethe bifurcation. Although autocorrelations and variances ofall variables increase before the bifurcation, some of themincrease only very near the bifurcation or the increase is notmonotonic. Hence, these variables are not useful indicators ofproximity to the bifurcation. In the three-bus system, autocor-relation in the differential equations was a better indicator ofproximity than autocorrelation in ∆Vl or ∆θl, which are notdirectly associated with the differential equations. Also, ∆Vlwas the only variable whose variance shows a gradual andmonotonic increase with the bifurcation parameter.
VI. CSD IN MULTI-MACHINE SYSTEMS
In order to compare these analytical results to results frommore practical power system, this section presents numericalresults for two multi-machine systems.
The first system was similar to the Three-bus system (caseB in Sec. V). The only difference was that instead of infinitebus, a generator similar to the other generator was used.The numerical simulation results were similar to the Three-bus system, except for the autocorrelation of ∆δ. Fig. 20shows that autocorrelation of ∆δ increases for one of themachines, while it decreases for the other one. This showsthat the autocorrelation of ∆δ is not a reliable indicator of theproximity to the bifurcation.
0 0.5 1 1.5 2 2.5 b 0.75
0.8
0.85
Auto
corr
elati
on
of
∆δ
Load level
G1
G2
Figure 20. Autocorrelation of ∆δ for two machines in the Three-bus systemwith two generators. G1 is the same generator as in the Three-bus systemand G2 is the new generator.
The second system we studied was the New England 39-bus system, using the system data from [41] We simulatedthis system for different load levels using the power systemanalysis toolbox (PSAT) [42]. Exciters and governors were notincluded in the results here, although subsequent tests indicatethat adding them do not substantially change the conclusions.
In order to change the system loading, each load was multi-plied by the same factor. At each load level, we added whitenoise to each load. As one would expect, increasing the loadsmoves the system towards voltage collapse. For solving thestochastic DAEs, we used the fixed-step trapezoidal solver ofPSAT with the step size of 0.01s. The noise intensity was keptconstant for all load levels.
The simulation results show that the variances and autocor-relations of bus voltage magnitudes increase with load. How-ever, similar to the Three-bus system, the autocorrelations ofvoltage magnitudes are very small, indicating that in practice,these variables would not be good indicators of proximity.The variances and autocorrelations of generator rotor anglesand speeds and bus voltage angles did not consistently showan increasing pattern. Figs. 21 and 22 show the variances andautocorrelations of the voltage magnitudes of five busbars andthe rotor angles of five generators of the system respectively.The buses and generators were arbitrarily chosen. As inprevious results, the autocorrelation time lag was chosen tobe 0.1s.
The results in this section suggest that autocorrelationsof differential variables show nonmonotic behavior in somecases, which limits their application as early waning signs ofbifurcation.
0 0.5 1 b 0
1
2
3
x 10−4
σ2 ∆
V
Load level 0 0.5 1 b
−0.01
0
0.01
0.02
0.03
0.04
0.05
Auto
corr
elati
on
of
∆V
Load level
Figure 21. The variances and autocorrelations of the voltage magnitudes offive busbars of the system. Load level is the ratio of the values of the system’sloads to their nominal values.
0 0.5 1 b 2
3
4
5
6
7x 10
−8
σ2 ∆
δ
Load level 0 0.5 1 b
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Auto
corr
elati
on
of
∆δ
Load levelFigure 22. The variances and autocorrelations of the rotor angles of fivegenerators of the system.
In many ways, this test case is a multi-machine version ofthe SMSL system. As with the SMSL and Three-bus systems,variances of bus voltage magnitudes are good early warning
12
signs. However, unlike in the SMSL system, autocorrelation involtage magnitudes increases, albeit only slightly, with systemload. Unlike in the SMSL system, voltage magnitudes in the39-bus case have non-zero autocorrelation for ∆t > 0. Thisresults from the fact that voltage magnitudes are coupled tothe differential variables in this system.
Results from this system, as with the SMSL system, suggestthat variance in voltage magnitudes is a useful early warningsign of voltage collapse. It is less clear from these results ifchanges in autocorrelation will be sufficiently large to providea reliable early warning of criticality.
VII. CONCLUSION
In this paper, we analytically and numerically solve thestochastic differential algebraic equations for three smallpower system models in order to understand critical slowingdown in power systems. The results from the single machineinfinite bus system and the Three-bus system models showthat critical slowing down does occur in power systems, andillustrate that autocorrelation and variance in some cases canbe good indicators of proximity to criticality in power systems.The results also show how non-linear dynamics influencethe observed changes in autocorrelation and variance. Forexample, linearity of the differential equation in the singlemachine single load system caused the autocorrelation of thedifferential variable to be constant. On the other hand, in theSMIB system and Three-bus system, the differential equationswere nonlinear and autocorrelations of the differential vari-ables increased with the bifurcation parameter.
Although the signs of critical slowing down do consistentlyappear as the systems approach bifurcation, only in a fewof the variables did the increases in autocorrelation appearsufficiently early to give a useful early warning of potentialcollapse. On the other hand, variance in load bus voltagesconsistently showed substantial increases with load, indicatingthat variance in bus voltages can be a good indicator of voltagecollapse in multi-machine power system models. This wasverified for the New England 39-bus system.
Together these results suggest that it is possible to obtainuseful information about system stability from high-samplerate time-series data, such as that produced by synchronizedphasor measurement units. Future research will focus ondeveloping an effective power system stability indicator basedon these results.
APPENDIX A
The derivation of (45) is presented in this section. Bylinearizing (40) around the equilibrium and replacing theobtained equation for Pg in (39), we obtained the following:
M∆δ +D∆δ = −C12∆Vl − C13 (∆δ −∆θl) (51)
where C12 and C13 are:
C12 = E′
a sin
(θl0 − δ0 − arctan
(GglBgl
))(52)
·√G2gl +B2
gl
C13 = Vl0E′
a cos
(θl0 − δ0 − arctan
(GglBgl
))(53)
·√G2gl +B2
gl
By linearizing (43) and (44) around the equilibrium, andsolving for ∆Vl and ∆δ −∆θl, we obtained the following:
∆Vl = C14η (54)∆δ −∆θl = C15η (55)
where C14 and C15 are:
C14 =C19Pd0 − C17Qd0
C17C18 − C16C19(56)
C15 =C18Pd0 − C16Qd0
C17C18 − C16C19(57)
where C16 − C19 are given below:
C16 = E′
a sin
(θl0 − δ0 + arctan
(GglBgl
))(58)
·√G2gl +B2
gl + 2GllVl0
C17 = Vl0E′
a cos
(θl0 − δ0 + arctan
(GglBgl
))(59)
·√G2gl +B2
gl
C18 = −E′
a cos
(θl0 − δ0 + arctan
(GglBgl
))(60)
·√G2gl +B2
gl − 2BllVl0
C19 = Vl0E′
a sin
(θl0 − δ0 + arctan
(GglBgl
))(61)
·√G2gl +B2
gl
Using (54) and (55), we rewrote (51) as (45) where C5 is asfollows:
C5 =(C13C18 + C12C19)Pd0 − (C13C16 + C12C17)Qd0
C16C19 − C17C18(62)
APPENDIX B
The derivation of C6, C7 is presented in this section. Byusing (1) and linearizing (47)-(49) around the equilibrium, wehave the following:
∆δ = −(D∆δ + C20∆Vl + C21 (∆δ −∆θl)
)/M(63)
0 = −Pd0η + C22∆Vl + C21∆δ + C23∆θl (64)0 = −∆Vl + C24∆δ + C25∆θl (65)
where C20 through C25 are as follows:
C20 (δ0, θl0) =E′aX
sin (δ0 − θl0) (66)
C21 (δ0, θl0, Vl0) =E′aVl0X
cos (δ0 − θl0) (67)
C22 (δ0, θl0) = C20 (δ0, θl0)− sin (θl0)
Xl2(68)
C23 (δ0, θl0, Vl0) = −C21 (δ0, θl0, Vl0) (69)
− Vl0Xl2
cos (θl0)
C24 (δ0, θl0) = −βE′a sin (δ0 − θl0) (70)C25 (δ0, θl0) = −C24 (δ0, θl0)− (1− β)
· sin (θl0) (71)
13
where β = Xl2/(X + Xl2). Using (64) and (65), we solvedfor ∆Vl and ∆θl:
∆Vl = C26∆δ + C27η (72)∆θl = C28∆δ + C29η (73)
where C26 through C29 are as follows:
C26 =C23C24 − C21C25
C22C25 + C23(74)
C27 =C25Pd0
C22C25 + C23(75)
C28 = −C21 + C22C24
C22C25 + C23(76)
C29 =Pd0
C22C25 + C23(77)
Equations (63), (72) - (77) lead to the following expressionsfor C6 and C7:
C6 = C21C28 − C20C26 − C21 (78)C7 = C21C29 − C20C27 (79)
ACKNOWLEDGMENT
The authors acknowledge Christopher Danforth for helpfulcontributions to this research, as well as the Vermont Ad-vanced Computing Core, which is supported by NASA (NNX-08AO96G), for providing computational resources.
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AUTHOR BIOGRAPHIES
Goodarz Ghanavati (S‘11) received the B.S. and M.S. degrees in ElectricalEngineering from Amirkabir University of Technology, Tehran, Iran in 2005and 2008, respectively. Currently, he is pursuing the Ph.D. degree in ElectricalEngineering at University of Vermont. His research interests include powersystem dynamics, PMU applications and smart grid.
Paul D. H. Hines (S‘96,M‘07) received the Ph.D. in Engineering and PublicPolicy from Carnegie Mellon University in 2007 and M.S. (2001) and B.S.(1997) degrees in Electrical Engineering from the University of Washingtonand Seattle Pacific University, respectively.He is currently an Assistant Professor in the School of Engineering, and theDept. of Computer Science at the University of Vermont, and a member of theadjunct research faculty at the Carnegie Mellon Electricity Industry Center.Formerly he worked at the U.S. National Energy Technology Laboratory, theUS Federal Energy Regulatory Commission, Alstom ESCA, and Black andVeatch. He currently serves as the vice-chair of the IEEE Working Group onUnderstanding, Prediction, Mitigation and Restoration of Cascading Failures,and as an Associate Editor for the IEEE Transactions on Smart Grid. He isNational Science Foundation CAREER award winner.
Taras I. Lakoba received the Diploma in physics from Moscow StateUniversity, Moscow, Russia, in 1989, and the Ph.D. degree in appliedmathematics from Clarkson University, Potsdam, NY, in 1996.In 2000 he joined the Optical Networking Group at Lucent Technologies,where he was engaged in the development of an ultralong-haul terrestrial fiber-optic transmission system. Since 2003 he has been with the Department ofMathematics and Statistics of the University of Vermont. His research interestsinclude multichannel all-optical regeneration, the effect of noise in fiber-opticcommunication systems, stability of numerical methods for nonlinear waveequations, and perturbation techniques.
Eduardo Cotilla-Sanchez (S‘08,M‘12) received the M.S. and Ph.D. degreesin electrical engineering from the University of Vermont, Burlington, in 2009and 2012, respectively. He is currently an Assistant Professor in the Schoolof Electrical Engineering and Computer Science at Oregon State University,Corvallis. His primary field of research is electrical infrastructure protection,in particular, the study of cascading outages. Cotilla-Sanchez is a member ofthe IEEE Cascading Failure Working Group.