Electrical Power and Energy Systems 32 (2010) 688–696

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Frequency induced risk assessment for a power system accounting uncertaintiesin operation of protective equipments

Y. Yu, D. Gan *, H. Wu, Z. HanCollege of Electrical Engineering, Zhejiang University, Hangzhou, Zhejiang, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 July 2007Received in revised form 25 December 2009Accepted 28 January 2010

Keywords:Frequency stabilityFrequency protectionLoad sheddingMonte Carlo simulationCascading outage

0142-0615/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.01.002

* Corresponding author. Tel.: +86 571 8795 1831; fE-mail addresses: yuyangzju@gmail.com (Y. Yu), d

vuhao@cee.zju.edu.cn (H. Wu).

A simple deterministic frequency stability model is developed in this work together with its stochasticcounterpart. While the deterministic model captures the fundamental characteristics of short-term fre-quency dynamics such as abnormal frequency tripping of generators, the stochastic model further main-tains the probabilistic characteristics of frequency dynamics, such as stochastic characteristics offrequency relays. The models are particularly suitable for studying system cascading failures introducedby frequency distortions. Our simulation results reveal that the relationship between blackout power lossand frequency has the characteristics of power law.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Frequency is one of the most important indicators of power sys-tem’s operation status. It reflects the active power equilibrium be-tween generation and load demand in power system. If generationmeets load demand, frequency is held at rated value. Otherwise,frequency rises or descends due to surplus or deficit of generation.Operation of generator (especially turbine), auxiliary machine andtransformer with large frequency deviation can result in malfunc-tions due to overheating or mechanical overstress [1]. Therefore,measures must be taken to stabilize frequency in acceptable range.

After occurrence of disequilibrium of generation and load theprimary frequency regulation will react to frequency deviationimmediately to try to keep frequency within required range. Ifthe frequency deviation still cannot meet system requirement,spinning reserves and rapid start generators will react subse-quently by increasing or decreasing generation in order to restorefrequency to nominal value [2,3]. Normally, frequency will be re-stored after these measures are applied.

However, if the response rate of reserves is too slow to take ef-fect or available reserve capacity is not enough after a large imbal-ance occurs, under-frequency (u.f.) load-shedding relays may betriggered according to their thresholds and time delays. Similarly,generators may be tripped if frequency exceeds its operation

ll rights reserved.

ax: +86 571 87952591.eqiang.gan@ieee.org (D. Gan),

limitation. When frequency does not satisfy system requirementafter the first round of u.f. load shedding, more loads shedding willbe further carried out. During operation within abnormal fre-quency range, generators will be tripped off with delays if any ofits cumulative operation time limitations is used up. The coopera-tion of u.f. load scheme and generators’ frequency protectionscheme in modern power system is not always effective. Thusthere exists a potential risk of cascading failure, even blackout,especially when a system transfers a great amount of power overlong distance or owns some generators that have very large capac-ity and heavily loaded. The loss of the main transmission lines orgenerators could introduce large deficiency of active power inreceiving power system and hence may initiate a rapid frequencydrop in a short period of time which increases the probability ofcascading outage.

Power system blackouts recently occurred in North Americanand Europe have attracted much attentions [4–6,27,30]. A funda-mental form of these blackouts is cascading outages of systemfacilities [4–8]. Analysis of NERC disturbances records from 1984to 1998 reveals that the probability of disturbance scale exhibitsa heavy tail and power law distribution [7–10], which is a clearindication of the high probability of blackout. This motivates anew research topic that tries to uncover the mechanism behindblackout. Different models, such as hidden failures with aim tostudy impact of hidden failure in system relays, CASCADE andOPA with the aim to study impact of cascading trips of transmis-sion lines have been proposed. All these simulation results verifiedthat the probability distribution of blackout size of modern power

ωΔ+

0mPΔ = 1

M s⋅

D

∑

ePΔ

Fig. 1. First-order system frequency response model.

Load

ωΔ+mPΔ 1

Ms∑

ePΔ

1

1 CHsT+1

1 GsT+vPΔ+

∑

Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696 689

system has similar stochastic characteristic [7,10–17]. However,the above results are mostly based on DC load flow models or uni-form network models and they do not take into account the impactof frequency which plays a key role in cascading outages.

This work presents deterministic models and their stochasticcounterparts for cascading failure investigation. Stochastic factorsassociated with generator frequency relay, u.f. load shedding andbreaker tripping time, etc. are explicitly modeled. Using a MonteCarlo simulation tool, the risk of frequency-induced cascading fail-ures is assessed. The influences of relevant factors on statistic char-acteristic of power system cascading failures, such as frequencyprimary regulation, AGC (spinning reserve), the size of initial dis-turbance and u.f. load shedding schemes, are also analyzed.Though the results presented are preliminary and no networkinformation is yet used, they already demonstrate interesting val-ues in identifying the factors that contribute to cascading failuresand blackouts.

reference set point

D

1 R

Fig. 2. Third-order system frequency response model.

2. A deterministic model for frequency dynamics

In most of the recent blackouts in the world, the time periodfrom the occurrence of dominating disturbance to the completeblackout is usually in order of minutes and involves several stabil-ity problems, such as frequency stability, voltage stability and tran-sient stability. Compared with the others, frequency stability is asystem-wide and crucial issue in blackout evolution period. With-out system frequency stability, reactive power balance and voltageprofile cannot be held and the overall system will crash quickly. In2003 Italy blackout [5], because of under-frequency, massive gen-erators tripping and loads shedding took place during only 2.5 minand eventually this lead to blackout. This illustrates that it is nec-essary to study the relationship between short-term frequency sta-bility and cascade outage.

Compared to the startup times of rapid start units and hot re-serve units, the frequency dynamics of modern power systems isshorter, thereby we mainly concern the frequency process withspinning reserve. In addition, generator and load restoration areneglected because these operations are taken after frequency isstabilized. In some cases, the primary regulation is not put in foreconomic reasons [4]. The impact of such ‘‘economic solution” isalso considered by assigning null primary regulation capability toall generators. Obviously, this is the worst situation in frequencydynamic progress.

Power system frequency characteristics can be modeled by asingle generating unit of non reheat type with good approximation[18–20]. For the sake of conciseness, the generator and loads inpower system are assumed to be ideally connected to a commonbus. Since the network is neglected, overloads on transmission lineare neglected in this work.

2.1. First-order frequency dynamics model

A linear model without considering primary regulation isshown in Fig. 1. The corresponding differential equations can bewritten as:

dDxdt¼ � D

MDx� DPe

Mð1Þ

where Dx is the speed deviation from nominal value; DPe the devi-ation in electrical power; M the angular momentum of the equiva-lent generator; D is the damping constant which characterizesdamping factor arising from both generator and load.

Given initial condition (Dx0), the analytical solution to Eq. (1)is:

Dx ¼ �DPe

Dþ Dx0 þ

DPe

D

� �e�

DMt ð2Þ

Generally, Dx is considered to be stable after 3s � 5s, wheretime constant s = M/D.

2.2. Third-order frequency dynamics model without AGC

The third-order model with fixed load reference set point isshown in Fig. 2. It can simulate the frequency dynamic responseduring the first a few seconds after a large disturbance [18–20].A set of differential equations can be constructed below:

dDxdt

dDPmdt

dDPvdt

2664

3775 ¼

� DM

1M 0

0 � 1TCH

1TCH

� 1TG �R

0 � 1TG

2664

3775

DxDPm

DPv

264

375þ

� DPeM

00

264

375 ð3Þ

where DPm is the deviation in mechanical power; DPv the per unitchange in valve position from nominal; R the speed regulation; TCH

the ‘‘charging time” time constant; TG is the main servo timeconstant.

The meanings of the other variables are the same as those inSection 2.1.

2.3. Fourth-order frequency dynamics mode

Spinning reserve is taken into account in this model. Fig. 3shows its block diagram, where k is a constant and B is called fre-quency bias factor [2]. The value of B can be calculated as follow:

B ¼ Dþ 1=R ð4Þ

The fourth-order differential equations are:

dDxdt

dDPmdt

dDPvdt

dDPLFdt

266664

377775 ¼

� DM

1M 0 0

0 � 1TCH

1TCH

0

� 1TG �R

0 � 1TG� 1

TG

Bk 0 0 0

266664

377775

DxDPm

DPv

DPLF

26664

37775þ

� DPeM

000

26664

37775 ð5Þ

where DPLF is the deviation in load reference.

690 Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696

The meanings of the other variables are the same as those inSections 2.1 and 2.2.

Although the models (1), (3) and (5) are rather standard, moresophisticated models could be used without introducing concep-tual difficulty in the proposed frequency stability analysis method.

1

for

gen

erat

or

2.4. The models for unit tripping and u.f. load shedding

After the occurrence of a disturbance DPe, system frequency fwould deviate from the rated value. Suppose unit i is operatingat output level gi (MW), the unit would be tripped if the system fre-quency drifts out of the interval ½f 0i; f 0i� within certain time. Simi-larly, load j absorbing power at level lj (MW) would beautomatically shed if system frequency is lower than f 0i with cer-tain time. Otherwise, no components would be disconnected.

Given a power system with NG generators and NL loads, the to-tal generation G and load L can be computed as:

G ¼XNG

i¼1

pigi; L ¼XNL

j¼1

qjlj ð6Þ

where

pi ¼0 f R ½f 0i; f 0i�1 f 2 ½f 0i; f 0i�

(ð7Þ

qj ¼1 f � f 00j0 f < f 00j

(ð8Þ

Given an initial disturbance DPe = G � L, the frequency dynamicprogress ðf̂ Þ can be computed by solving the differential Eqs. (1),(3), (5). If f̂ meets the following conditions:

f̂ 2TNG

i¼1½f 0i; f 0i�

f̂ > maxðf 00j Þ; j ¼ 1;2; . . . ;NL

8><>: ð9Þ

no generator and load would be disconnected, otherwise, genera-tors tripping or load shedding would happen.

After the ignition of the first round of load-shedding or genera-tor tripping, the new system frequency dynamic progress can becomputed. Re-evaluating Eq. (9) again would allow one to findout if another round of load-shedding or generator tripping wouldneed to be carried out. Repeating the above steps until the systemfrequency is restored to normal range, we can get the amount ofoverall load-shedding. If frequency runs under a critical threshold(such as 47.5 Hz [1]), frequency collapses and the system understudy is blackout. The former situation corresponds to a stable casewhile the latter is regarded as a system blackout hereafter.

The amount of total load-shedding and unit tripping can be ob-tained after finishing the above simulation. The model also servesas a basis for quantifying the risk of cascading failures introducedby frequency distortions.

LFPΔ ωΔ+mPΔ 1

M S⋅

D

∑

ePΔ

1

1 CHsT+1

1 GsT+vPΔ

1 R

+∑

B

1

k s

−⋅

Fig. 3. Fourth-order system frequency response model with AGC (spinningreserve).

3. Stochastic factors affecting frequency dynamics

During the frequency dynamic progress, the frequency relaysettings of generators are typically set by manufactures and plantoperators thus are stochastic in nature [29]. Some stochastic fac-tors, such as generator relay tripping [21], u.f. load shedding relayand time of breaker tripping are modeled, respectively.

3.1. Random behavior of generator frequency relays

Frequency range can be divided into three ranges according tooperation limitation of generator, turbine, power plant auxiliaries,etc. [1]:

(1) Continuous operation range: generators can operate contin-uously without time restriction.

(2) Restricted time operation range: continuous and accumula-tive operation times of generators are both restricted. Gener-ators should be tripped if the time limits are used up.

(3) Prohibited operation range: generators should be tripped atonce.

During most of time, system frequency stays in the continuousoperation range and the tripping probability of generators has norelationship with system frequency but is related to fault rate ofgenerator and its frequency relay, which can be obtained from his-torical statistical data. During frequency dynamic progress, gener-ator may run outside of continuous range and its accumulativerestricted operation time may reach to the threshold. Therefore,some generators may be tripped by relays in restricted operationrange and some do not even at the same frequency. As the distancefrom continuous operation range increases, the tripping probabil-ity of frequency protection increases. Furthermore, relay havemeasurement errors and can have design flaws in nature [21,22].

The simplified stochastic characteristic of generator frequencyprotection is shown in Fig. 4. where f0 is the nominal frequency;Probgi the fault rate of generator i; Prob0gi the fault rate of frequencyrelay of generator i; ½f 01; f 02� the continuous operation j range;

ðf 0i; f 01ÞSðf 02; f 0iÞ the restricted time operation range of generator i;

ðf 0i; f 01Þ is the prohibited operation range of generator i.According to Fig. 4, the tripping probability of generator i fre-

quency relay can be written as Eq. (10):

piðf Þ ¼

probgi þ prob0gi f 2 ðf 01; f 02Þð1�probgi�prob0giÞðf�f 02Þ

f 0i�f 02

f 2 ½f 02; f 0i�ð1�probgi�prob0giÞðf�f 01Þ

f 0i�f 0

1f 2 ½f 0i; f 01�

1 f R ½f 0i; f 0i�

8>>>>>>><>>>>>>>:

ð10Þ

2f ′0f1f ′

if ′if ′

gi giprob prob ′+

Frequency (Hz) Tri

ppin

g pr

obab

ility

of

u.f.

rel

ay

Fig. 4. Tripping probability of generator frequency relay as a function of systemfrequency.

Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696 691

3.2. Random behavior of u.f. load-shedding relays

When the system frequency is lower than a pre-set threshold f 00j ,step j of u.f. load shedding should act to disconnect loads.

Similar to generator frequency relays, the tripping probability ofu.f. relays increase while frequency approaches to its thresholdvalue.

Fig. 5 illustrates the probabilistic characteristic of such relays.According to Fig. 5, the tripping probability of u.f. load shedding

relay can be written as Eq. (11):

pjðf Þ ¼

problj f > f 00jð1�probljÞðf�f 00

jÞ

f 00j�f 00

j

f 2 ½f 00j ; f 00j �

1 f < f 00j

8>>>><>>>>:

ð11Þ

where problj is the fault rate of u.f. relay j; f > f 00j the continuousoperation range of u.f. relay j; f 00j is the setting of u.f. relay j.

Actually, in each step load shedding is implemented at differentlocations separately. To simulate the phenomenon, the load to beshed is divided into equal portions which is shed by different re-lays. According to Eq. (11), some u.f. relays trip, the others maynot at the same frequency with the same relay setting.

3.3. Random behavior of breaker’s tripping time

There exists time delay (T1) [22] due to measurement error offrequency relay. It also needs time (T2) [23] to disconnect compo-nent from power system by breaker. The tripping time

jf ′′

Tri

ppin

g pr

obab

ility

of

u.f.

load

she

ddin

g re

lay

1

ljprob

0fjf ′′ Frequency (Hz)

Fig. 5. Tripping probability of u.f. load shedding relay as a function of systemfrequency.

Tμ

Prob

abili

ty o

f tr

ippi

ng ti

me

2~ ( , )T N Tμ σ

Tripping Time (s)

Fig. 6. The distribution of component’s tripping time.

T(T = T1 + T2) of component is a random variable. We assume in thiswork that tripping time follows normal distribution shown inFig. 6.

Mathematically, this is equivalent to:

T � NðTl;r2Þ ð12Þ

where Tl and r are the mean and standard deviation of T,respectively.

Despite of the fact that the tripping times vary only slightly, thetripping sequence and frequency dynamic progress may howeverbe quite different after the same initial disturbance.

4. A Monte Carlo simulation algorithm

Monte Carlo method is widely used to simulate power systemoperation and calculate its probabilistic characteristics [20,24].Here, it is used to simulate frequency dynamic progress and calcu-late the probability of total generation loss after the occurrence of arandom imbalance between load and generation.

Before the simulation starts, the initial disturbance needs to bedetermined, thus the simulation comprises of two parts: the Mainand the Subroutine. The algorithm is described briefly below. Fig. 7shows the flowchart of Monte Carlo simulation.

Main:

M1

Set the maximum simulation number: Nmax

M2

Set the initial disturbance: select generator

outage randomly

M3

Pass all system parameters and initial

conditions to slave program

M4

Run the Subroutine algorithm, and calculate

load and generator losses

M5

Return to M1 to start a new loop until Nmax loops

are completed

Subroutine:

S1

Modify model’s parameters, calculate initial

conditions of Eqs. (1), (3) or (5)

S2

Solve Eqs. (1), (3) or (5) numerically, get the

frequency dynamics solution

S3

Check the frequency constraints in Eq. (9).

Determine the index set of all the units and

loads to be tripped (under-frequency or over-

frequency) according to Eqs. (10) and (11).

Calculate the probability and their tripping

time according to Eqs. (12)

S4

If no more generator trips or load sheds or

frequency reaches the collapse threshold,

return to Main

S5

Return to S1

5. Simulation results of a test system

5.1. Introduction of the simulation system

A test system is used for testing our developed method. The sys-tem generation data are shown in Table 1 [20]. It is worth mention-ing that in 2001 a blackout occurred in this system after initialtripping of two generators and frequency collapsed [20].

The test system’s base frequency is 50 Hz. The base capacity ofthe test system is set to online capacity (1 p.u.). The damping

N

Y

N

N

Y

N

Y

Y

Set the max simulation number maxN

Random select initial disturbances

Form disturbance list

Calculate the time ( TΔ ) to reach a new stabile frequency after t

Set delay time for each load shedding step based on eq.(12)

1i i= +

maxi N≤ ?

Modify parameters of system response model

Set initial time of disturbance ( 0t = )

Calculate frequency dynamic progress in TΔ

Input dates of test system

Is there any load to be shed? Random select partial load to be shedding based on ljprob

1i =

Recalculate frequency dynamic progress in TΔ

Select one component with maximum probability

Set delay time of its relay based on eq.(12)

1t t=

Does the frequency in TΔdrift out of [0.95,1.03]

Random select one point of frequency progress in TΔ

Is there any generators or loads to be

shed based on eq.(10) and eq.(11)

Form disturbance list

Log the tripped components during

the whole frequency progress

End

Modify parameters of system response model

All generators are tripped and system blackout

Subroutine

Fig. 7. Flowchart of Monte Carlo simulation.

692 Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696

constant (D) is set to 1.5. The number of maximal simulation times(Nmax) is set to 50,000.

5.2. Settings of frequency relays

Although generators of the test system have different frequencycharacteristic, different limitations in accumulation time and fre-

quency of occurrence within different frequency ranges, the allow-able frequency operation ranges (continuous and restricted) ofgenerators are quite similar [1]. For simplicity, generators of thetest system have the same continuous and restricted frequencyoperation ranges. According to IEC60034-3:1996, the restrictedfrequency operation range is set to (0.95, 0.98)\(1.02, 1.03) andthe continuous and prohibited operation ranges are set to

Table 1System generation data.

Unitno. (i)

Nominalpower (MW)

Failure rate(occ./year)

M (s) R (Hz/MW) TCH (s) TG (s)

1 23.5 5.1 4.5 0.22 1.263 0.07192 23.5 5.2 4.5 0.09 1.263 0.07193 23.5 5.0 4.5 0.09 1.263 0.07194 14.1 6.1 4.0 0.09 1.422 0.10925 14.1 6.2 4.0 0.10 4.422 0.10926 5.9 7.0 4.0 0.08 1.146 0.13187 11.5 5.5 5.0 0.07 1.014 0.078 11.5 5.6 5.0 0.10 1.014 0.079 11.5 5.4 5.0 0.07 1.014 0.07

10 11.5 5.7 5.0 0.07 1.014 0.0711 42.0 5.0 4.0 0.06 0.806 0.060312 20.0 3.0 4.0 0.55 1.796 0.186513 42.0 5.1 4.0 0.06 0.806 0.060314 20.0 3.0 4.0 0.55 1.796 0.186515 40.0 7.0 3.5 0.04 0.806 0.060316 57.0 4.5 4.5 0.15 0.806 0.060317 57.0 4.4 4.5 0.15 0.806 0.060318 16.0 6.0 4.0 0.06 0.566 0.078319 15.0 5.6 3.0 0.06 0.566 0.078320 15.0 5.7 3.0 0.06 0.566 0.078321 19.0 6.1 3.5 0.04 0.902 0.052322 15.0 6.2 3.5 0.04 0.902 0.052323 13.0 6.0 3.0 0.13 0.566 0.0783

Table 2Parameters of generation frequency continuous operation range.

f 0i f 01 f0 f 02 f 0i

0.95 0.98 1 1.02 1.03

Table 3Load shedding scheme (no. 1).

Step (j) 1 2 3 4 5

f 00j (p.u.) 0.985 0.98 0.975 0.97 0.965

f 00j (p.u.) 0.98 0.975 0.97 0.965 0.96

Delay time (s) 0.5 0.5 0.5 0.5 0.5Load dropped (%) 5 5 5 5 5

Fig. 8. A typical frequency dynamics of the first-order model (initial disturbance:tripping of one generator).

Fig. 9. The cumulative shedded loads and tripping generators corresponding to 8.

Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696 693

[0.98, 1.02] and ð0:95;1:03Þ, respectively [1]. The relay parametersare listed in Table 2. In the fourth-order model, spinning reserve isconsidered and is assigned to each generator equally.

Different systems have different schemes of u.f. load shedding[1,18,25–28]. For simplicity, a simple load shedding scheme isadopted (Table 3). The setting of the first round u.f. load sheddingis set at 0.98 and the one of the last round is 0.96. The total amountof load shedding is 0.25 p.u.

In the numerical example, the failure rate probgi of each gener-ator frequency relay in Fig. 4 and problj of each u.f. relay in Fig. 5 isset to 3 occ./years.

In consideration of relay measurement delay and breaker trip-ping delay, Tl is set to 0.25 s and r to 0.01 s.

Fig. 10. A typical frequency dynamic simulation of the third-orders model (onegenerator initial disturbance).

5.3. Simulation analysis

Fig. 8 shows a typical frequency dynamic transient of the first-order model and Fig. 9 shows components tripping sequencecorresponding to Fig. 8. After an initial disturbance (unit no. 13tripping), frequency descends rapidly and the first step frequencysetting (0.98 p.u.) is activated at 2.952 s with 0.045 p.u. load shed-ding because of a u.f. relay fault at a location. At 4.8998 s, another

generator (no. 15) trips because its time limitation in restrictiverange is used up. After the generator trips, step 2 and step 3 ofu.f. load shedding are triggered, respectively, at 5.489 s and

Fig. 13. The cumulative loads shedded and generators tripped corresponding toFig. 12.

694 Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696

5.907 s with total 10% of load shedding. Eventually, frequency re-stores to nominal range without any more generators trippingand load shedding. During the dynamic process, 0.1572 p.u. gener-ation and 0.145 p.u. load are disconnected.

Figs. 10 and 11 show a typical frequency dynamics of the third-order model and its components tripping sequence respectively.After an initial disturbances (unit no. 16 tripping), frequency des-cends. Sequentially, another unit (no. 14) trips at 6.307 s for unitfault and frequency descends again. The first step of u.f. load shed-ding is triggered at 7.792 s with 5% of loads shedding. Eventually,frequency meets Eq. (9) and system frequency arrives at a new sta-ble value. During the whole dynamic period, 0.1927 p.u. generationcapacity and 0.05 p.u. load are shed.

Figs. 12 and 13 show a typical frequency dynamic process of thefourth-order model with 15% spinning reserve and its componentstripping sequence, respectively. After an initial disturbance (unitno. 17 tripping), a unit (no. 13) trips at 7.1642 s for relay fault dur-ing frequency restoration. After the two units tripped, the first stepof u.f. load shedding is triggered at 10.719 s. Eventually, frequencyreaches to 0.9975 p.u. without any more components tripped orshed. During the whole transient, system loses 0.1898 p.u. genera-tion capacity and 0.05 p.u. Comparing Fig. 12 to Fig. 10, final fre-quency is closer to rated value due to the presence of spinningreserves.

Fig. 11. The cumulative loads shedded and generators tripped corresponding to 10.

Fig. 12. A typical frequency curve of the fourth-orders model (one generator initialdisturbance, 15% spinning reserve).

Fig. 14. Distribution of blackout size with different model (initial disturbance:tripping of one generator, the fourth-order model with 15% spinning reserve, u.f.load shedding scheme 1).

Fig. 15. Distributions of blackout size (p.u.) with different initial disturbance sizes(fourth-orders model with 15% spinning reserve, load shedding scheme 1).

Table 4Additional load shedding schemes.

Scheme no. 2 3

Step (j) 1 2 3 4 1 2 3 4 5

f 00j 0.985 0.98 0.975 0.97 0.985 0.98 0.975 0.97 0.975

f 00j 0.98 0.975 0.97 0.965 0.98 0.975 0.97 0.965 0.96

Delay time (s) 0.5 0.5 0.5 0.5 0.3 0.3 0.3 0.3 0.3Load trip (%) 9 6 5 5 9 7 7 6 6

Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696 695

We performed simulation studies at different situations usingMonte Carlo method. The power loss of each simulation is calcu-lated and analyzed. The relationship between power loss x andits cumulative distribution function F(x) is analyzed. The functionP(x) is considered:

PðxÞ ¼ 1� FðxÞ � xb ð13Þ

Taking logarithms to both sides:

Fig. 17. Distributions of blackout size (p.u.) with different u.f. load sheddingschemes (fourth-order model with 5% spinning reserve, initial disturbance: trippingof two generators).

lg PðxÞ � b lg x ð14Þ

If we draw the log–log plot of P(x) vs. x (power loss), we can ob-tain a good linear fit following Eq. (14), which means power lawdistribution [10].

Based on 50,000 simulations, distributions of the first, third andfourth-order model are illustrated in Fig. 14. Blackout size of thefourth-order model is smaller and its probability is lower thanthose of third-order and first-order model. Proportion of blackoutsto total simulation number based on the first-order, third-orderand fourth-order model are 5.34%, 0.01% and 0.004%, respectively.The effect of primary regulation is obvious.

When some generators’ primary regulations are not put in andsome generators do not have spinning reserve, the distribution ofsystem loss should be between that of the first- and fourth-orderin Fig. 14. Obviously, the shutdown of primary regulation for eco-nomic reason is not ‘‘economic”.

The above results may be used to quantify the minimumrequirement of spinning reserves that is necessary for maintainingsystem reliability. In fact, with system topology being considered,much more valuable results can be obtained.

Given different initial disturbances, their distributions are com-pared and shown in Fig. 15. Obviously, larger initial disturbance re-

Fig. 16. Distributions of blackout size (p.u.) with different spinning reserve (fourth-order model with u.f. load shedding scheme 1, initial disturbance: tripping of twogenerators).

sults in lager blackout size and higher probability, however, it haslittle influence on the value of b.

With different spinning reserve levels (5%, 15% and 25%), thelog–log plots of P(x) vs. power loss (x) are shown in Fig. 16. Withmore spinning reserves, the probability of power loss is smaller.At the same time, the benefit of committing too much spinningis not obvious.

Two additional u.f. load shedding schemes are present in Table4. The log–log plots of P(x) vs. power loss (x) with u.f. load sheddingschemes 1–3 are shown in Fig. 17. Compared to schemes 1 and 2with the same amount of load shedding but different arrangementsat each step has better effect (less probability of power loss). Com-pared to scheme 2, the probability with scheme 3 (more amount ofload shedding) is almost no different to that with scheme 2.

6. Conclusions

A simple deterministic frequency stability model has been pre-sented in this work together with its stochastic counterpart. Whilethe deterministic model captures the fundamental characteristicsof short-term frequency dynamics such as abnormal frequencytripping of generators, the stochastic model further maintains theprobability parameters of frequency dynamics. The model de-scribes frequency cascading outage and its statistic characteristiceffectively. A power law distribution of power system blackout sizeis revealed.

Acknowledgements

This work is jointly supported by theNational Basic ResearchProgram (973 Program) (Project No: 2004CB217902) and the Nat-ural Science Foundation of China (Project 50595411).

696 Y. Yu et al. / Electrical Power and Energy Systems 32 (2010) 688–696

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