On the frequency stability of power systems and its statistic characteristics Y. Yu, H. Wu, H. Xin, D. Gan, Z. Han

College of Electrical Engineering Zhejiang University, Hangzhou, China

yuyangzju@gmail.com, vuhao@cee.zju.edu.cn, eexinhh@gmail.com, deqiang.gan@ieee.org, hanzx@zju.edu.cn

Abstract - A simple deterministic frequency stability model is developed in this work together with its stochastic counterpart. While the deterministic model captures the fundamental characteristics of short-term frequency dynamics such as abnormal frequency tripping of generators, the stochastic model further maintains the probabilistic characteristics of frequency dynamics, such as stochastic characteristics of frequency relays. The models are particularly suitable for studying system cascading failures introduced by frequency distortions. Our simulation results reveal that the relationship between blackout scale and frequency has the characteristics of power law.

Keywords: Frequency stability; Frequency protection; Load shedding; Monte Carlo simulation; Cascading outage

1. INTRODUCTION

Frequency is one of the most important indicators of power system’s operation status. It reflects the active power balance between generation and load demand in power system. If generation meets 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 and transformer with large frequency deviation can result in malfunctions due to overheating or mechanical overstress [1]. Therefore, measures must be taken to stabilize frequency in acceptable range.

After occurrence of imbalance of generation and load the primary frequency regulation will react to frequency deviation immediately to try to keep frequency within required range. If the frequency deviation still can not meet system requirement, spinning reserves and rapid start generators will react subsequently by increasing or decreasing generation in order to restore frequency to nominal value [2-3]. Normally, frequency will be restored after these measures are applied.

However, if the response rate of reserves is too slow to take effect or available reserve capacity is not enough after a large imbalance occurs, underfrequency (u.f.) load-shedding relays may be triggered according to their thresholds and time delays. Similarly, generators may be tripped if frequency exceeds its operation limitation. When frequency does not satisfy system requirement after the first round of u.f. load shedding, more loads shedding will be further carried out. During operation within abnormal frequency range, generators will be tripped off with delays if any of its cumulative operation time limitations is used up. The cooperation of u.f. load scheme and generators’ frequency protection scheme in modern power system is not always effective. Thus there exists a potential risk of cascading failure, even blackout, especially when a system transfers a great amount of

power over long distance or owns some generators that have very large capacity and heavily loaded. The loss of the main transmission lines or generators could introduce large deficiency of active power in receiving power system and hence may initiate a rapid frequency drop in a short period of time which increases the probability of cascading outage.

Power system blackouts recently occurred in North American and Europe have attracted much attention [4-6]. A fundamental form of these blackouts is cascading outages of system facilities [4-7]. Analysis of NERC disturbances records from 1984 to 1998 reveals that the probability of disturbance scale exhibits a heavy tail and power law distribution [7-9], which is a clear indication of the high probability of blackout. This motivates a new research topic that tries to uncover the mechanism behind blackout. Different models, such as Hidden Failures, CASCADE and OPA have been proposed. All these simulation results verified that the probability distribution of blackout size of modern power system has similar stochastic characteristic [7, 9-11]. However, the above results are mostly based on DC load flow models or uniform network models and they do not take into account the impact of frequency which plays a key role in cascading outages.

This work presents deterministic models and their stochastic counterparts for cascading failure investigation. Stochastic factors associated with generator frequency relay, u.f. load shedding and breaker tripping time, etc. are explicitly modeled. Using a Monte Carlo simulation tool, the risk of frequency-induced cascading failures is assessed. The influences of relevant factors on statistic characteristic of power system cascading failures, such as frequency primary regulation, AGC (spinning reserve), the size of initial disturbance and u.f. load shedding schemes, are also analyzed. Though the results presented are preliminary and no network information is yet used, they already demonstrate interesting values in identifying the factors that contribute to cascading failures and blackouts.

2. A DETERMINISTIC MODEL FOR FREQUENCY DYNAMICS

In most of the recent blackouts in the world, the time period from the occurrence of dominating disturbance to the complete blackout is usually in order of minutes and involves several stability problems, such as frequency stability, voltage stability and transient stability. Compared with the others, frequency stability is a global and crucial issue in blackout evolution period. Without system frequency stability, reactive power balance and voltage profile can not be held and the overall system will crash quickly. Fig.1 shows the frequency variation

16th PSCC, Glasgow, Scotland, July 14-18, 2008 Page 1

curve in Italy after the disconnection of Italy from the UCTE grid at 9-28-2003 [5]. Because of under frequency, massive generators tripping and loads shedding took place during only 2.5 minutes and eventually this lead to blackout. This illustrates that it is necessary to study the relationship between short-term frequency stability and cascade outage.

Figure 1: Frequency variation of Italy power system at 9-28-2003

Compared to the launch times of rapid start units and hot reserve units, the frequency dynamics of modern power systems is shorter, thereby we mainly concern the frequency process with spinning reserve. In addition, generator and load restoration are neglected because these operations are taken after frequency is stabilized. In some cases, the primary regulation is not put in for economic reasons [4]. The impact of such “economic solution” is also considered by assigning null primary regulation capability to all generators. Obviously, this is the worst situation in frequency dynamic progress.

Power system frequency characteristics can be modeled by a single generating unit of non reheat type with good approximation [12-14]. For the sake of conciseness, the equivalent generator and loads in power system are assumed to be ideally connected to a common bus. Since the network is neglected, overloads on transmission line are neglected in this work.

2.1. First-order frequency dynamics model

Figure 2: First-order system frequency response model

A linear model without considering primary regulation is shown in Fig. 2. The corresponding differential equations can be written as:

ePd Ddt M Mω ω

ΔΔ= − Δ − (1)

where ωΔ : speed deviation from nominal value

ePΔ : deviation in electrical power M : angular momentum of the equivalent generator D : damping constant which characterizes damping

factor arising from both generator and load Given initial condition ( 0ωΔ ), the analytical solution

to eq.(1) is:

0( )D te e MP P

eD D

ω ω−Δ Δ

Δ = − + Δ + (2)

Generally, ωΔ is considered to be stable after 3 5τ τ∼ , where time constant M Dτ = .

2.2. Third-order frequency dynamics model without AGC

Figure 3: Third-order system frequency response model

The third-order model with fixed load reference set point is shown in Fig. 3. It can simulate the frequency dynamic response during the first a few seconds after a large disturbance [12-14]. A set of differential equations can be constructed below:

1 0

1 100

1 1 00

e

mm

CH CHv

v

G G

DdM M Pdt

Md P PT TdtPd P

T R Tdt

ω

ω

⎡ ⎤Δ⎡ ⎤ −⎢ ⎥⎢ ⎥ Δ⎡ ⎤⎢ ⎥ Δ −⎡ ⎤⎢ ⎥ ⎢ ⎥Δ ⎢ ⎥ ⎢ ⎥⎢ ⎥ − ⎢ ⎥= Δ +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ⎢ ⎥ ⎣ ⎦ ⎢ ⎥Δ ⎢ ⎥ ⎣ ⎦⎢ ⎥ − −⎢ ⎥⎢ ⎥ ⋅⎣ ⎦ ⎣ ⎦

(3)

where mPΔ : deviation in mechanical power

vPΔ : per unit change in valve position from nominal R : speed regulation

CHT : “charging time” time constant

GT : main servo time constant The meanings of the other variables are the same as

those in section 2.1.

2.3. Fourth-order frequency dynamics mode

Figure 4: Fourth-order system frequency response model with AGC

Spinning reserve is taken into account in the fourth-order model. Fig. 4 shows its block diagram, where k is a constant and B is called frequency bias factor [2]. The value of B can be calculated as follow:

1B D R= + (4) The fourth-order differential equations are:

16th PSCC, Glasgow, Scotland, July 14-18, 2008 Page 2

1 0 0

1 10 00

1 1 1 000

0 0 0

em

CH CH m

vv

G G G LF

LF

DdM Mdt P

d P MT T PdtPd P

T R T Tdt Pd P B

dt k

ω

ω

⎡ ⎤Δ⎡ ⎤ −⎢ ⎥⎢ ⎥⎢ ⎥ Δ⎡ ⎤⎢ ⎥ Δ −⎡ ⎤Δ ⎢ ⎥ ⎢ ⎥⎢ ⎥ − ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ Δ⎢ ⎥⎢ ⎥= + ⎢ ⎥⎢ ⎥ ⎢ ⎥ΔΔ ⎢ ⎥ ⎢ ⎥⎢ ⎥ − − − ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⋅ Δ⎣ ⎦⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦Δ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(5)

where LFPΔ : deviation in load reference. The meanings of the other variables are the same as those in section 2.1 and 2.2.

Although the models (1), (3) and (5) are rather standard, more sophisticated models could be used without introducing conceptual difficulty in the proposed frequency stability analysis method.

2.4. The models for unit tripping and u.f. load shedding After the occurrence of a disturbance ePΔ , system

frequency f would deviate from the rated value. Suppose unit i is operating at output level ig (MW), the unit would be tripped if the system frequency drifts out of the interval ,i if f⎡ ⎤′ ′⎣ ⎦ within certain time. Similarly, load j

absorbing power at level jl (MW) would be automatically shed if system frequency is lower than jf ′′ with certain

time. Otherwise, no components would be disconnected. Given a power system with NG generators and NL

loads, the total generation G and load L can be computed as:

1 1,

NG NL

i i j ji j

G p g L q l= =

= =∑ ∑ (6)

where

0 ,

1 ,

i i

i

i i

f f fp

f f f

⎧ ⎡ ⎤′ ′∉⎪ ⎣ ⎦= ⎨⎡ ⎤′ ′∈⎪ ⎣ ⎦⎩

(7)

1

0

jj

j

f fq

f f

⎧ ′′≥⎪= ⎨′′⎪ <⎩

(8)

Given an initial disturbance eP G LΔ = − , the

frequency dynamic progress ( f̂ ) can be computed by

solving the differential eq.(1), (3) or (5). If f̂ meets the following conditions:

1

ˆ ,

ˆ 1,2,max( ),

NG

i ii

j

f f f

j NLf f=

⎧ ⎡ ⎤′ ′∈⎪ ⎣ ⎦⎨⎪ ′′ =>⎩

∩ (9)

no generator and load would be disconnected, otherwise, generators tripping or load shedding would happen.

After the ignition of the first round of load-shedding or generator tripping, the new system frequency dynamic progress can be computed. Re-evaluating eq.(9) again would allow one to find out if another round of load-shedding or generator tripping would need to be carried out. Repeating the above steps until the system

frequency is restored to normal range, we can get the amount of overall load-shedding. If frequency runs under a critical threshold (such as 47.5Hz [1]), frequency collapses and the system under study is blackout. The former situation corresponds to a stable case while the latter is regarded as a system blackout hereafter.

The amount of total load-shedding and unit tripping can be obtained after finishing the above simulation. The model also serves as a basis for quantifying the risk of cascading failures introduced by frequency distortions.

3. STOCHASTIC FACTORS AFFECTING FREQUENCY DYNAMICS

During the frequency dynamic progress, the frequency relay settings of generators are typically set by manufactures and plant operators thus are stochastic in nature. Some stochastic factors, such as generator relay tripping [15], u.f. load shedding relay and time of breaker tripping are modeled respectively. 3.1. Random behavior of generator frequency relays

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

1) Continuous operation range: Generators can operate continuously without time restriction.

2) Restricted time operation range: Continuous and accumulative operation times of generators are both restricted. Generators should be tripped if the time limits are used up.

3) Prohibited operation range: Generators should be tripped at once.

During most of time, system frequency stays in the continuous operation range and the tripping probability of generators has no relationship with system frequency but is related to fault rate of generator and its frequency relay, which can be obtained from historical statistical data. During frequency dynamic progress, generator may run outside of continuous range and its accumulative restricted operation time may reach to the threshold. Therefore, some generators may be tripped by relays in restricted operation range and some don’t even at the same frequency. As the distance from continuous operation range increases, the tripping probability of frequency protection increases. Furthermore, relays have measurement errors and can have design flaws in nature [15-16].

Figure 5: Tripping probability of generator frequency relay as a function of system frequency

2f ′0f 1f ′ if ′ if ′

1

gi giprob prob ′+

Trip

ping

pro

babi

lity

of u

.f. re

lay

fo

r gen

erat

or

Frequency (Hz)

16th PSCC, Glasgow, Scotland, July 14-18, 2008 Page 3

The simplified stochastic characteristic of generator frequency protection is shown in Fig. 5. where

0f : the nominal frequency.

giprob : fault rate of generator i .

giprob ′ : fault rate of frequency relay of generator i .

1 2[ , ]f f′ ′ : continuous operation range.

1 2( , ) ( , )i if f f f′ ′ ′ ′∪ : restricted time operation range of generator i .

( , )i if f′ ′ : prohibited operation range of generator i . According to Fig. 5, the tripping probability of

generator i frequency relay can be written as eq.(10):

( )

( )1 2

22

2

11

1

,

(1 )( ),

(1 )( ),

1 ,

gi gi

gi gii

ii

gi gii

i

i i

prob prob f f f

prob prob f ff f f

f fp f

prob prob f ff f f

f f

f f f

⎧ ′ ′ ′+ ∈⎪⎪ ′ ′− − −⎪ ⎡ ′ ′⎤∈ ⎣ ⎦⎪ ′ ′−⎪= ⎨

′ ′− − −⎪⎡ ⎤′ ′∈⎪ ⎣ ⎦′ ′−⎪

⎪ ⎡ ⎤′ ′∉⎪ ⎣ ⎦⎩

(10)

3.2. Random behavior of u.f. load shedding relays When the system frequency is lower than a pre-set

threshold jf ′′ , step j of u.f. load shedding should act to disconnect loads.

Similar to generator frequency relays, the tripping probability of u.f. relays increase while frequency approaches to its threshold value.

Fig. 6 illustrates the probabilistic characteristic of such relays. According to Fig. 6, the tripping probability of u.f. load shedding relay j can be written as eq.(11):

( )(1 )( )

,

1

lj j

lj jj j j

j j

j

prob f f

prob f fp f f f f

f f

f f

⎧ ′′>⎪

′′⎪ − −⎪ ⎡ ⎤′′′′= ∈⎨ ⎣ ⎦′′′′ −⎪⎪ ′′<⎪⎩

(11)

where ljprob : fault rate of u.f. relay j .

jf f ′′> : continuous operation range of u.f. relay j .

jf ′′ : setting of u.f. relay j

Actually, in each step load shedding is implemented at different locations separately. To simulate the phenomenon, the load to be shed is divided into equal portions which are shed by different relays. According to eq.(11), some u.f. relays trip, the others may not at the same frequency with the same relay setting.

3.3. Random behavior of breaker’s tripping time There exists time delay ( 1T ) [16] due to measurement

error of frequency relay. It also needs time ( 2T ) [17] to

Figure 6: Tripping probability of u.f. load shedding relay as a function of system frequency

disconnect component from power system by breaker. The tripping time T ( 1 2T T T= + ) of component is a random variable [17]. We assume in this work that tripping time follows normal distribution. Mathematically, this is equivalent to:

2( , )T N Tμ σ∼ (12)

where Tμ and σ are the mean and standard deviation of T , respectively.

4. A MONTE CARLO SIMULATION ALGORITHM

Monte Carlo method is widely used to simulate power system operation and calculate its probabilistic characteristics. Here, it is used to simulate frequency dynamic progress and calculate the probability of total generation loss after the occurrence of a random imbalance between load and generation.

Before the simulation starts, the initial disturbance needs to be determined, thus the simulation comprises of two parts: the Main and the Subroutine. The algorithm is described briefly below.

Main: M1 Set the maximum simulation number: maxN ;

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

maxN loops are completed.

Subroutine: S1 Modify model’s parameters, calculate

initial conditions of eq.(1), (3) or (5); S2 Solve eq.(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 eq.(10) and (11). Calculate the probability and their tripping time according to eq.(12).

S4 If no more generator trips or load sheds or frequency reaches the collapse threshold, return to Main.

S5 Return to S1.

jf ′′

1

ljprob

0fjf ′′ Frequency (Hz)

Trip

ping

pro

babi

lity

of u

.f. lo

ad

shed

ding

rela

y

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5. SIMULATION RESULTS AND ANALYSIS

5.1. Introduction of the simulation system A test system is used for testing our developed method.

The system generation data are shown in Table 1 [14]. It is worth mentioning that in 2001 a blackout occurred in this system after initial tripping of two generators and frequency collapsed [14].

Unit No. ( i )

Nominal power (MW)

Failure rate (occ./year)

M (s)

R (Hz/MW)

CHT(s)

GT(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 0.07 8 11.5 5.6 5.0 0.10 1.014 0.07 9 11.5 5.4 5.0 0.07 1.014 0.07 10 11.5 5.7 5.0 0.07 1.014 0.07 11 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 1: Test system generation date

The test system’s base frequency is 50Hz. The Base Capacity of the test system is set to online capacity (1 p.u.). The damping constant ( D ) is set to 1.5. The number of maximal simulation times ( maxN ) is set to 50,000.

5.2. Settings of frequency relays Although generators of the test system have different

frequency characteristic, different limitations in accumulation time and frequency of occurrence within different frequency ranges, the allowable frequency operation ranges (continuous and restricted) of generators are quite similar [1]. For simplicity, generators of the test system have the same continuous and restricted frequency operation ranges. According to IEC60034-3:1996, the restricted frequency operation range is set to ( ) ( )0.95,0.98 1.02,1.03∩ and the continues and prohibited operation rang are set to [0.98,1.02] and ( )0.95,1.03 respectively [1]. The relay parameters are listed in Table 2. In the fourth-order model, spinning reserve is considered and is assigned to each generator equally.

if ′ 1f ′ 0f 2f ′ if ′

0.95 0.98 1 1.02 1.03 Table 2: Parameters of generation frequency operation rang

Different systems have different schemes of u.f. load shedding [1, 12, 18]. For simplicity, a simple load shedding scheme is adopted (Table 3). The setting of the

first round u.f. load shedding is set at 0.98 and the one of the last round is 0.96. The total amount of load shedding is 0.25 p.u..

Step ( j ) 1 2 3 4 5

jf ′′ (p.u.) 0.985 0.98 0.975 0.97 0.975

jf ′′ (p.u.) 0.98 0.975 0.97 0.965 0.96 Delay time (s) 0.5 0.5 0.5 0.5 0.5

Load dropped (%) 5 5 5 5 5

Table 3: Load shedding scheme (No. 1)

In the numerical example, the failure rate giprob of each generator frequency relay in Fig. 5 and ljprob of each u.f. relay in Fig. 6 is set to 3 occ./year.

In consideration of relay measurement delay and breaker tripping delay, Tμ is set to 0.25s and σ to 0.01s.

5.3. Simulation analysis Fig. 7 shows a typical frequency dynamic progress of

the first-order model. After an initial disturbance (unit No. 13 tripping), frequency descends rapidly and the first step frequency setting (0.98 p.u.) is activated at 2.952s with 0.045p.u. load shedding because of a u.f. relay fault at a location. At 4.8998s, another generator (No. 15) trips because its time limitation in restrictive range is used up. After the generator trips, step 2 and step 3 of u.f. load shedding are triggered respectively at 5.489s and 5.907s with total 10 percent of load shedding. Eventually, frequency restores to nominal range without any more generators tripping and load shedding. During the dynamic process, 0.1572 p.u. generation and 0.145 p.u. load are disconnected.

Figure 7: A typical frequency dynamics of the first-order model (initial disturbance: tripping of one generator)

Fig. 8 shows a typical frequency dynamics of the third-order model. After an initial disturbances (unit No. 16 tripping), frequency descends. Sequentially, another unit (No. 14) trips at 6.307s for unit fault and frequency descends again. The first step of u.f. load shedding is triggered at 7.792s with 5 percent of loads shedding. Eventually, frequency meets eq.(9) and system frequency arrives at a new stable value. During the dynamic period, 0.1927 p.u. generation capacity and 0.05 p.u. load are shed.

16th PSCC, Glasgow, Scotland, July 14-18, 2008 Page 5

Figure 8: A typical frequency dynamic simulation of the third-orders model (one generator initial disturbance)

Fig. 9 shows a typical frequency dynamic process of the fourth-order model with 15% spinning reserve. After an initial disturbance (unit No.17 tripping), a unit (No. 13) trips at 7.1642s for relay fault during frequency restoration. After the two units tripped, the first step of u.f. load shedding is triggered at 10.719s. Eventually, frequency reaches to 0.9975p.u. without any more components tripped or shed. During the dynamic progress, system loses 0.1898 p.u. generation capacity and 0.05 p.u. Comparing Fig. 9 to Fig. 8, final frequency is closer to rated value due to the presence of spinning reserves.

Figure 9: A typical frequency curve of the fourth-orders model (one generator initial disturbance, 15% spinning reserve)

We performed simulation studies at different situations using Monte Carlo method. The power loss of each simulation is calculated and analyzed. The relationship between power loss x and its Cumulative Distribution Function ( )F x is analyzed. The function

( )P x is considered: ( ) 1 ( ) ~P x F x xβ= − (13)

Taking logarithms to both sides: lg ( ) ~ lgP x xβ (14)

This means power law distribution as reported in the literature [9].

Figure 10: 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)

Based on 50,000 simulations, distributions of the first, third and fourth order model are illustrated in Fig. 10. Blackout size of the fourth-order model is smaller and its probability is lower than those of third-order and first-order model. Proportion of blackouts to total simulation number based on the first-order, third-order and 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 and some generators do not own spinning reserve, the distribution of system should be between that of the first- and fourth-order in Fig.10. Obviously, the shutdown of primary regulation for economic reason is not economic.

Figure 11: Distributions of blackout size (p.u.) with different initial disturbance sizes (fourth-orders model with 15% spinning reserve, load shedding scheme 1)

Given different initial disturbances, their distributions are shown in Fig. 11. Obviously, larger initial disturbance results in lager blackout size and higher probability, however, it has little influence on the value of β .

With different spinning reserve levels (5%, 15% and 25%), the log-log plots of ( )P x vs. power loss ( x ) are shown in Fig. 12. With more spinning reserves, the probability of power loss is smaller. At the same time, the benefit of committing too much spinning is not obvious.

16th PSCC, Glasgow, Scotland, July 14-18, 2008 Page 6

Figure 12: Distributions of blackout size (p.u.) with different spinning reserve (fourth-order model with u.f. load shedding scheme 1, initial disturbance: tripping of two generators).

Scheme No. 2 3 Step ( j ) 1 2 3 4 1 2 3 4 5

jf ′′ 0.985 0.98 0.975 0.97 0.985 0.98 0.975 0.97 0.975

jf ′′ 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%

Table 4: Additional load shedding schemes

Two additional u.f. load shedding schemes are present in Table 4. The log-log plots of ( )P x vs. power loss ( x ) with u.f. load shedding scheme 1-3 are shown in Fig. 13. Compared to scheme 1, scheme 2 with the same amount of load shedding but different arrangements at each step has better effect (less probability of power loss). Compared to scheme 2, the probability with scheme 3 (more amount of load shedding) is almost no different to that with scheme 2.

Figure 13: Distributions of blackout size (p.u.) with different u.f. load shedding schemes (fourth-order model with 5% spinning reserve, initial disturbance: tripping of two generators)

6. CONCLUSIONS A simple deterministic frequency stability model has

been presented in this work together with its stochastic counterpart. While the deterministic model captures the fundamental characteristics of short-term frequency dynamics such as abnormal frequency tripping of generators, the stochastic model further maintains the

probability parameters of frequency dynamics. The model describes frequency cascading outage and its statistic characteristic effectively. A power law distribution of power system blackout size is revealed. The system topology, which plays an important role during cascading failures, will be considered in our future work.

7. ACKNOWLEDGEMENT This work is jointly supported by the National Basic

Research Program (973 Program) (project number: 2004CB217902), the Natural Science Foundation of China (Project 50595411) and China Postdoctoral Science Foundation (20070420224)..

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