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Abstract— This paper presents a study on the short-term

impact of wind generation on Loss of Load Probability, an important generation adequacy measure. Although the capacity provided by wind generation improves the generation adequacy in general, the intermittent and variable wind generation causes short-term variation of LOLP which may impose immediate risks on the system reliability.

In this study, variable wind generation is represented by an instantaneous multi-state model and the derivation of its impact on LOLP is provided. A discrete method by Markov chain is also derived so that LOLP can be computed based on metered wind output. The method is applied to calculation of LOLP based on 10-minute-interval wind generation data. Various instantaneous LOLP profiles at different initial wind levels, different wind penetration levels, and different wind output “smoothness” are presented.

This study quantifies the convergence of instantaneous LOLP and the significance of wind generation forecast in short-term LOLP estimation. The method can be extended to the assessment of short-term risks of variable wind generation in terms of generation adequacy.

Index Terms—Wind Generation, Generation Adequacy, Loss

of Load Probability, Markov Chain.

I. INTRODUCTION

he intermittent and variable wind generation in the future electric energy supply portfolio challenges the power

system reliability. One of the important issues is to understand the impact of wind generation on generation adequacy. This is because, although the capacity provided by wind generation improves the generation adequacy in general, the intermittent and variable wind generation changes the generation adequacy condition in the short-term, which may impose immediate operational risk on the system reliability.

One of the essential measures of generation adequacy is Loss of Load Probability (LOLP). LOLP is a widely used index to measure the probability of the total generation capacity that cannot meet the request of the peak load for the time horizon of interest, i.e., for any time t∈[0, T] and the peak load L during this period of T. The LOLP can be expressed as follows,

})()(Pr{)( LtCtCtLOLP WT ≤+= (1)

The authors are associated with the School of Electrical and Computer

Engineering at the University of Oklahoma. Email: jnjiang@ou.edu.

where CT(t) is a random variable representing the total generation availability offered by all conventional generation at time t, CW(t) is a random variable representing the wind generation at the same time, t.

Conventionally, the calculation of LOLP only takes into account the uncertainty caused by generation forced outage because of the low penetration level of wind generation. The parameters of the model of generation forced outages are considered time invariant so LOLP is usually estimated with some probability convolution methods at the planning stage of power system operation. Such conventional ways of estimation of system adequacy indices, such as LOLP, is no longer satisfactory because of the lack of consideration of short-term variation of wind generation.

How to incorporate the wind generation in LOLP calculation is explored in several recent studies. [1-2] used the Monte Carlo method to simulate hourly stochastic processes of generation availability, taking into account the auto-correlation and dynamics of wind speeds, the random generation failures, etc. An auto-regressive and moving average (ARMA) time series model is used in [3] to simulate the hourly wind speeds and thus the available wind power considering chronological characteristics, which requires a large amount of historical wind speed data. [4-5] presented analytical methods using multi-state models of wind speed to estimate the wind generation with the power curve at various output states.

Although the short-term impact of wind generation on LOLP has been discussed, the existing techniques mentioned above focus on the long-term reliability evaluation of power systems ranging from several months to several years, characterizing the stationary LOLP and cannot be directly used to quantify the change of LOLP due to variable wind generation. Since the short-term generation availability depends on the current and future wind generation level, it is important to develop new techniques to assess the short-term LOLP.

In this paper, a new method for calculation of short-term LOLP is presented. An instantaneous multi-state model is constructed to characterize the output of wind generation. The instantaneous state probabilities are estimated by solving differential equations in the Markov process. A discrete approximation is derived to simplify the continuous stochastic process for the calculation of the state probabilities at different moments. Finally, a study of short-term LOLP is presented

A Study of Short-Term Impact of Wind Generation on LOLP

John N. Jiang Senior Member, Chenxi Lin Student Member, Thordur Runolfsson Senior Member, IEEE

T

978-1-4244-6547-7/10/$26.00 © 2010 IEEE

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based on the proposed method with simulated wind power data.

The remainder of the paper is organized as follows: Section II introduces an instantaneous multi-state model and the corresponding instantaneous LOLP is derived. Section III presents the differential equations of a continuous Markov process to estimate the instantaneous state probabilities. The discrete approximation of the Markov process model is also derived so that the state probabilities can be estimated based on meter wind output data. In Section IV, a study based on various instantaneous LOLP profiles estimated at different initial wind levels, under different wind penetration conditions, and with different wind output “smoothness” is presented. Section V concludes the paper.

II. LOLP WITH MULTI-STATE WIND GENERATION MODEL

A. Instantaneous Multi-State Wind Generation Model Time-invariant multi-state Markov chain has been used to

study the impact of wind generation on long-term LOLP [4]. More specifically, the wind generation is assumed to have multiple output states, while conventional generation usually has only two levels: “on” for available state and “off” for failure state. The advantage of the multi-state model is that the convolution method can still be used to estimate the LOLP with wind generation.

In order to estimate the short-term impact of wind generation on LOLP, an extended multi-state model is proposed in which the transition probabilities between output states can be time variant. This instantaneous k-state wind generation model at time t is described by Equation 2.

⎪⎪⎩

⎪⎪⎨

⎧

=

− 1)(y probabilitwith

(1)y probabilitwith (0)y probabilitwith

)(

1

1

0

k-C

CC

tC

tkW

tW

tW

W

μ

μμ

M

(2)

where CWj, j = 0,…,k-1 are discretized states for wind

generation levels, while μt(j) are the corresponding probabilities for the output states at time t. Such probabilities are time varying and defined as follows,

1,...,2,1,0)],,()(Pr[)( 1 −=∈= + kjCCtCj jW

jWWtμ (3)

where CWk= CW

max, the maximum output of wind generation. We can represent μt(j) with a probability row vector, i.e. μt = [μt(0),…, μt(k-1)] to describe the distributions of various wind generation states at time t.

B. Instantaneous LOLP

The LOLP can be found through probability convolution. Let’s firstly consider the cumulative probability after adding a two-state conventional generation with capacity C. Let FOR be the generation forced outage rate, then the probabilities of available state and outage state are (1-FOR) and FOR, respectively. Thus the cumulative probability of having a generation capacity less than or equal to X after adding this

conventional generation, P(X) can be calculated as follows,

)(ˆ)1()(ˆ)( CXPFORXPFORXP −−+⋅= (4)

where )(ˆ XP is the cumulative probability for having a system capacity less than or equal to X before adding generation capacity C.

Similar to the convolution of a two-state conventional generation, a time-variant cumulative probability, Pt(X), for having a system capacity less than or equal to X after adding a k-state wind generation can be found as follows,

∑−

=

−=≤+=1

0)(ˆ)(])()(Pr[)(

k

j

jWttWTt CXPjXtCtCXP μ (5)

where μt(j) is the instantaneous state probabilities of wind generation for state j at time t, while )(ˆ XPt

is the instantaneous cumulative probability for having conventional generation less than or equal to X before adding any wind generation at time t.

Based on Equation 5, the instantaneous LOLP at time t∈[0,T] for peak load L can be expressed by,

∑∑−

=

−

=

−=−==1

0

1

0

)(ˆ)()(ˆ)()()(k

j

jWt

k

j

jWttt CLPjCLPjLPtLOLP μμ

(6)

Equation 6 assumes that the system has only two types of supplies, conventional generation (two-states) and wind generation (k-states). Since the generation forced outage rate (FOR) is usually assumed time-invariant, we can replace

)(ˆ •tP with )(ˆ •P , which can be computed iteratively based on Equation 4. Therefore, the key to compute LOLP described by Equation 6 is to find the instantaneous state probabilities, i.e., μt(j), for all j at time t.

In the study of long-term LOLP [4-5], the state probabilities in the multi-state model, which is the probabilities for wind output level, are computed by using the wind turbine power curve and the Weibull distribution of hourly wind speeds.

For instantaneous LOLP calculation, since we want to compute the instantaneous state probabilities, the steady-state methods used in the current literature cannot be used.

In the next section, a computational algorithm using the properties of Markov chain is derived for the estimation of instantaneous state probabilities of multi-state wind generation.

III. COMPUTATION OF SHORT-TERM LOLP

The properties of stochastic Markov processes, e.g. a multi-state Markov process, make it a good candidate for modeling wind generation for reliability analysis. Particularly, a multi-state Markov process describes a time varying random variable with the state probabilities and transient probability based on the information from previous states. Once appropriate levels and orders of states are selected, a Markov chain can capture the dynamics of a variety of random

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processes.

In recent studies, such as [6], a first-order Markov chain is used for generating synthetic series of wind speed with various lengths. The results shown that short-term dynamics described by Markov chain is very close to the measured values of actual wind speed: it is reported that the first-order Markov chain captures more than 90% of information based on the experiment conducted at 10 stations. In [7], the authors also proposed a method for the direct generation of synthetic time series of wind power output by Markov chain Monte Carlo. The simulation results confirmed that Markov chain is an excellent model and provide satisfactory solutions.

A Markov process for wind generation in which the transitions only happen between adjacent output levels is proposed in [8]. In this paper, in order to obtain a better representation of power generation from the wind, we further developed a more general Markov process which includes the transitions between non-adjacent states to make it even more flexible to represent the intermittency of wind generation. In addition, the underling empirical formulation for estimation of the transition probability is developed as well.

A. Derivation of Transition Probability of Wind Generation

Let wind generation, CW(t), be a homogenous Markov process with states described by Equation (2). Thus, the instantaneous state probabilities, μt(j), can be evaluated by solving the following difference equations,

1,...,2,1,0,)(])([)( 1

0

1

0−=−= ∑∑

−

≠=

−

≠=

kjjidt

jd k

jii

jit

k

jii

ijtt αμαμμ

(7)

where αij is the transition rate from state i to state j.

[9] shows that, by solving Equation 7 under initial condition μ0(j), j=1,2,…,k-1, the instantaneous state probability, μt(j), can be calculated for any time instant t∈[0, T]. The challenge is that, in order to have high computational accuracy, a large number of states may be needed result in difficulties in solving Equation 7.

Here we develop a discrete approximation so that Equation 7 can be solved easily without compromising much accuracy. Let Δt be a very small time step compared to the minimum residence time in all states of wind generation. Then the time period of interest, [0,T], can be divided to N intervals with equal length of Δt. The first and the last intervals are [0, Δt] and [(N-1)Δt, NΔt], and NΔt =T. During each small time interval of Δt, the state probability, μt(j), is assumed constant. Let )(ˆ jnμ be the probability that wind generation is in j state during the (n+1)th time interval, and )()(ˆ jj TN μμ = . Since Δt is very small, Equation 7 can be approximated with the Euler formula,

1,...,2,1,0,)()()( −=Δ

−= Δ− kjt

jjdt

jd tttt μμμ

Thus, a discrete approximation of the Markov model can be obtained for interval n = 0,1,…,N-1,

1

1 10

ˆ ˆ ˆ( ) [ ( ) ] [ ( ) ], 0,1, 2,..., 1k

n n ij n jjii j

j i p j p j kμ μ μ−

− −=≠

= + = −∑ (8)

where pij is the transition probability defined by, 1

0

1 , for

, for

k

ijj

ij j i

ij

t i jp

t i j

α

α

−

=≠

⎧ − Δ =⎪⎪= ⎨⎪ Δ ≠⎪⎩

∑ (9)

Letnμ̂ be the row vector representation of state

probabilities )(ˆ jnμ , j=1,2,…,k-1. The state distribution at the nth step of the Markov chain can be characterized by,

1 0ˆ ˆ ˆ... nn nμ μ μ−= ⋅ = = ⋅P P (10)

where P is a k×k stochastic transition matrix with entries [P]ij=pij.

With the results developed above, once the initial distribution 0μ̂ and the transition probability matrix P is given, the state

distribution of the Markov Chain for any step can be found.

B. Formula for Empirical Estimation of Markov Distributions

In this section, formulas for estimation of distributions of the Markov chain of wind generation will be developed so that we can conduct an empirical study on the short-term LOLP with actual metered data of wind generation.

Note that the transition rate between two states describes the frequency of occurrence of a transition per unit time independent from the time step Δt. For a metered wind generation in time period Tt, the maximum likelihood estimator for the transition rate from state i to state j is given by,

t

ijij T

n=α (11)

where nij is the number of transitions of samples from state i to state j observed in the time period of Tt. For instance, given a wind power time series {Wt,t=0,1,..,N} in time interval ΔT , the transition rate αij can be estimated by,

NTCCWCCWt j

Wj

WtiW

iWt

ij ⋅Δ∈∈=

++

+ )},(),,(:{ ofnumber the 11

1

α (12)

Since the time period of interest is divided into N intervals with equal length of Δt, there must exist a unique n∈{0,1,…,N-1} with which the corresponding row vector of state probabilities,

nμ̂ , that can characterize the instantaneous state distribution, μt for any t∈[0, T]. In other words, we have,

0ˆ ˆtt

t tt

μ μ μ⎢ ⎥⎢ ⎥Δ⎣ ⎦

⎢ ⎥⎢ ⎥Δ⎣ ⎦

= = ⋅ P (13)

where “ ⎣ ⎦• ” is the floor function mapping a real number to the

4

next smallest integer.

Equation 12 enables us to compute the transition matrix P with complexity reduction, further to estimate the state probability at different steps by Equation 10.

C. Empirical Formula for Estimation of Short-Term LOLP

Once the instantaneous state probabilities of wind generation are known, the short-term LOLP can be found. Consider the instantaneous multi-state model of wind generation, LOLP expressed by Equation 6, which can be written as follows,

∑−

= ⎥⎦⎥

⎢⎣⎢

Δ

−==1

0

)(ˆ)(ˆ)()(k

j

jW

ttt CLPjLPtLOLP μ

(14)

Note that the row vector of state distribution of wind generation

⎣ ⎦•μ̂ can be estimated with Equation 13, and the

cumulative probability, )(ˆ •P , for conventional generation can be found by Equation 4. Therefore, once the time step Δt is determined, the stochastic transition matrix can be obtained by Equation 9, and short-term LOLP during the time period [0,T] for a given initial wind generation condition can be obtained by Equation 14.

As t →∞, it can be proved that the state distribution ⎣ ⎦•μ̂ will

converge to the stationary distribution π of the Markov chain. The corresponding steady state LOLP is given by,

∑−

=

−=∞1

0

)(ˆ)()(k

j

jWCLPjLOLP π (15)

The steady state LOLP can be used for long-term generation adequacy analysis.

In the next section, we will study the characteristics of short-term LOLP based on the results of the above theoretical analysis.

IV. STUDY ON SHORT-TERM LOLP

In this section, four case studies on short-term LOLP are presented. Case 1 only considers conventional generation in estimation of short-term LOLP. Case 2 and Case 3 study short-term LOLP when wind generation is incorporated. The wind generation profile in Case 3 is “smoother” than that in Case 2. Case 4 studies the impact of wind penetration level on short-term LOLP.

Case 1: The short-term LOLP of conventional generation is a constant because FOR of generation is time-invariant in adequacy analysis.

Assume that the power system has 6 conventional generating units, each with 250 MW of capacity and an FOR of 0.08. The convolution algorithm, Equation 4, is applied to the calculation LOLP by adding one generator at a time. The results are shown in Table I.

If the peak load L is assumed to be 1000 MW, the LOLP with

conventional generation is only 0.077, and it won’t change over time.

TABLE I. SHORT-TERM LOLP WITH CONVENTIONAL GENERATION Available Gen. Forced Outage Gen. Prob.

1500 MW 0 MW 1.000 1250 MW 250 MW 0.394 1000 MW 500 MW 0.077 750 MW 750 MW 0.009 500 MW 1000 MW 0.001 250 MW 1250 MW 0.000 0 MW 1500 MW 0.000

Case 2: The short-term LOLP will converge to its steady state level in the long run. The profile of short-term LOLP depends on the initial wind generation conditions: with low initial wind generation, the LOLP gradually reduces to its long-term level, while increasing to its long-term value with high initial wind generation. Figure 2 shows the LOLP profile calculated with equations derived from previous sections.

In Case 2, 500MW wind generation is added to the conventional generation described in Case 1. The wind generation consists of 100 5MW turbines. The wind generation is simulated based on a synthetic wind speed model proposed in [10] and a simple cubic wind generation power curve. The power curve for one wind turbine is defined as follows,

3

0 ,( / )

t ci t co

t t t ci t

t co

w w w wP P w w w w w

P w w wτ τ

τ τ

< ≥⎧⎪= ⋅ ≤ <⎨⎪ ≤ <⎩

(16)

where Pt is the wind generation at time t, wt is wind speed at time t, wτ and Pτ are the rated speed and power of the wind turbine, wci and wco are the cut-in wind speed and cut-out wind speed, respectively. In the study, we choose wci=3m/s, wτ=14m/s, wco=25m/s. The generated wind power time series is shown in Figure 1 (a). Figure 1 (b) is a smoother wind speed profile for Case Study 3. Once the simulated time series of wind speed and the power curve are given, the time series of wind generation can be obtained.

5

0 50 100 150 200 250 300 350 400 450 5000

100

200

300

400

500Wind power time series (rough)

(a)

Pow

er o

utpu

t (M

W)

0 50 100 150 200 250 300 350 400 450 5000

100

200

300

400

500

(b)

Pow

er o

utpu

t (M

W) Wind power time series (smooth)

Fig. 1. Time Series of Wind Generation

In this study, the wind generation is modeled with 10 state with an equal increment of 50MW, i.e., CW

0=0, CW1=50MW,…,

CW9=450MW. The time step Δt=1 minute is chosen. Based on

the formula provided in previous sections, the transition rates between different wind generation states can be found based on the simulated time series of wind power. The corresponding 10×10 transition matrix P can also be obtained.

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=Ρ

8671.00712.00092.00054.01649.06476.00157.00087.0

0009.00004.05618.02081.00002.00001.00980.07829.0

L

L

MMOMM

L

For the initial state, three different initial distribution profiles are selected to represent low wind speed, medium wind speed and high wind speed situations with state probability vectors

I0μ̂ =[1,0,…,0], II

0μ̂ =[0,…,1,…0] and I0μ̂ =[0,0,…,0,1],

respectively. The time horizon of interest, T, is set to be 6 hours. The peak load and the forced outage rates of conventional generation are the same as those in Case 1.

0 5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time/10-min

LOLP

(t)

LOLP for rough wind power process

low speedmedian speedhigh speed

Fig. 2. Short-Term LOLP Profile

As shown in Figure 2, all instant state distributions with different initial probabilities converge to their stationary

distributions, and the steady state LOLP, or LOLP(∞), is equal to 0.0543. Note that time period T=6h is quite long since the short-term LOLP converges to the steady state level in 2 hours as shown in Figure 2.

Case 3: “Smoother” wind generation slows down the convergence speed of LOLP to its steady state level. For a smoother wind generation, longer horizon is needed to estimate the short-term LOLP.

In Case 3, a smoother time series of wind generation shown in Figure 1 (b) is used, and its Markov transition matrix is,

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=Ρ

9961.00005.00003.00001.00024.09253.00006.00003.0

0007.00002.08721.00087.00003.00006.00103.09045.0

L

L

MMOMM

L

0 5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time/10-min

LOLP

(t)

LOLP for smooth wind power process

low speedmedian speedhigh speed

Fig. 3. Short-term LOLP profile with Smoother Wind Generation

The initial distributions of wind generation for Case Study 3 are the same as those in Case 2. Comparing to Case 2, the second largest eigenvalue λ*=0.84, while in Case 2 it is 0.46. The increase in the second largest eigenvalue indicates that the short-term LOLP will converge to its steady state level at a slower speed, as shown in Figure 3.

It is also shown that with a smoother wind generation, the steady state LOLP is improved due to the fact that the wind generation in Case 3 remains in the high capacity level for longer time than that in Case 2.

Case 4: The profile of short-term LOLP with lower penetration of wind generation is presented.

In Case 4, the short-term LOLP profile of lower penetration with the same wind source as Case 2 but reduced number of turbines is shown in Figure 4. The maximum capacity of wind generation is 150 MW. Comparing to Case 2 and Case 3 with 25% wind penetration level, the penetration level in Case 4 is less than 10%. The ratio of changes in the steady state LOLP is 29.7%, 49.5% for Case 2 and Case 3, while the ratio of capacity change is the same 33.3%. Therefore, adding 300MW wind capacity brings more long-term LOLP benefit in Case 2 than that in Case 3, due to the “smoother” wind generation profile.

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0 5 10 15 20 25 30 350

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time/10-min

LOLP

(t)

LOLP for low wind power penetration level

low speedmedian speedhigh speed

Fig. 4. Short-term LOLP with Low Wind Penetration

It is shown in Figure 4 that there are a few improvements in the steady state LOLP compared to the results in Case 1. It is also shown that wind penetration affects the short-term LOLP profile. We will confirm these observations in our future research.

V. CONCLUSIONS AND FUTURE RESEARCH

This paper presents a study on the short-term impact of wind generation on LOLP. A number of formulas for calculation of instantaneous LOLP are derived. In this study, variable wind generation is represented by a multi-state Markov Chain and the derivation of its impact on LOLP is provided. A discrete method is also derived so that LOLP can be computed based on metered wind output. Based on various instantaneous LOLP profiles at different initial wind levels, different wind penetration levels, and different wind output “smoothness”, we show that the quantitative methods developed can improve our understanding of the impact of wind generation on short-term LOLP. The study affirms the significance of wind generation forecast on short-term generation adequacy. The method can also be extended to the assessment of short risks of variable wind generation in terms of generation adequacy.

One of the future research projects is to select the optimal time period for calculation of short-term LOLP. Since matrix P is a stochastic matrix, there always exists a stationary distribution π, which is the left eigenvector of the transition matrix associated with the eigenvalue 1; i.e. π=πP. Moreover, the convergence rate is determined by the second largest absolute eigenvalue of the transition matrix. More specifically, if the second largest absolute eigenvalue is small, the short-term LOLP for different initial wind power will quickly converge to the steady state LOLP. Hence, choosing a suitable time period [0,T] is important for calculation of the short-term LOLP to better describe its impact on generation adequacy.

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John N. Jiang (SM’05) was born in Beijing, China. He currently is an assistant professor in the School of Electrical and Computer Engineering at the University of Oklahoma. He obtained his B.S. and M.E. degrees in the power engineering from Tsinghua University and the Chinese Academy of Sciences respectively, in China. He received his M.S. and Ph.D. degrees in power engineering, economics and finance area from the University of Texas at Austin. His current research interest is risk analysis in renewable energy, power system operation and electric energy markets. Chenxi Lin (S’09) was born in Fujian, China. He received his B.S. degree in Electrical Engineering in 2006 from Xiamen University in China. He is presently a Ph.D. candidate in the School of Electrical and Computer Engineering at the University of Oklahoma. His research interest is in control systems, uncertainty analysis, power system operation and reliability, and power markets. Thordur Runolfsson (S’87, M’89, SM’00) is a native of Iceland. He received his BS in Electrical and Computer Engineering and Mathematics at the University of Wisconsin in 1983 and his Ph.D. in EE-Systems from the University of Michigan, Ann Arbor in 1988. He joined the School of Electrical and Computer Engineering at the University of Oklahoma in 2003 as a Professor. Dr. Runolfsson’s research interests include systems and control, complex dynamical systems, uncertainty analysis, modeling and simulation, power systems and energy efficiency .