978-1-4244-5721-2/10/$26.00 ©2010 IEEE PMAPS 2010

Non-sequential Simulation Methods for Reliability Analysis of Power Systems with Photovoltaic

Generation Zhen Shu

Electrical and Computer Engineering National University of Singapore

21 Lower Kent Ridge Road, 119077, Singapore. eleshuz@nus.edu.sg

Panida Jirutitijaroen Electrical and Computer Engineering

National University of Singapore 21 Lower Kent Ridge Road, 119077, Singapore.

elejp@nus.edu.sg

Abstract—This paper investigates non-sequential simulation methods for single-area electric power system reliability evaluation with time dependent sources such as photovoltaic (PV) generation. Three random sampling techniques are proposed, considering the system load and PV generation as correlated random variables. Reliability indices such as expected loss of load and loss of load probability are estimated from the simulation. The comparisons are made between direct application of sequential sampling and the proposed random sampling methods. Results show that the indices from the proposed simulation methods are as accurate as those from sequential simulation while requiring much less CPU time. The test system is taken from Singapore Energy Market Authority (EMA) data in 2008.

Index Terms—Monte Carlo simulation, single-area power system reliability, renewable energy, correlation analysis.

I. INTRODUCTION HE increasing cost and environmental impact of

conventional electric power systems has brought considerable attention and utilization of renewable energy resources. As the renewable energy market continues to grow, the power plants with these sources [1]-[4] are increasingly built and integrated into the existing power system for environmental and economic benefits. For reliability analysis of the entire system, incorporating the renewable generation into conventional generating systems is required. Moreover, there is a need to consider the correlation between time dependent sources - system load and renewable generating capacity. As both the fluctuating load and renewable sources, such as PV and wind, are dependent on the time and weather, the independent assumption among them is no longer applicable since it will inevitably affect system reliability evaluation.

A number of approaches have been reported to evaluate the reliability of power systems including renewable sources, which can technically classified into two categories, namely, analytical methods and simulation methods [5], [6].

With analytical methods, several approaches have been developed for adequacy assessment based on the renewable generation modeling or load modeling, such as the wind multi-state model [7]-[10], discrete wind speed distribution model [11] and load modification technique [12]. However, the correlation between the load and renewable sources is not considered in these approaches. To overcome this problem, the clustering algorithm [13] is utilized to represent the correlation among loads [14] or between load and renewable sources [15], [16]. In [15] and [16], all the generating units are grouped into subsystems, one containing conventional units and each of the others containing one type of renewable units. According to a measure of similarity, clustering method then classifies the observations into groups, each one of them expressed as a state containing the mean value of load, mean outputs of renewable units, as well as the probability of this group. When a state is identified, the outputs of renewable sources are found based on a given load value, and all the subsystems and clusters are combined to obtain the reliability indices. This approach preserves and incorporates the required correlation efficiently, however, the clustering algorithm imposes fixed combination of observations and the result accuracy is sensitive to the number of groups produced by clustering.

Analytical methods become impractical and cumbersome for systems with higher complexity or higher reliability, and particularly when time dependent sources are included. Monte Carlo (MC) simulation methods [6] offer an alternative solution as it creates the artificial system history by simply selecting the states of components and flexibly generating states of time varying sources such as wind and PV.

MC random sampling technique is applied in the Auto-regressive and moving average (ARMA) time series model to generate wind speed series for generating adequacy assessment [17]. In [18], random sampling is also utilized for reliability analysis of renewable systems by generating wind speed and PV power according to Weibull distribution and PV time series model respectively. Although these approaches accurately simulate the states of the time dependent renewable sources, they do not consider these sources and load correlated along the time intervals. For random sampling method, in particular, reference [19] proposes a correlation sampling technique considering bus loads as correlated random variables. Load

T

This work is supported in part by Singapore Ministry of Education-Academic Research Fund, Grant No. WBS R-263-000-487-112 and by Singapore National Research Foundation, Grant No. NRF2007EWT-CERP01-0954.

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states are sampled and correlation among them is preserved by the covariance matrix. However, this sampling method assumes that all the random variables are normally distributed, which may not be applicable to other types of distributions.

MC sequential sampling is practical and commonly used to incorporate correlations between random variables. Based on their time dependency, the sampling is done chronologically and individually [20]-[23]. For example, in [22], the hourly output of PV and wind is generated from their chronological variation models and combined with the conventional units, which are generated by simulating failures and repairs according to the transition time distributions. Together with the generating capacity, the corresponding hourly load is generated chronologically for reliability indices evaluation. When compared with non-sequential sampling, sequential sampling methods need more space to store chronological information and may require longer time to converge.

This paper proposes three non-sequential sampling methods for single-area power systems reliability analysis with consideration of renewable generation sources. The proposed methods efficiently incorporate correlations between generating capacity and load while sampling out each pair of states of the time dependent variables with less computational time than sequential sampling method. We focus our analysis on solar energy sources in this paper as an example of the renewable generations. The test system is taken from Singapore energy system where PV has high potential to be employed.

This paper is organized as follows. Section II describes the MC methods for single-area power system reliability analysis. Section III explains the proposed sampling methods. Section IV shows the performances and comparisons of the methods proposed and section V finally gives the conclusion.

II. SINGLE-AREA POWER SYSTEM RELIABILITY EVALUATION WITH MONTE CARLO SIMULATION TECHNIQUES

MC simulation for system reliability evaluation can be conducted by sequential sampling or non-sequential sampling. Sequential sampling simulates the history of system and preserves the chronological states over time while non-sequential sampling performs random draws over all the possible states that the system assumes without considering the duration of generation states or the sequential variations of the load with time. Each simulation method as well as their convergence criterion is described in details in the followings.

A. Sequential Sampling In sequential sampling, a state of the system is simulated as

a sequence of time. Each component in the system is represented by a Markov model. The time can be advanced by two methods, namely, fixed time interval or next event. In fixed time interval method, a state of a component is drawn based on its transition probability. In next event method, a time to change state of each component is sampled from the transition time distribution. In this paper, we utilize fixed-time

method and the simulation steps for single-area reliability evaluation are given in the following.

1. Initialize the system state by assuming that all generators are in working states and calculate the transition probability matrix of each component.

2. Determine the status of each generator in the next transition and combine the capacity of all the generators to give system generating capacity, G . In Fig. 1, the transition diagram of a generator with failure rate iλ and repair rate iμ is shown. To approximate the two state Markov process, discrete time step tΔ is chosen for simulation and the transition probability matrix is also given in Fig. 1. With the present state of a generator, by generating a random number, the next state is obtained.

iλ

iμ

Up Down

11

i i

i i

t tt t

λ λμ μ− Δ Δ⎛ ⎞

⎜ ⎟Δ − Δ⎝ ⎠

Up DownUp

Down

Fig. 1. Two-state transition diagram and probability matrix

3. Generate the load level, L . 4. Evaluate the reliability index f of the state. 5. Calculate the expected value of reliability indices

based on the evaluation results obtained in step 4. 6. Repeat Steps 2-5 until the stopping rule is reached.

B. Non-sequential Sampling For non-sequential sampling, known as random sampling,

all the states of the random variables are sampled according to their probability distribution. The steps of random sampling are as follows.

1. Initiate the stopping rule and number of samples，0sN = .

2. Select the states of each generator by generating uniform random numbers and find the combined generating capacity, G .

3. Generate the load level, L . 4. Evaluate the reliability index f of the state. 5. Calculate the expected value of reliability indices

based on the evaluation results obtained in step 4. 6. Repeat Steps 2-5 for each iteration 1s sN N= + . Stop

when the stopping rule is reached.

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C. Convergence Criterion 1. The expected value of reliability index f is [ ]E f ,

and its estimator is:

[ ]1

1ˆsi N

iis

E f fN

=

=

= ∑ (1)

2. The coefficient of variation for the uncertainty of the

estimate is calculated in (2) and used as the acceptable value. By defining an epsilon ε , the simulation should stop when COV ε< . Typically, the COV less than 0.05 is considered sufficient.

[ ]( )[ ] [ ]

ˆ 1 ( )ˆ ˆ

s

COVVar E f Var f

NE f E f= = (2)

The steps of sequential and non-sequential simulation above

do not take into account the generating capacity from unconventional sources. To include the capacity of renewable energy sources, we can combine the generating capacities chronologically through time, which can be easily implemented in the sequential sampling. However, for non-sequential sampling where a generating capacity state is randomly sampled from its distribution, the generating capacity from renewable energy sources and system load should be selected according to the time of day and weather condition.

In this paper, we focus our analysis to solar energy, denoting pvG as the PV generating capacity. To include this dependency in non-sequential simulation, we need to select

pvG together with load L according to certain correlation between them in the 3rd step of non-sequential sampling, as shown in section II. Then, the total generating capacity is updated by total pvG G G= + for reliability evaluation. The three proposed methods of selecting both generating capacity and system load states are proposed in the next section.

III. PROPOSED RANDOM SAMPLING TECHNIQUES INCLUDING CORRELATION BETWEEN LOAD AND PV

We propose three non-sequential simulation methods that properly select load level according to the generating capacity of the renewable energy sources. The first method utilizes load duration curve where the PV generating capacity and load are randomly selected from the same time interval. The second method employs linear regression model [24] by solving derivative equations, and the third method uses polynomial regression model to describe the correlation.

A. Load Duration Method (LDM) LDM preserves all levels of load and PV generation

chronologically in 24 hours. In the load duration curve [25],

we divide the time horizon into equal intervals and randomly select a sample from the time intervals with identical probability. For example, in one year, there are totally 8760 hours, and the probability for any individual hour to be sampled is 1/8760. As in (3), for a number of time intervals t , the probability of sampling interval , {1,2,..., }i i t∈ is:

( ) 1, 1, 2,3...p i i tt

= = (3)

The cumulative distribution function (CDF) is written as:

( ) ( ) ( ) ( ) , 1,2,3...n n

n ni i i i

iF i P I i P I i p i i tt≤ ≤

= ≤ = = = = =∑ ∑ (4)

where ni is the sampling interval i for the thn hour.

It is shown in Fig. 2 that the random sampling is conducted on the CDF of time with load and PV states corresponding to the sampling time interval. As the CDF of time intervals is always the same for any load or PV curve, this method is efficient by simply sampling time intervals to find each pair of PV and load values. In this paper, the sampling space of time intervals is 8760 hours of one year. The states of conventional generators are randomly sampled according to their failure probabilities.

Fig. 2. CDF of one day (24 hours)

B. Linear Regression Method (LRM) In LRM, the regression analysis between two random

variables is applied in the simulation, where the load and PV are sampled according to their linear regression calculated through least - squares approach.

The relationship between random variables x and y may be linear or nonlinear. Regardless of the true pattern of association, the linear model offers the first and simple approximation as the following:

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y ax b e= + + (5)

where a is the slope, b is the intercept, and e is the residual error of the line.

The value of y predicted is:

y ax b= + (6)

Use least squares solution to minimize the variation which is:

[ ] ( )

( ) ( )

22

22

ˆ

1 ( ) 2 ( , )

e Var y y E ax b y

n Var y aCov x y a Var x ax b yn

⎡ ⎤= − = + −⎣ ⎦−⎛ ⎞ ⎡ ⎤= − + + + −⎜ ⎟ ⎣ ⎦⎝ ⎠

(7)

Under the constraint [ ] 0E ax b y+ − = , equations (8) and (9)

take partial derivatives of a and b , and set them to zero:

( )( ) ( )

212 , ( ) 0

e n Cov x y aVar x x ax b ya n

∂ ⎡ − ⎤⎛ ⎞= − + + + − =⎡ ⎤⎜ ⎟⎢ ⎥⎣ ⎦∂ ⎝ ⎠⎣ ⎦ (8)

( )( )

2

2 0e

ax b yb

∂= + − =

∂ (9)

where [ ]E is the expected value, n is the number of observations, Cov is covariance,Var is variation, and x , y are averages of observations.

The solutions are as follows.

( )( )

,Cov x ya

Var x= (10)

b y ax= − (11)

With linear relationship between PV generation and load, the sampling procedure is conducted for one variable and mapped into the other variable.

C. Non-linear Regression Method (NRM) Instead of the linear regression analysis, this method

computes the polynomial parameters of PV and load curves straightforwardly by solving the partial derivatives.

With the given data set ( ) ( ) ( )1 1 2 2, , , ... ,n nx y x y x y , use thm degree polynomial to approximate prediction of y as:

2

0 1 2ˆ ... mmy a a x a x a x= + + + + (12)

The residual is given by:

( ) 22 20 1 2

1

... , 1n

mi i i m i

i

R y a a x a x a x n m=

⎡ ⎤= − + + + + ≥ +⎣ ⎦∑ (13)

The partial derivatives are set to zeros and written as:

( ) ( )

2

20 1 2

0 12 ... 0

nm

i i i m ii

Ry a a x a x a x

a =

∂= − − + + + + =

∂⎡ ⎤⎣ ⎦∑ (14)

( ) ( )2

20 1 2

1 12 ... 0

nm

i i i m i ii

Ry a a x a x a x x

a =

∂= − − + + + + =

∂⎡ ⎤⎣ ⎦∑ (15)

•••

( ) ( )2

20 1 2

12 ... 0

nm m

i i i m i iim

Ry a a x a x a x x

a =

∂= − − + + + + =

∂⎡ ⎤⎣ ⎦∑ (16)

( ) ( ) ( )

( ) ( )1

2 ...0 11

11

2

... 0

n m mi i m i m i i

im

i i i im

Ry a a x a x a In x x

m

a In x x a In x x

= − − + + +=

−−

∂⎡⎡ ⎤⎣ ⎦ ⎣∂

⎤+ + + =⎦

∑ (17)

where y is the predicted value of y , 0 1, ,..., ma a a are polynomial

coefficients, 2R is the residual of y , ix , iy are the observations, and n is the number of observations.

To obtain degree m beforehand, instead of solving non-linear equation (17), numerical calculation is employed as follows.

For each , {1,2,..., 1}m m n∈ − , solve equations (14)-(16) to obtain 0 1, ,..., ,ma a a use equation (13) to compute residuals

2 2 21 2 1, ,..., ,nR R R − and find the minimum residual donated

by 2MinR , where Min is the optimal degree, so let m Min= .

The obtained degree m , together with the corresponding coefficients 0 1, ,..., ma a a will establish the polynomial function with minimized variation. In this paper, load is expressed as the polynomial function of PV during 24hours. Each load sample value can be predicted simply by the least squares “best fit” curve according to the equation (12) obtained.

IV. CASE STUDIES The case studies compare the sequential MC simulation

method with three proposed non-sequential MC simulation methods for single-area reliability evaluation. A test system is modified from the Singapore power generation and demand data from Energy Market Authority of Singapore [26], and the corresponding PV data is taken from a Weather Station in National University of Singapore (NUS) [27]. The reliability indices in this study are loss of load probability (LOLP) and expected unserved energy (EUE). The coefficient of variation of LOLP is given 3%LOLPε = as the convergence criteria for both sequential and non-sequential simulations.

The sequential MC simulation is performed by controlling the state transition of component by its recent past information,

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including time varying load and PV generation straightforwardly. This implies that the correlations among random variables are preserved and that the solution from sequential simulation can be considered as the most accurate among the solutions from various simulation methods. We use the solution from sequential simulation as a benchmarking. The percentage error of a reliability index from the proposed random sampling methods is calculated by the percentage deviation from the index given by sequential simulation, which is shown in (18).

100%rs ssReliability Index

ss

Index IndexError

Index−

= × (18)

where rsIndex and ssIndex are indices obtained from random and sequential simulation respectively.

In order to perform case studies, we simulate the system with significant amount of generation from renewable sources. Since the PV generating capacity from NUS weather station is in the order of Watts which is insignificant when combined with conventional generation, the unit of PV generation is then increased to be in the order of Megawatts. The incoming radiation ( 2/W m ) for each hour is aggregated and averaged for each day, and considered as the total output PV power ( MW ) over all the radiation areas of the system. The normalized average levels of load and PV in each hour are shown in Fig. 3, where the separate load curves represent different hourly load pattern during weekday, Saturday and Sunday.

Of the methods proposed, LRM and NRM both describe load as a function of PV based on average hourly value. For LRM, load is predicted by PV based on the data through one whole year. For NRM, to facilitate the polynomial function calculation, load is expressed as the functions of PV for 24 hours on weekday, Saturday and Sunday respectively.

Fig. 3. Hourly average values of load and PV Currently in Singapore, the total capacity of the system

generation is 10453.03MW with 39 generators installed and the hourly peak load is 5953.3MW. According to [28], Singapore load demand is projected to grow at an annual rate of 2.8% from 2002 to 2030. The system load is then increased by 64.4% to reflect energy demand forecast in 2026. Each simulation method is replicated 10 times to ensure the precision, with mean value (Mean) and standard deviation (Std) generated. The simulations are conducted with Matlab2009a on a PC with Intel Core2 2.0GHz CPU and 2GB RAM.

The reliability indices of simulations by random sampling not taking PV into account are shown in Table I below.

TABLE I

RELIABILITY INDICES FROM RANDOM SAMPLING METHOD WITHOUT PV

Simulation Method

Indices

LOLP (%)

EUE (MW)

No. States

CPU Time

(s)

RS without PV

Mean 3.88 14.65 439816 105.06

Std 0.022 0.14 300046 73.76

The other simulations are conducted considering PV generation. The results from random sampling PV and load independently (RS) are shown in Table II. The results from sequential sampling (SS), together with those from the proposed methods are also shown in Table II for comparisons.

TABLE II

RELIABILITY INDICES FROM SEQUENTIAL SAMPLING METHOD AND RANDOM SAMPLING METHODS INCLUDING PV POWER

Simulation Method

Indices

LOLP (%) EUE (MW) No. States CPU Time (s)

SS Mean 1.776 6.13 60671109 11800

Std 0.033 0.099 23362313 7145.43

RS Mean 2.99 11.24 285612 125.64 Std 0.029 0.15 69814 30.68

LDM Mean 1.772 6.01 535006 235.90 Std 0.018 0.13 190527 91.48

LRM Mean 1.812 6.45 509198 174.98 Std 0.019 0.12 179151 61.80

NRM Mean 1.804 6.38 553196 190.44 Std 0.014 0.12 210777 73.50

The errors of indices from the proposed methods are

calculated using (18) based on the results from sequential sampling as reference and shown in Table III.

TABLE III

ERRORS OF INDICES FROM RANDOM SAMPLING METHOD WITHOUT CORRELATION AND PROPOSED METHODS

Errors (%)

Simulation Methods RS LDM LRM NRM

LOLP 68.36 -0.23 2.03 1.58 EUE 83.36 -1.96 5.22 4.08 Results show that, compared with Table I, LOLP and EUE

in Table II decreases due to the contribution of PV generation, and number of sample states increases due to smaller LOLP. However, compared with SS, the reliability index error of RS

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is much larger than that generated by the proposed methods, indicating that the precision is highly affected by the correlation between load and PV.

As seen from Table II, LDM, LRM and NRM require much less computational time than SS, and the LOLP and EUE errors of LDM, LRM and NRM from Table III show that all the proposed methods provide acceptable accuracy. It is also found that LDM provides more accurate performance indices. This is because the correlation model in LDM can better represent the actual data pattern while the models in LRM and NRM approximate the correlated pattern by linear or polynomial assumptions.

The fact that independent sampling of load and PV generation brings about higher degree of error shows that the correlation between load and PV generation contributes to the increase of system reliability and therefore the simulation considering this correlation evidently provides more accuracy.

We repeat the simulation with different system load. According to [28], in year 2028, 20 years after 2008, the total load will be increased by 73.7%. Results are shown in Table IV, Table V and Table VI respectively.

TABLE IV

RELIABILITY INDICES FROM RANDOM SAMPLING METHOD WITHOUT PV WHEN LOAD IS INCREASED

Simulation Method

Indices

LOLP (%)

EUE (MW)

No. States

CPU Time

(s)

RS without PV

Mean 11.53 54.78 57351 16.78

Std 0.17 1.13 28327 8.32

TABLE V RELIABILITY INDICES FROM SEQUENTIAL SAMPLING METHOD AND RANDOM

SAMPLING METHODS INCLUDING PV POWER WHEN LOAD IS INCREASED

Simulation Method

Indices LOLP (%)

EUE (MW)

No. States

CPU Time (s)

SS Mean 6.64 26.70 21734392 5795.49

Std 0.067 0.34 11956273 2973.35

RS Mean 9.31 42.62 103952 45.18 Std 0.15 0.74 78818 34.41

LDM Mean 6.58 26.66 167714 41.69 Std 0.083 0.53 156862 39.41

LRM Mean 6.66 27.35 160088 55.07 Std 0.088 0.51 136344 46.89

NRM Mean 6.68 27.25 134326 45.70 Std 0.12 0.39 78562 26.82

TABLE VI

ERRORS OF INDICES FROM RANDOM SAMPLING METHOD WITHOUT CORRELATION AND PROPOSED METHODS WHEN LOAD IS INCREASED Errors

(%) Simulation Methods

RS LDM LRM NRM LOLP 40.21 -0.90 0.30 0.60 EUE 59.63 -0.15 2.43 2.06

Under this situation, it can be observed that the addition of

PV generation alleviates LOLP and EUE, and the correlation between PV and load further alleviates LOLP and EUE. Under

different load levels, comparing the LOLP results of each method in Table II to those in Table V, it is found that the relationship between reliability index and number of sample is non-linear, which can be explained by the convergence equation in (2).

In particular, the errors of RS in Table III and Table VI show that the RS method considerably underestimates the system reliability, producing the error of EUE up to 83.36% and 59.63%. This will most likely lead to misleading conclusion and unnecessary cost in system planning. Additionally, the errors in Table VI are even smaller than those in Table III, revealing that the proposed methods could be more accurate in systems with lower reliability.

V. CONCLUSIONS This paper presents three random sampling simulation

methods for reliability evaluation of single-area power systems considering the correlation between renewable generation and load. These methods are proposed including the actual correlated fluctuations of time varying sources-PV and load to simulate the power system comprehensively and accurately. The simulation results indicate that the MC simulation techniques considering such correlation would give more accurate estimation of reliability indices. The proposed three methods, when compared with sequential simulation, can provide high accuracy while save large amount of computational time. The comparison of the three methods shows that they perform quite closely both in calculation speed and indices accuracy. These methods are simple yet effective according to the case studies. It can be seen that because of their simplicity, efficiency and flexibility, the methods proposed can be applied into many non-sequential samplings with correlated random variables.

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