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A new approach of point estimate method for probabilistic load flow
Electrical Power and Energy Systems 51 (2013) 54–60
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A new approach of point estimate method for probabilistic load flow M. Mohammadi ⇑, A. Shayegani, H. Adaminejad Department of Power and Control Eng., School of Electrical and Computer Eng., Shiraz University, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 20 August 2012 Received in revised form 7 February 2013 Accepted 22 February 2013 Available online 27 March 2013 Keywords: Probabilistic load flow Monte Carlo simulations Point estimate method New point estimate method
a b s t r a c t This paper analyses the power system load flow using new point estimate method considering uncertainties, which may happen in the power system. These uncertainties may arise from different sources, such as load demands or generation unit outages. A novel probabilistic load flow based on new point estimate method has been proposed in this paper. The proposed method surpasses the previous point estimate methods when generalizing the approach to multiple random variables with various probabilistic distributions. Addressing the results of Monte Carlo simulation method as reference, the new point estimate method is applied to IEEE 14-bus test system and the advantages of this new method have been presented. The results reveal that the proposed point estimate method has less computational burden and time comparing than Monte Carlo simulation method while the accuracy remains at a high level. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Probabilistic Load Flow (PLF) is an important issue in power system analysis. During the last decade, by increasing the importance of power quality for sensitive loads, the uncertainties in a power system, such as load variation, faults of lines and generator trip have been in great attention. As these events have probabilistic behavior, they are called Random Variables (RVs) and can highly influence on the load flow through the system. It is so important to know these parameters so that protective actions could be done. Some of these parameters are bus voltages, generated active and reactive in slack bus, line currents and some other important factors. All the mentioned parameters are obtained from the load flow calculations. In order to explain the behavior of RV, a Probability Density Function (PDF) is assigned to each RV. These PDFs can be used in corresponding equations. In such cases, the traditional deterministic load flow does not have reliable results because some parameters like active or reactive power or even bus voltages do not have a deterministic amount. In order to consider these uncertainties, PLF must be utilized to gain reliable results. We should know that the output results of load flow equations are also random parameters, which are described with special PDFs according to the input parameters. The final goal of a PLF is to define these PDFs. In deterministic methods, system parameters such as loads, network configurations and other parameters are considered to be constant, while in an actual grid these time-varying parameters have ⇑ Corresponding author. Tel.: +98 711 6133278; fax: +98 711 2303081. E-mail addresses: [email protected] (M. Mohammadi), amir.shaye [email protected] (A. Shayegani), [email protected] (H. Adaminejad). 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.02.019
random nature. Uncertainty of system parameters cannot be studied by deterministic methods; so probabilistic methods are more suitable for this purpose. Various techniques such as Monte Carlo Simulation (MCS) method, analytical methods and approximate methods have been proposed to deal with these uncertainties based on probability theory [1,2]. MCS [3] generates random values for uncertain input variables, and they are used in deterministic routines to solve the problem in each simulation [4]. When the simulation method is used, the computation efficiency is very important because usually the sample set is very large [5]. This technique has been widely used in power systems analysis to model uncertainty. One of its major disadvantages includes its high cost in terms of computer time and stringent programming requirements to achieve a practical level of program efficiency. For this reason, the development of analytical methods was carried out [6]. In contrast, analytical methods are computationally more effective [4]. The analytical approaches were proposed to reduce the workload of calculation in solving PLF problem. These approaches generally include two components. The first component is to simplify the traditional PLF formulations. The second component is to calculate the convolution of different probabilistic variables by their linear relationships [7]. When the analytical method is used, the accuracy is very important because simplifications in the model of the analytical method may cause corresponding errors in the results [8]. In [9,10], dc-load flow model was used to find the PDFs of network parameters. In [11], a linearized model was used PLF. [12] Used a linearized model of load flow equation by using McLaren’s series of sine and cosine functions and an approximation in conversion of multiply to
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M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
Nomenclature Ij Yij Vj
Hi Si
Hij Pi Qi Z(k) xkn
injected current to bus j admittance between bus j and j voltage of bus j voltage angle of bus i apparent power of bus i angle of ijth element of admittance matrix active power of bus i reactive power of bus i kth iteration random variable of Monte Carlo simulation nth input variable of kth iteration of Monte Carlo simulation
summation. The author in [13] used another method for simplification of the load flow equations. The main disadvantage of these techniques is that they need mathematical assumptions to simplification [7]. Approximate methods provide an approximate description of the statistical properties of output RV. This method was used in many papers relating to PLF. A number of studies [14–17] have been conducted to investigate PLF via approximate methods. In particular, all the papers of the previous list use the Point Estimate Method (PEM) to solve some probabilistic problems. In [18] the PEM is used to analyze the steady state operating conditions of an unbalanced power system. In [19] the valuation of the steady state operating condition of an unbalanced three-phase power system is performed in presence of uncertainties and wind farms. In [20] the analysis was performed considering correlated and uncorrelated input random variables. In [21], the Third-Order Polynomial Normal Transformation (TPNT) technique was employed to transform a multivariate non-normal dependent random variables group into a multivariate standard normal independent one. Incorporating three-point estimate method with TPNT technique, a TPNT-based three-point estimate method is proposed to solve PLF problems with nonnormal dependent variables. In [22], finally, the PEM replace the probability distribution of the random parameters of a model with a finite number of discrete points in sample space selected in such a way to preserve limit probabilistic information of involved RVs. Like MCS method, PEMs use deterministic procedures in order to solve probabilistic problems by using only few first statistical moments of probability functions such as mean, variance, skewness and kurtosis coefficients. The PEM in this procedure is somehow better than MCS method as it has less computational efforts and consumed time since a smaller volume of data is required. The PEMs have some drawbacks associated with them. Increasing the number of estimating points is necessary to improve accuracy and preciseness in the previous PEMs. In the previous PEMs the estimated points may be outside the region in which the RV is defined especially for some RV with relatively large standard deviation, such as variables having a lognormal or exponential distribution. So, in these situations, PEM does not show appropriate results in comparison with MCS. The new Point Estimate Method (new PEM) is an approach to remove the above limitation associated with previous PEMs in which it is easier to increase the number of estimated points since the estimating points are independent of the RV in its original space [23]. In this paper, a new PEM has been presented to solve PLF, since in an actual power grid; there are many uncertain parameters with various probabilistic distributions and relatively large standard deviations. Thus, the new PEM is the approach to gain preciseness while the computational issues are also taken into account.
f E(Z) VAR(Z) Z xn
random function mean value variance value random variable of point estimate method nth input variable of point estimate method expected value of jth variable standard deviation of jth variable the kth central moment of the jth variable probability density function
lj rj Mk(xj) g(xj)
2. Probabilistic Load Flow (PLF) 2.1. Problem formulation Load flow equations can be derived by using the network configurations, bus voltages and current injections. The current injected into bus can be written as:
½Ii ¼ ½Y ij ½V j ¼
X Y ij V j
ð1Þ
j
where Yj is the network admittance matrix, Ij is the matrix of injected current to each bus and Vj is the bus voltage matrix [24]. Complex power can be calculated as follows:
Si ¼ V i Ii ¼ V i
X
jY ij V j
¼
X j
jV i jjV j jjY ij jejðhi hj hij Þ
ð2Þ
where hi is the angle of Vi, hj is the angle of Vj and hij is the angle of the ijth element of the admittance matrix. By expansion of the above equation and separating the real part and imaginary part of the equation:
Pi ¼ Qi ¼
X j
jV i jjV j jjY ij j cosðhi hj hij Þ
ð3Þ
jV i jjV j jjY ij j sinðhi hj hij Þ
ð4Þ
X j
Eqs. (3) and (4) can be solved by using iteration technique such as Gauss–Seidel and Newton Raphson. It is important to mention that this equation must be written only for PV and PQ busses. 2.2. Monte Carlo Simulation method (MCS method) The complexity of the analytical relations between the conversion system parameters, and in particular, their nonlinearity, addresses to the use of the well-known Monte Carlo method [25,26]. Usually, Monte Carlo methods are used to simulate a prescribed random behavior of the network loads. That is, random number generators are used to assign specific probability distributions to certain parameters of the loads, thus reflecting the RVs in the load’s operating condition. In this way, deterministic models of the load can then be used to generate the random active and reactive power. The advantage of this approach is in the possibility of simulating a wide variety of random load characteristics until the resulting statistics agree with available field measurements. The disadvantage is that this method is computationally intensive and time consuming since it is difficult to determine how to adjust the load random models in order to produce desired results. In the MCS method variables are calculated for each randomly generated set of input variables as below:
ðkÞ ðkÞ Z ðkÞ ¼ f x1 ; x2 ; . . . ; xnðkÞ
ð5Þ
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M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
Finally, the mean and variance of outputs are calculated as: n 1X EðZÞ ¼ Z ðkÞ k k¼1
EðZÞ ¼ ð6Þ
" # n 1X ðkÞ 2 VARðZÞ ¼ ðZ Þ ðEðZÞÞ2 k k¼1
ð7Þ
n X
½pm;1 f ðl1 ; . . . ; xm;1 ; . . . ; ln Þ
m¼1
þ pm;2 f ðl1 ; . . . ; xm;2 ; . . . ; ln Þ VARðZÞ ¼
n X
fpm;1 ½f ðl1 ; . . . ; xm;1 ; . . . ; ln Þ2
m¼1
þ pm;2 ½f ðl1 ; . . . ; xm;2 ; . . . ; ln Þ2 g ½EðZÞ2 2.3. Point Estimate Method (PEM) Let Z to be a function of n independent RV as follow:
Z ¼ f ðXÞ ¼ f ðx1 ; x2 ; . . . ; xn Þ
ð8Þ
Assume that the expected value and standard deviation of the jth variable are lj and rj respectively. The kth is central moment of the jth variable with PDF of g(xj) is calculated as follow:
M k ðxj Þ ¼
Z
þ1
1
k
ðxj lj Þ g j ðxj Þdxj ;
k ¼ 1; 2; . . .
ð9Þ
And kj;k is defined as follow:
kj;k ¼
Mk ðxj Þ
ð10Þ
rkj
where kj;k is known as coefficient of skewness. The expected value of the function can be calculated using the following equation:
lZ ¼ Eðf ðxÞÞ
ð11Þ
Using Taylor series, expanding the function at the expected values results in:
lZ ¼ f ðl1 ; l2 ; . . . ; ln Þ þ
1 X n X 1 @if ðl1 ; l2 ; . . . ; ln Þkm;i rim i i! @x m i¼1 m¼1
ð12Þ
Let Xj, 1 = lj + nj,1rj and Xj,2 = lj nj,2rj be two predefined concentration points and pj,1 and pj,2 be the probability concentrations at points xj,1 and Xj,2, respectively. nj,1 And nj,2 are four constants which should be determined. By estimating the mean of points, it can be said that: n X 2 X X X 1 @if pm;k þ i! @xim m¼1 k¼1 ðl1 ; l2 ; . . . ; ln Þ pm;1 nim;1 þ pm;2 nim;2 rim
2.4. New Point Estimate Method (new PEM) As discussed before improving accuracy of the current PEM requires increasing the number of estimating points, thus calculating high-order moments of a RV [23]. Even if the high-order moments of an arbitrary RV can be computed, it would be almost impossible to attain a general expression for estimated points and their corresponding weights as in. Furthermore, because the moments are dependent on the PDF of the RV, it is difficult to avoid the problem of estimated points since the estimated points tend to move outside the region on which the RV is defined. A solution to the above problems is to obtain the estimating points in the standard normal space having the knowledge that any set of RV can be transformed into a set of standard normal RV through the Rosenblatt transformation [23].
lZ ¼ S g ½T 1 ðUÞhðUÞdU
ð20Þ
MkZ ¼ Sðg½T 1 ðUÞ lZ Þk hðUÞdU
ð21Þ
where h is the PDF of standard normal RVs. By this reformulation, the estimating points and their corresponding weights can be directly calculated by integration, thus no central moment of the original RV is required when obtaining
ð13Þ
Now, by making first three orders of (12) and (13) equal, find the three required equations as follow:
i ¼ 1; 2; 3
ð14Þ
As a fact the sum of the probability concentrations should be equal to one: n X 2 X pm;k ¼ 1
ð15Þ
m¼1 k¼1
Thus, the constant parameters can be derived:
nm;k
km;3 ¼ þ ð1Þ3k 2
pm;k ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 km;3 nþ 2
nm;k 1 ð1Þk rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi n k 2 n þ m;3 2
m ¼ 1; 2; . . . ; n
ð16Þ
ð17Þ
Finally, mean and variance of function Z of several RVs, X can be found:
ð19Þ
It is clear that for problems with discrete set of solutions, the sum of each set of answers’ probabilities equals the expected value of that set. So, in (18), f(l, . . . , Xm,k, . . . , ln) indicates the discrete answer corresponding to point xm,k with probability Pm,k. Flowchart of the PEM can be summarizing as Fig. 1.
lZ ¼ f ðl1 ; l2 ; . . . ; ln Þ
pm;1 nim;1 þ pm;2 nim;2 ¼ km;i ;
ð18Þ
Fig. 1. Flowchart of the point estimate method.
M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
lh ¼
m X pj h½T 1 ðuj Þ
57
ð22Þ
j¼1
Mkh ¼
m X pj ðh½T 1 ðuj Þ lh Þk
ð23Þ
j¼1
where T1(uj) is the inverse of the transformation which maps arbitrary RV to standard normal variables which requires to be evaluated only at estimating points. 2.4.1. Generalizing new PEM for function of N variables The procedure described in the previous section can be generalized to a function of N variables Z = g(X). When the RV are mutually independent, g(X) can be approximated while the accuracy remains in a high level. Here g(X) is approximated by the following function:
gðXÞ g 0 ðXÞ ¼
N X ðg i g l Þ þ g l
ð24Þ
i¼1
where
g i ¼ g½T 1 ðU i Þ Fig. 2. Flowchart of the new point estimate method.
the estimating points in the standard normal space. Furthermore, because a normal RV is defined within the whole range (1, +1), the infeasibility problem of an estimating point is avoided. After deriving the estimating points u1, . . . , um and their corresponding weights p1, . . . , pm, the kth central moment of any function h = h(x) can be calculated by the following equations:
ð25Þ
gl is the function evaluated at the variable means and Ui means ui is the only RV, with the other variables set equal to mean values transformed into standard normal space; hence gi becomes a function of only ui. Because U = T(X) are mutually independent, primal RV X need not to be mutually independent and using the point estimate described in the previous section the following equations can be derived for calculating standard deviation and moments of Z.
lZ ¼
N X ðg i g l Þ þ g l i¼1
Fig. 3. The single line diagram of modified IEEE 14 bus test system.
ð26Þ
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M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
Table 1 Nodal probabilistic data for bus. Bus no.
Active power (MW)
Reactive power (MVAR)
Normal Mean value = 90, variance = 25
Uniform Upper band = 12, lower band = 8
11
Exponential Exponential parameter = 20
Normal Mean value = 10, variance = 4
13
Uniform Upper band = 34, lower band = 26
Exponential Exponential parameter = 20
2
Table 2 Injection bus data. Bus no.
2
Injected active power (MW)
Voltage magnitude (PU)
A
B
A
B
10
0.15
1.00
0.001
A: mean value. B: standard deviation.
rZ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN
a3Z ¼
a4Z ¼
r2i
ð27Þ
a3i r3i
ð28Þ
i¼1
N 1 X
r3Z
i¼1
1
N X
r4Z
i¼1
4 i
a4i r þ 6
N 1 X N X
! 2 i
rr
2 j
ð29Þ
i¼1 j>1
The flowchart of the new PEM has been presented in Fig. 2. 3. Case study In order to demonstrate the effectiveness of the proposed method, it has been applied to the modified IEEE 14 bus test system. Fig. 3 shows the single line diagram of the system. This system
contains 14 buses, two synchronous generators and 18 lines. MCS algorithm has been applied on this system and the accuracy of new PEM results is compared with the MCS. In the network, these parameters were supposed to be stochastic with particular PDFs. Table 1 presents nodal probabilistic data for different load buses of 14 bus test system. Synchronous generator at bus 1 (G1) has been considered as the slack bus. Table 2 shows the probabilistic data for power injected and voltage magnitude of synchronous generator at bus 2 (G2). It has been assumed that these RV have normal PDF. MCS with 1000 iterations have been done. Results of these simulations have been used as reference for comparison with other methods. A selection of typical value for the mean and variance of the power flow through the lines calculated via five different methods has been shown in Tables 3 and 4. Also a selection of typical values for mean and variance of bus voltage amplitude and angle are shown in Table 5. In Table 6 active and reactive power of slack bus has been presented. The results show that the proposed new PEM has small and acceptable errors, although the uncertain loads have different unlike probability distributions. It can also be seen that the new PEM is still much faster than MCS. As the number of RV increases more points will be needed to gain suitable results. In contrast with traditional PEMs, increasing the number of estimating points is not a limitation in the new PEM since the estimating points are independent of the RV in the original space and the use of highorder moments are not required. As it is obvious, by applying new PEM, results are totally good and close to the results of MCS, while the elapsed time decreases. Furthermore, by existence of PDFs such as exponential distribution, results have acceptable accuracy and none of the estimated points were outside the defined region. The results are compared with MCS, PEM and new PEM from the viewpoint of both accuracy and time consumption in this case. It is worth nothing that MCS method may take CPU time of 52.5 s to obtain results, while the 2PE, New-2PE, New-3PE, New-5PE takes a CPU time of 1.16, 1.16, 2.67, 5.52 s on the same processor. These simulations have been conducted on a Pentium 4. 2.88 GHz dual core PC.
Table 3 Mean and variance of active powers flow through lines via five different methods. From bus
To bus
Active power Mean (PU)
1 2 3 6 10 13
5 3 4 11 11 14
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
80.1100 53.4417 13.5517 50.1718 39.8332 10.7560
82.1200 55.4400 15.4630 51.2291 40.5681 11.7382
81.5500 54.2201 14.2300 51.0101 40.2203 11.2204
80.9000 53.8801 13.7601 50.6900 39.9903 10.9101
80.1000 53.4420 13.5518 50.1718 39.8900 10.7700
0.2260 0.0843 0.2039 0.0025 0.2012 0.0019
0.2333 0.0944 0.2089 0.0339 0.2900 0.0100
0.2235 0.0856 0.2071 0.0021 0.2051 0.0021
0.2235 0.0856 0.2071 0.0021 0.2051 0.0021
0.2234 0.0855 0.2061 0.0020 0.2031 0.0020
Table 4 Mean and variance of reactive powers flow through lines via five different methods. From bus
To bus
Reactive power Mean (PU)
1 2 3 6 10 13
5 3 4 11 11 14
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
9.5026 12.5917 1.9027 7.4145 1.9718 11.7321
10.1100 11.6100 2.1101 7.9991 2.2212 12.0111
9.9919 11.9919 2.0001 7.8767 2.0010 12.8901
9.6666 12.3243 1.9999 7.6923 1.9999 11.7898
9.5011 12.601 1.9011 7.4142 1.9717 11.7322
0.0043 0.0037 0.0063 0.0006 0.0025 0.0014
0.0055 0.0031 0.0050 0.0057 0.0035 0.0010
0.0047 0.0035 0.0055 0.0005 0.0033 0.0015
0.0043 0.0035 0.0057 0.0006 0.0040 0.0019
0.0046 0.0033 0.0051 0.0006 0.0029 0.0014
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M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60 Table 5 The results of some buses voltage via five different methods. Bus
Item
Voltage amplitude Mean (PU)
1 2 3 5 7 8 9 11
Voltage amplitude
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
1.0024 1.0001 1.0014 1.0005 1.0023 0.9560 1.0475 1.0001
1.115 1.010 1.090 1.001 1.020 0.880 1.090 0.900
1.0110 1.0010 1.0020 1.0090 1.0036 0.9200 1.0880 0.9900
1.0090 1.0070 1.0090 1.0009 1.0029 0.0990 1.0666 1.0001
1.0040 1.0030 1.0011 1.0006 1.0025 0.0980 1.0477 1.0003
0.0032 0.0007 0.0017 0.0037 0.0032 0.0038 0.0027 0.0002
0.0036 0.0011 0.0012 0.0023 0.0030 0.0042 0.0022 0.0005
0.0036 0.0010 0.0015 0.0030 0.0030 0.0040 0.0024 0.0002
0.0036 0.0010 0.0015 0.0031 0.0030 0.0036 0.0029 0.0002
0.0036 0.0010 0.0015 0.0036 0.0030 0.0040 0.0024 0.0002
Table 6 The results of active and reactive power of slack bus via five methods. Item
Slack generator Mean (PU)
Active power Reactive power
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
89.223 8.7571
92.21 9.222
90.101 8.79
90.077 8.777
90.013 8.7649
0.1333 0.1543
0.08 0.1
0.08 0.1
0.05 0.09
0.01 0.01
4. Conclusion
Table 7 Mean value for different iteration of MCS. Output
P slack Q slack V1 V2 V3 V5 V7 V8
Number of iterations 100
500
1000
3000
10,000
88.991 8.555 0.999 1.0023 0.9902 0.9989 0.9912 0.9706
84.959 8.578 1.0014 0.9985 0.9995 0.9981 1.0008 0.9499
89.223 8.7571 1.0024 1.0001 1.0014 1.0005 1.0023 0.9560
89.999 8.5650 1.0022 1.0005 1.0008 0.9990 1.0021 0.9608
90.001 8.869 1.0022 1.0000 1.0010 1.0005 1.0020 0.9512
In this paper a new PEM has been proposed to evaluate the probability moments of load flow results. The new PEM has advantages of previous PEMs in comparison with MCS method while it overcomes the limitations of the PEMs in actual power grids. With these new PEM, increasing the number of estimated points is easier because the estimating points are independent of the RV in the original space and the use of high-order moments of the RV is not required. The results of the new PEM with those of MCS method show the accuracy for the proposed method. References
Table 8 Variance value for different iteration of MCS. Output
P slack Q slack V1 V2 V3 V5 V7 V8
Number of iterations 100
500
1000
3000
10,000
0.1103 0.1543 0.0031 0.0005 0.0011 0.0033 0.0033 0.0033
0.1103 0.1543 0.0029 0.0005 0.0011 0.0035 0.0034 0.0033
0.1333 0.1543 0.0032 0.0007 0.0017 0.0037 0.0032 0.0038
0.1339 0.1543 0.0033 0.0008 0.0018 0.0038 0.0032 0.0038
0.1331 0.1543 0.0030 0.0006 0.0016 0.0036 0.0031 0.0035
3.1. Sensitivity analysis In this subsection sensitivity of the number of iteration for proposed MCS has been investigated. The PLF has been done for different iterations of MCS (100, 500, 1000, 3000, 10,000). It should be mentioned that by using the method presented in [27], almost sane results have been obtained. Mean and variance have been used for as Indies to measure the MCS accuracy. Results have been shown in Tables 7 and 8. It can be seen that 1000 is suitable number and increasing the number of iteration only makes the simulation more time consuming.
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[22] Wang Y, Tung Y. Improved probabilistic point estimation schemes for uncertainty analysis. Appl Math Model 2008;33:1042–57. [23] Zhao YG, Ono T. New point estimates for probability moments. J Eng Mech 2000:433–6. [24] Grainger JJ, Stevenson WD. Power system analysis. New York: McGraw-Hill; 1994. [25] Rubinstein RY. Simulation and the Monte Carlo method. New York: John Wiley and Sons; 1981. [26] Allan RN, Billinton R. Reliability evaluation of engineering systems. 2nd ed. New York: Plenum Press; 1992. [27] Ruiz-Rodriguez FJ, Hernandez JC, Jurado F. Probabilistic load flow for photovoltaic distributed generation using the Cornish–Fisher expansion. Electr Power Syst Res 2012;89:129–38.
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
A new approach of point estimate method for probabilistic load flow M. Mohammadi ⇑, A. Shayegani, H. Adaminejad Department of Power and Control Eng., School of Electrical and Computer Eng., Shiraz University, Shiraz, Iran
a r t i c l e
i n f o
Article history: Received 20 August 2012 Received in revised form 7 February 2013 Accepted 22 February 2013 Available online 27 March 2013 Keywords: Probabilistic load flow Monte Carlo simulations Point estimate method New point estimate method
a b s t r a c t This paper analyses the power system load flow using new point estimate method considering uncertainties, which may happen in the power system. These uncertainties may arise from different sources, such as load demands or generation unit outages. A novel probabilistic load flow based on new point estimate method has been proposed in this paper. The proposed method surpasses the previous point estimate methods when generalizing the approach to multiple random variables with various probabilistic distributions. Addressing the results of Monte Carlo simulation method as reference, the new point estimate method is applied to IEEE 14-bus test system and the advantages of this new method have been presented. The results reveal that the proposed point estimate method has less computational burden and time comparing than Monte Carlo simulation method while the accuracy remains at a high level. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction Probabilistic Load Flow (PLF) is an important issue in power system analysis. During the last decade, by increasing the importance of power quality for sensitive loads, the uncertainties in a power system, such as load variation, faults of lines and generator trip have been in great attention. As these events have probabilistic behavior, they are called Random Variables (RVs) and can highly influence on the load flow through the system. It is so important to know these parameters so that protective actions could be done. Some of these parameters are bus voltages, generated active and reactive in slack bus, line currents and some other important factors. All the mentioned parameters are obtained from the load flow calculations. In order to explain the behavior of RV, a Probability Density Function (PDF) is assigned to each RV. These PDFs can be used in corresponding equations. In such cases, the traditional deterministic load flow does not have reliable results because some parameters like active or reactive power or even bus voltages do not have a deterministic amount. In order to consider these uncertainties, PLF must be utilized to gain reliable results. We should know that the output results of load flow equations are also random parameters, which are described with special PDFs according to the input parameters. The final goal of a PLF is to define these PDFs. In deterministic methods, system parameters such as loads, network configurations and other parameters are considered to be constant, while in an actual grid these time-varying parameters have ⇑ Corresponding author. Tel.: +98 711 6133278; fax: +98 711 2303081. E-mail addresses: [email protected] (M. Mohammadi), amir.shaye [email protected] (A. Shayegani), [email protected] (H. Adaminejad). 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.02.019
random nature. Uncertainty of system parameters cannot be studied by deterministic methods; so probabilistic methods are more suitable for this purpose. Various techniques such as Monte Carlo Simulation (MCS) method, analytical methods and approximate methods have been proposed to deal with these uncertainties based on probability theory [1,2]. MCS [3] generates random values for uncertain input variables, and they are used in deterministic routines to solve the problem in each simulation [4]. When the simulation method is used, the computation efficiency is very important because usually the sample set is very large [5]. This technique has been widely used in power systems analysis to model uncertainty. One of its major disadvantages includes its high cost in terms of computer time and stringent programming requirements to achieve a practical level of program efficiency. For this reason, the development of analytical methods was carried out [6]. In contrast, analytical methods are computationally more effective [4]. The analytical approaches were proposed to reduce the workload of calculation in solving PLF problem. These approaches generally include two components. The first component is to simplify the traditional PLF formulations. The second component is to calculate the convolution of different probabilistic variables by their linear relationships [7]. When the analytical method is used, the accuracy is very important because simplifications in the model of the analytical method may cause corresponding errors in the results [8]. In [9,10], dc-load flow model was used to find the PDFs of network parameters. In [11], a linearized model was used PLF. [12] Used a linearized model of load flow equation by using McLaren’s series of sine and cosine functions and an approximation in conversion of multiply to
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M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
Nomenclature Ij Yij Vj
Hi Si
Hij Pi Qi Z(k) xkn
injected current to bus j admittance between bus j and j voltage of bus j voltage angle of bus i apparent power of bus i angle of ijth element of admittance matrix active power of bus i reactive power of bus i kth iteration random variable of Monte Carlo simulation nth input variable of kth iteration of Monte Carlo simulation
summation. The author in [13] used another method for simplification of the load flow equations. The main disadvantage of these techniques is that they need mathematical assumptions to simplification [7]. Approximate methods provide an approximate description of the statistical properties of output RV. This method was used in many papers relating to PLF. A number of studies [14–17] have been conducted to investigate PLF via approximate methods. In particular, all the papers of the previous list use the Point Estimate Method (PEM) to solve some probabilistic problems. In [18] the PEM is used to analyze the steady state operating conditions of an unbalanced power system. In [19] the valuation of the steady state operating condition of an unbalanced three-phase power system is performed in presence of uncertainties and wind farms. In [20] the analysis was performed considering correlated and uncorrelated input random variables. In [21], the Third-Order Polynomial Normal Transformation (TPNT) technique was employed to transform a multivariate non-normal dependent random variables group into a multivariate standard normal independent one. Incorporating three-point estimate method with TPNT technique, a TPNT-based three-point estimate method is proposed to solve PLF problems with nonnormal dependent variables. In [22], finally, the PEM replace the probability distribution of the random parameters of a model with a finite number of discrete points in sample space selected in such a way to preserve limit probabilistic information of involved RVs. Like MCS method, PEMs use deterministic procedures in order to solve probabilistic problems by using only few first statistical moments of probability functions such as mean, variance, skewness and kurtosis coefficients. The PEM in this procedure is somehow better than MCS method as it has less computational efforts and consumed time since a smaller volume of data is required. The PEMs have some drawbacks associated with them. Increasing the number of estimating points is necessary to improve accuracy and preciseness in the previous PEMs. In the previous PEMs the estimated points may be outside the region in which the RV is defined especially for some RV with relatively large standard deviation, such as variables having a lognormal or exponential distribution. So, in these situations, PEM does not show appropriate results in comparison with MCS. The new Point Estimate Method (new PEM) is an approach to remove the above limitation associated with previous PEMs in which it is easier to increase the number of estimated points since the estimating points are independent of the RV in its original space [23]. In this paper, a new PEM has been presented to solve PLF, since in an actual power grid; there are many uncertain parameters with various probabilistic distributions and relatively large standard deviations. Thus, the new PEM is the approach to gain preciseness while the computational issues are also taken into account.
f E(Z) VAR(Z) Z xn
random function mean value variance value random variable of point estimate method nth input variable of point estimate method expected value of jth variable standard deviation of jth variable the kth central moment of the jth variable probability density function
lj rj Mk(xj) g(xj)
2. Probabilistic Load Flow (PLF) 2.1. Problem formulation Load flow equations can be derived by using the network configurations, bus voltages and current injections. The current injected into bus can be written as:
½Ii ¼ ½Y ij ½V j ¼
X Y ij V j
ð1Þ
j
where Yj is the network admittance matrix, Ij is the matrix of injected current to each bus and Vj is the bus voltage matrix [24]. Complex power can be calculated as follows:
Si ¼ V i Ii ¼ V i
X
jY ij V j
¼
X j
jV i jjV j jjY ij jejðhi hj hij Þ
ð2Þ
where hi is the angle of Vi, hj is the angle of Vj and hij is the angle of the ijth element of the admittance matrix. By expansion of the above equation and separating the real part and imaginary part of the equation:
Pi ¼ Qi ¼
X j
jV i jjV j jjY ij j cosðhi hj hij Þ
ð3Þ
jV i jjV j jjY ij j sinðhi hj hij Þ
ð4Þ
X j
Eqs. (3) and (4) can be solved by using iteration technique such as Gauss–Seidel and Newton Raphson. It is important to mention that this equation must be written only for PV and PQ busses. 2.2. Monte Carlo Simulation method (MCS method) The complexity of the analytical relations between the conversion system parameters, and in particular, their nonlinearity, addresses to the use of the well-known Monte Carlo method [25,26]. Usually, Monte Carlo methods are used to simulate a prescribed random behavior of the network loads. That is, random number generators are used to assign specific probability distributions to certain parameters of the loads, thus reflecting the RVs in the load’s operating condition. In this way, deterministic models of the load can then be used to generate the random active and reactive power. The advantage of this approach is in the possibility of simulating a wide variety of random load characteristics until the resulting statistics agree with available field measurements. The disadvantage is that this method is computationally intensive and time consuming since it is difficult to determine how to adjust the load random models in order to produce desired results. In the MCS method variables are calculated for each randomly generated set of input variables as below:
ðkÞ ðkÞ Z ðkÞ ¼ f x1 ; x2 ; . . . ; xnðkÞ
ð5Þ
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M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
Finally, the mean and variance of outputs are calculated as: n 1X EðZÞ ¼ Z ðkÞ k k¼1
EðZÞ ¼ ð6Þ
" # n 1X ðkÞ 2 VARðZÞ ¼ ðZ Þ ðEðZÞÞ2 k k¼1
ð7Þ
n X
½pm;1 f ðl1 ; . . . ; xm;1 ; . . . ; ln Þ
m¼1
þ pm;2 f ðl1 ; . . . ; xm;2 ; . . . ; ln Þ VARðZÞ ¼
n X
fpm;1 ½f ðl1 ; . . . ; xm;1 ; . . . ; ln Þ2
m¼1
þ pm;2 ½f ðl1 ; . . . ; xm;2 ; . . . ; ln Þ2 g ½EðZÞ2 2.3. Point Estimate Method (PEM) Let Z to be a function of n independent RV as follow:
Z ¼ f ðXÞ ¼ f ðx1 ; x2 ; . . . ; xn Þ
ð8Þ
Assume that the expected value and standard deviation of the jth variable are lj and rj respectively. The kth is central moment of the jth variable with PDF of g(xj) is calculated as follow:
M k ðxj Þ ¼
Z
þ1
1
k
ðxj lj Þ g j ðxj Þdxj ;
k ¼ 1; 2; . . .
ð9Þ
And kj;k is defined as follow:
kj;k ¼
Mk ðxj Þ
ð10Þ
rkj
where kj;k is known as coefficient of skewness. The expected value of the function can be calculated using the following equation:
lZ ¼ Eðf ðxÞÞ
ð11Þ
Using Taylor series, expanding the function at the expected values results in:
lZ ¼ f ðl1 ; l2 ; . . . ; ln Þ þ
1 X n X 1 @if ðl1 ; l2 ; . . . ; ln Þkm;i rim i i! @x m i¼1 m¼1
ð12Þ
Let Xj, 1 = lj + nj,1rj and Xj,2 = lj nj,2rj be two predefined concentration points and pj,1 and pj,2 be the probability concentrations at points xj,1 and Xj,2, respectively. nj,1 And nj,2 are four constants which should be determined. By estimating the mean of points, it can be said that: n X 2 X X X 1 @if pm;k þ i! @xim m¼1 k¼1 ðl1 ; l2 ; . . . ; ln Þ pm;1 nim;1 þ pm;2 nim;2 rim
2.4. New Point Estimate Method (new PEM) As discussed before improving accuracy of the current PEM requires increasing the number of estimating points, thus calculating high-order moments of a RV [23]. Even if the high-order moments of an arbitrary RV can be computed, it would be almost impossible to attain a general expression for estimated points and their corresponding weights as in. Furthermore, because the moments are dependent on the PDF of the RV, it is difficult to avoid the problem of estimated points since the estimated points tend to move outside the region on which the RV is defined. A solution to the above problems is to obtain the estimating points in the standard normal space having the knowledge that any set of RV can be transformed into a set of standard normal RV through the Rosenblatt transformation [23].
lZ ¼ S g ½T 1 ðUÞhðUÞdU
ð20Þ
MkZ ¼ Sðg½T 1 ðUÞ lZ Þk hðUÞdU
ð21Þ
where h is the PDF of standard normal RVs. By this reformulation, the estimating points and their corresponding weights can be directly calculated by integration, thus no central moment of the original RV is required when obtaining
ð13Þ
Now, by making first three orders of (12) and (13) equal, find the three required equations as follow:
i ¼ 1; 2; 3
ð14Þ
As a fact the sum of the probability concentrations should be equal to one: n X 2 X pm;k ¼ 1
ð15Þ
m¼1 k¼1
Thus, the constant parameters can be derived:
nm;k
km;3 ¼ þ ð1Þ3k 2
pm;k ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 km;3 nþ 2
nm;k 1 ð1Þk rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi n k 2 n þ m;3 2
m ¼ 1; 2; . . . ; n
ð16Þ
ð17Þ
Finally, mean and variance of function Z of several RVs, X can be found:
ð19Þ
It is clear that for problems with discrete set of solutions, the sum of each set of answers’ probabilities equals the expected value of that set. So, in (18), f(l, . . . , Xm,k, . . . , ln) indicates the discrete answer corresponding to point xm,k with probability Pm,k. Flowchart of the PEM can be summarizing as Fig. 1.
lZ ¼ f ðl1 ; l2 ; . . . ; ln Þ
pm;1 nim;1 þ pm;2 nim;2 ¼ km;i ;
ð18Þ
Fig. 1. Flowchart of the point estimate method.
M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
lh ¼
m X pj h½T 1 ðuj Þ
57
ð22Þ
j¼1
Mkh ¼
m X pj ðh½T 1 ðuj Þ lh Þk
ð23Þ
j¼1
where T1(uj) is the inverse of the transformation which maps arbitrary RV to standard normal variables which requires to be evaluated only at estimating points. 2.4.1. Generalizing new PEM for function of N variables The procedure described in the previous section can be generalized to a function of N variables Z = g(X). When the RV are mutually independent, g(X) can be approximated while the accuracy remains in a high level. Here g(X) is approximated by the following function:
gðXÞ g 0 ðXÞ ¼
N X ðg i g l Þ þ g l
ð24Þ
i¼1
where
g i ¼ g½T 1 ðU i Þ Fig. 2. Flowchart of the new point estimate method.
the estimating points in the standard normal space. Furthermore, because a normal RV is defined within the whole range (1, +1), the infeasibility problem of an estimating point is avoided. After deriving the estimating points u1, . . . , um and their corresponding weights p1, . . . , pm, the kth central moment of any function h = h(x) can be calculated by the following equations:
ð25Þ
gl is the function evaluated at the variable means and Ui means ui is the only RV, with the other variables set equal to mean values transformed into standard normal space; hence gi becomes a function of only ui. Because U = T(X) are mutually independent, primal RV X need not to be mutually independent and using the point estimate described in the previous section the following equations can be derived for calculating standard deviation and moments of Z.
lZ ¼
N X ðg i g l Þ þ g l i¼1
Fig. 3. The single line diagram of modified IEEE 14 bus test system.
ð26Þ
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M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60
Table 1 Nodal probabilistic data for bus. Bus no.
Active power (MW)
Reactive power (MVAR)
Normal Mean value = 90, variance = 25
Uniform Upper band = 12, lower band = 8
11
Exponential Exponential parameter = 20
Normal Mean value = 10, variance = 4
13
Uniform Upper band = 34, lower band = 26
Exponential Exponential parameter = 20
2
Table 2 Injection bus data. Bus no.
2
Injected active power (MW)
Voltage magnitude (PU)
A
B
A
B
10
0.15
1.00
0.001
A: mean value. B: standard deviation.
rZ ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN
a3Z ¼
a4Z ¼
r2i
ð27Þ
a3i r3i
ð28Þ
i¼1
N 1 X
r3Z
i¼1
1
N X
r4Z
i¼1
4 i
a4i r þ 6
N 1 X N X
! 2 i
rr
2 j
ð29Þ
i¼1 j>1
The flowchart of the new PEM has been presented in Fig. 2. 3. Case study In order to demonstrate the effectiveness of the proposed method, it has been applied to the modified IEEE 14 bus test system. Fig. 3 shows the single line diagram of the system. This system
contains 14 buses, two synchronous generators and 18 lines. MCS algorithm has been applied on this system and the accuracy of new PEM results is compared with the MCS. In the network, these parameters were supposed to be stochastic with particular PDFs. Table 1 presents nodal probabilistic data for different load buses of 14 bus test system. Synchronous generator at bus 1 (G1) has been considered as the slack bus. Table 2 shows the probabilistic data for power injected and voltage magnitude of synchronous generator at bus 2 (G2). It has been assumed that these RV have normal PDF. MCS with 1000 iterations have been done. Results of these simulations have been used as reference for comparison with other methods. A selection of typical value for the mean and variance of the power flow through the lines calculated via five different methods has been shown in Tables 3 and 4. Also a selection of typical values for mean and variance of bus voltage amplitude and angle are shown in Table 5. In Table 6 active and reactive power of slack bus has been presented. The results show that the proposed new PEM has small and acceptable errors, although the uncertain loads have different unlike probability distributions. It can also be seen that the new PEM is still much faster than MCS. As the number of RV increases more points will be needed to gain suitable results. In contrast with traditional PEMs, increasing the number of estimating points is not a limitation in the new PEM since the estimating points are independent of the RV in the original space and the use of highorder moments are not required. As it is obvious, by applying new PEM, results are totally good and close to the results of MCS, while the elapsed time decreases. Furthermore, by existence of PDFs such as exponential distribution, results have acceptable accuracy and none of the estimated points were outside the defined region. The results are compared with MCS, PEM and new PEM from the viewpoint of both accuracy and time consumption in this case. It is worth nothing that MCS method may take CPU time of 52.5 s to obtain results, while the 2PE, New-2PE, New-3PE, New-5PE takes a CPU time of 1.16, 1.16, 2.67, 5.52 s on the same processor. These simulations have been conducted on a Pentium 4. 2.88 GHz dual core PC.
Table 3 Mean and variance of active powers flow through lines via five different methods. From bus
To bus
Active power Mean (PU)
1 2 3 6 10 13
5 3 4 11 11 14
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
80.1100 53.4417 13.5517 50.1718 39.8332 10.7560
82.1200 55.4400 15.4630 51.2291 40.5681 11.7382
81.5500 54.2201 14.2300 51.0101 40.2203 11.2204
80.9000 53.8801 13.7601 50.6900 39.9903 10.9101
80.1000 53.4420 13.5518 50.1718 39.8900 10.7700
0.2260 0.0843 0.2039 0.0025 0.2012 0.0019
0.2333 0.0944 0.2089 0.0339 0.2900 0.0100
0.2235 0.0856 0.2071 0.0021 0.2051 0.0021
0.2235 0.0856 0.2071 0.0021 0.2051 0.0021
0.2234 0.0855 0.2061 0.0020 0.2031 0.0020
Table 4 Mean and variance of reactive powers flow through lines via five different methods. From bus
To bus
Reactive power Mean (PU)
1 2 3 6 10 13
5 3 4 11 11 14
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
9.5026 12.5917 1.9027 7.4145 1.9718 11.7321
10.1100 11.6100 2.1101 7.9991 2.2212 12.0111
9.9919 11.9919 2.0001 7.8767 2.0010 12.8901
9.6666 12.3243 1.9999 7.6923 1.9999 11.7898
9.5011 12.601 1.9011 7.4142 1.9717 11.7322
0.0043 0.0037 0.0063 0.0006 0.0025 0.0014
0.0055 0.0031 0.0050 0.0057 0.0035 0.0010
0.0047 0.0035 0.0055 0.0005 0.0033 0.0015
0.0043 0.0035 0.0057 0.0006 0.0040 0.0019
0.0046 0.0033 0.0051 0.0006 0.0029 0.0014
59
M. Mohammadi et al. / Electrical Power and Energy Systems 51 (2013) 54–60 Table 5 The results of some buses voltage via five different methods. Bus
Item
Voltage amplitude Mean (PU)
1 2 3 5 7 8 9 11
Voltage amplitude
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
1.0024 1.0001 1.0014 1.0005 1.0023 0.9560 1.0475 1.0001
1.115 1.010 1.090 1.001 1.020 0.880 1.090 0.900
1.0110 1.0010 1.0020 1.0090 1.0036 0.9200 1.0880 0.9900
1.0090 1.0070 1.0090 1.0009 1.0029 0.0990 1.0666 1.0001
1.0040 1.0030 1.0011 1.0006 1.0025 0.0980 1.0477 1.0003
0.0032 0.0007 0.0017 0.0037 0.0032 0.0038 0.0027 0.0002
0.0036 0.0011 0.0012 0.0023 0.0030 0.0042 0.0022 0.0005
0.0036 0.0010 0.0015 0.0030 0.0030 0.0040 0.0024 0.0002
0.0036 0.0010 0.0015 0.0031 0.0030 0.0036 0.0029 0.0002
0.0036 0.0010 0.0015 0.0036 0.0030 0.0040 0.0024 0.0002
Table 6 The results of active and reactive power of slack bus via five methods. Item
Slack generator Mean (PU)
Active power Reactive power
Variance (PU)
MCS
2PE
N2PE
N3PE
N5PE
MCS
2PE
N2PE
N3PE
N5PE
89.223 8.7571
92.21 9.222
90.101 8.79
90.077 8.777
90.013 8.7649
0.1333 0.1543
0.08 0.1
0.08 0.1
0.05 0.09
0.01 0.01
4. Conclusion
Table 7 Mean value for different iteration of MCS. Output
P slack Q slack V1 V2 V3 V5 V7 V8
Number of iterations 100
500
1000
3000
10,000
88.991 8.555 0.999 1.0023 0.9902 0.9989 0.9912 0.9706
84.959 8.578 1.0014 0.9985 0.9995 0.9981 1.0008 0.9499
89.223 8.7571 1.0024 1.0001 1.0014 1.0005 1.0023 0.9560
89.999 8.5650 1.0022 1.0005 1.0008 0.9990 1.0021 0.9608
90.001 8.869 1.0022 1.0000 1.0010 1.0005 1.0020 0.9512
In this paper a new PEM has been proposed to evaluate the probability moments of load flow results. The new PEM has advantages of previous PEMs in comparison with MCS method while it overcomes the limitations of the PEMs in actual power grids. With these new PEM, increasing the number of estimated points is easier because the estimating points are independent of the RV in the original space and the use of high-order moments of the RV is not required. The results of the new PEM with those of MCS method show the accuracy for the proposed method. References
Table 8 Variance value for different iteration of MCS. Output
P slack Q slack V1 V2 V3 V5 V7 V8
Number of iterations 100
500
1000
3000
10,000
0.1103 0.1543 0.0031 0.0005 0.0011 0.0033 0.0033 0.0033
0.1103 0.1543 0.0029 0.0005 0.0011 0.0035 0.0034 0.0033
0.1333 0.1543 0.0032 0.0007 0.0017 0.0037 0.0032 0.0038
0.1339 0.1543 0.0033 0.0008 0.0018 0.0038 0.0032 0.0038
0.1331 0.1543 0.0030 0.0006 0.0016 0.0036 0.0031 0.0035
3.1. Sensitivity analysis In this subsection sensitivity of the number of iteration for proposed MCS has been investigated. The PLF has been done for different iterations of MCS (100, 500, 1000, 3000, 10,000). It should be mentioned that by using the method presented in [27], almost sane results have been obtained. Mean and variance have been used for as Indies to measure the MCS accuracy. Results have been shown in Tables 7 and 8. It can be seen that 1000 is suitable number and increasing the number of iteration only makes the simulation more time consuming.
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