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10
LOAD PROFILES SIMULATION FOR EVALUATION ENERGGY LOSSES IN
DISTRIBUTION NETWORKS
Balametov A.B., Halilov, E.D.
e-mail: balametov. [email protected]
Introduction. When calculating the energy losses in distribution networks using performance load profiles. Known methods of calculating energy losses in electric networks [1] are based on normal operating conditions and functioning of the electrical network, uninterrupted electricity supply to consumers.
Methods of calculating the losses of electricity use charts on duration. Earlier load profiles have stable characteristics and allow you to calculate the energy loss to adapt to these conditions, the simplified formulas. Currently, load profiles feeders 6-10 kV are many different forms, there were changes in the structure of energy consumption. Transition economies characterized by: non-uniformity in the daily load profiles; disconnection associated with non-payment for electricity; limitations associated with the overload of network elements, etc.
Most informative are the load profiles of individual groups of consumers for whom the known types of graphics. Load profiles of feeder formed of different consumer groups. The combination of load profiles generated total schedule feeder. Information about the probabilistic characteristics of load profiles is generally little known.
Calculation of energy losses by the method of medium loads is the use of expressions with the form factor. A calculation of the form factor reduces to obtaining the expressions having a clear and simple for hand calculation of the form. This is largely possible in a simple way to explain the patterns of influence on the profile, load losses of electricity. However, the pace of development of computer technology with great potential and their application in all spheres of government allow the use of complex computational algorithms, more flexible and accurate simulation. At the same time with this to some extent this may lose the simplicity and clarity of representation formulas for calculating the load losses.
Deterministic methods of calculation do not take into account the inaccuracy of initial data for plotting load. To overcome these deficiencies have developed methods for calculating the energy loss, based on the probabilistic representation of the graphs of electrical loads. These methods can be divided into: methods of submission of the load as a random variable and regression methods for calculating the losses.
Staging. Energy loss in the elements of an electric network is a function of the characteristics of load profiles. For the calculation of load losses in distribution networks using the method of average loads
awr = ap kit ,
L av f '
and the method of the number of hours the greatest losses
AWr =AP t ,
L max '
where APav - loss of power in the network at an average load of nodes (or networks in general) for the time T; k2f - a square form factor graphics power or current; APmax - loss of power in the network at maximum loads of nodes; t - the number of hours the maximum losses.
Key indicators of load profiles in the calculation of loss are: the number of hours of peak load Tmax, the fill factor loading schedule. Another important characteristic of load is the ratio of minimum load to maximum kmin=Pmin/Pmax. In the calculations of energy losses characterize the shape of load profiles parameters: the number of hours of the greatest losses t and form factor graphics power
k2f. The most accurate values of t and k f can be identified by well-known load
profiles. Research aimed at obtaining a more accurate dependency on the parameters characterizing k2f load profiles led to a set of design formulas. With unknown load schedule k2f value can be determined using various empirical formulas [1].
The most commonly used in technical literature are the following calculation formula k2f depending on two parameters, kZ and kmin [3]:
k2 = i + (1 -ka)2(k3 -KJ when A< 1;
f (2 kfill kmin Jkfll
k2 = i + (^W-^/ -/ when À>! , (1)
( 1 + kfill - 2kmin)kfill
where X = km
1 kfill
In [1] based on approximations performed alternative calculations for all possible configurations of the load duration profile in increments of 0.1 in both axes (t and kfill), a formula for the average expected value
k2 = 1+2^ (2)
f 3k
3kfiii
The error in the formula (1) using two parameters (kfill and kmin) is estimated [1] is about 10.8%, and the error formula (2) uses only one parameter, kfill is about 13%.
Assuming that the specification obtained by the two small parameters, in [1] proposed to use the formula (2). In this case, also noted that kmin has less credibility.
In [4] the error of the known empirical expression of certain k f, as well as the formula (1). Accuracy in determining k2f can increase the input except used two parameters kfill and kmin, additional parameters characterizing the load profiles. As an additional parameter that can be taken during the duration of the maximum and minimum load graphics, etc. For example, the duration of the off-peak schedule is one of the parameters characterizing the performance load profiles. Normally, advance information on the duration of the minimum load can be obtained. Since the total load profile is formed from the sum of standard load profiles, we can evaluate the length of the minimum load. As an additional parameter taken during the duration of treatment with minimal impact - Tmin or in relative units ktmin.
It is known that more significantly affect the parameters are taken into account in determining k f, the greater accuracy of simulation can be achieved. Thus, the load profile is proposed to characterize the three parameters kfill, kmin and ktmin.
In connection with the foregoing, the article examines the issues of error estimation k2f taking into account the dependence on two parameters, kfill and kmin and three kfill, kmin and ktmin
To assess the calculation errors k f in this article are considered: the use of simulation modeling of discrete possible configurations load duration profile and simulation load profiles analytical dependences in time.
Load profiles on duration are smoothly varying. In this regard, following the technique of estimating the loss of electricity distribution networks performance load duration profile as a continuously decreasing function of time. Calculations of energy losses are usually made on the PC. Therefore, excessive efforts to simplify the formulas for calculating k2f in modern conditions of development of computer technology due to the loss of accuracy unreasonable.
Simulation discrete simulation of the possible configurations of the load duration profile. Modeling of possible configurations of the load duration profile is reduced to the equivalent problem of simulation, a combination of levels of the columns of n * n and solving the following integral equations with inequalities.
Formed equation for specified values kfill and kmin in the form of equation
11 + I2 + ^
I = n • k
where In- the current value or the power corresponding n-th stage of load profile. Sets values of the first and the n-th column
I
n,
I = n • k
Formed by inequalities of the form
I2 * Ii:
I3 * I2,
I * I
n-1 •
(3)
(4)
(5)
Solution of equations (3), (4) and (5) are integer variables I2 ^ In-1. Search all the options for given values of kfill and kmin is produced by changing the values in columns (a decrease by one unit)
2
from the load profile corresponding to the maximum k f toward its reduction. With respect to the discrete change of stress level AAI=1/n and the time duration of the AAT=1/n discrete step changes in the level kfill has a value of AAKfill=1/n .
In accordance with the algorithm (3), (4) and (5) developed a mathematical model and simulation program for the possible configurations of the load duration profile. For example, Fig. 1 shows a histogram k f whenn=10, A=0.01 and 340 possible choice of load profiles when kfill=0.4, kmin=0.1 and its comparison with the normal distribution profile.
l43J
50 45 40 35
f 30 IL
-> 25
F(int) - 20
15
10
5
0 0
1 1 J.002
_ I I
Histogram
- Normal distribution
2
Fig. 1. Histogram k ffor load profiles kfill=0.4, kmin=0.1.
The results of processing histograms k f at different steps of discreteness show that the choice of the step discontinuity, having a sufficient number of possible load profiles, which allows to reliably determine characteristics of the distribution k2f have parameters close to the normal law. As shown in Fig. 1 case, a normal law with mean 1,609 and standard deviation of 0.127.
Table 1 shows the average k2f for possible configurations of load profiles for the duration in increments of 0.1 according to the kfill and kmin.
Analysis of the results shows that the use of formula (1) and (2) is associated with large systematic errors. Analysis of the results of the discrete simulation of characteristics of load profiles for various kfill, depending on kmin shows the possibility of increasing the accuracy by obtaining appropriate and adequate dependency formulas.
Equation (2), obtained by averaging the form factor k2f all possible load profiles, can not completely eliminate the error. For example, for values of kmin = 0.1 (1) and (2) have a negative error k2f. For values of kmin = 0.2 and kmin = 0.3, (1) is negative, and (2) positive systematic errors. For kfiu = 0.4, kmin = 0.1, (1) has a systematic error reaches up to -35.2%, and (2) to - 19.82%.
Using discrete simulation with kmjn> 0.1 with a limited step Aai=0.1 leads to a systematic error modeling. For a preliminary comparative evaluation of the error histograms of the distribution k2f. For example, if k3=0.19 and kmin=0.1 with a step of discreteness AAI=0.1 and the time duration of the AAT=0.1 there is a possibility k f =3.02, then kmin=0.1 and a discrete step c AAI=0.05, AAt=0.05, n=400, a step change in the level of discreteness k3 AAKfill=0.0025, we have over 300 options with an average k f =2.22. Thus, the average value, as determined in step discrete 0.1 in this case has an error of 36%.
2
Table 1. Results comparing the values k f by (1) and (2), depending on the kfill at kmin = 0.1.
№ Results of the load profiles simulation Results of the calculation according to the formulas
The fill factor, kfill Number of variants Form factor, k2f By formule (1) By formule (2) Error of formula %
(1) (2)
1 0.4 340 1.58 1.45 1.50
2 0.39 298 1.60 1.47 1.52 -8.36 -5.14
3 0.38 253 1.63 1.49 1.54 -8.59 -5.31
4 0.37 218 1.66 1.51 1.57 -8.81 -5.44
5 0.36 186 1.68 1.53 1.59 -8.8 -5.29
6 0.35 155 1.71 1.56 1.62 -8.94 -5.27
7 0.34 127 1.75 1.58 1.65 -9.53 -5.68
8 0.33 104 1.78 1.60 1.68 -9.9 -5.80
9 0.32 82 1.82 1.63 1.71 -10.47 -6.1
10 0.31 66 1.86 1.65 1.74 -11.02 -6.31
11 0.3 50 1.91 1.68 1.78 -11.8 -6.69
12 0.29 39 1.95 1.71 1.82 -12.54 -6.97
13 0.28 29 2.02 1.74 1.86 -13.96 -7.89
To ensure the adequacy of the discrete simulation problem arises of selecting a rational step, discrete, depending on the values of kfill and kmin. For example to load profiles with kZ <0.3 0.1 acceptance of discreteness step is coarse, in terms of number of charting options in terms of compliance with the normal distribution law.
Reduction of discrete steps increases the accuracy of the simulation. However, the sharply rising number of possible loads profiles. In this regard, along with a complete discrete simulation of characteristics of production load profiles according to the algorithm (3), (4) and (5) with a selectable discrete steps suggested below, the proposed use of a simplified simulation algorithm, which is based on the assumption that the distribution of k2f possible load profiles for the normal law.
Discrete simulation graphs of electrical loads for the duration of the choice of discrete steps, depending on the kfill and kmin and receive library approximated by improving the accuracy of modeling technical energy losses in distribution networks.
Load profiles on duration are smoothly varying. Using the full discrete model leads to a systematic error and the relatively time-consuming to model. Therefore, further consider the use of simulation load profiles analytic functions, which has certain advantages dimension of the task, speed and visibility, and can explain the causes of systematic error in formula (1) and their elimination.
In this regard, following the technique of estimating the loss of electricity distribution networks performance load profiles on duration as a continuously decreasing function and obtaining the necessary characteristics of the graph by direct integration.
Method of determining k f simulation load profiles analytical dependences in time. Energy losses in the elements of an electric network are a function of the characteristics of load
profiles. Load profiles on duration can be expressed in different functions: parabolic when kflii>0.7; linear at кял=0.5^0.7; exponential with кш =0.25^0.5; hyperbolic linear at кш < 0.25 etc. [2].
Equation (1) obtained an approximation of load profiles for the duration of the following analytical dependences in time:
(t Y
I = I - (I -1 . ) — при X> 1 (6)
max V max min / r-p ± \ у
V T у
i
( t V
I = I ■ + (I - I , ) 1 - - при X < 1 (7)
min V max min / гр r \ у
V T У
where - Imax, Imin values of maximum and minimum currents for the settlement period of time T. Auxiliary factor X determined as follows:
о I - 1 ■
2 _ av_mm
i - i
max av
In deriving (1) the following assumptions [3]: load profiles the load as a random variable has a beta - distribution; load profiles on duration represented by analytical dependences in time form (6) and (7). Given the fact that the analytical dependence (6) and (7) are inferable, the parameters for the beta - the distribution and, accordingly, (1).
Approximation load profiles analytical dependences of the form (6) and (7), although much more accurate simulation of energy loss, but does not completely eliminate the systematic errors [1]. In this regard, in [4, 5], attempts were made to obtain empirical relationships that eliminate these shortcomings. In [5], the choice of approximating functions load profiles different analytical dependences.
Next, we consider obtaining empirical approximation for k fload profiles exponential dependence of the form
I = I . + (I -1 . ) • e-(tt2t)P (8)
min V max min / v s
Here, a and p - zoom options, determined by approximation.
1.1
ii(t) I2(t) I3(t) I4(t) I5(t) I6(t) I7(t) I8(t) I9(t)
1.1 1
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
чС4' \ \
ч v \
\ '
■ До \\ M
ч. v\
V\ \\
î^y.
...
0 0.1 0.2 0.3 0.4 0.5 0. 0 t
0.7 0.8 0.9 1 1
Fig. 2. Family load profiles with kmi =0.4 , Imin=0.1 and ^=0.5 represented by a power function of the form (8) and the exponential form (7) for different p and a.
0
Improving the accuracy of modeling the energy loss and reduction of systematic errors can be achieved by selecting the type of approximating dependence (p and a) production load profiles. Modeling load profiles exponential form (8) by choosing p and a gives a family of graphs and kf are close to the real (Fig. 2).
Formulation of the problem of modeling the characteristics of production load profiles for the duration. Define the parameters a and p approximation load profiles dependence (8), which are also an implicit function of the parameters load profiles kfm , kmin and ktmin .
i
kflU =J(I mm + (Imax - I min) " *(9) 0
Dispersion for the given load profiles is determinates by expression
1 2 D = f{l . + (I -1 . ) • e-(at)P ) dt - k2f„ (10)
il \ min \ max mm/ I fill v '
0
To calculate the definite integral (9) and (10) used numerical integration methods, in particular, Simpson method by selecting the corresponding a and p. In general, the load profiles kmi = const have many solutions ai and pi and different dispersions.
Selection coefficients ai and pi, providing multiple solutions is a solution of the problem of minimizing the function
n2
1
fI .+ (I -1 ) • e-(at) dt - kf„
I min \ max min / fill
0
0 (11)
subject to the restrictions on kmin and ktmin as
0 < kmin < 1,
mm
0 < ktmin < 1.
Direct integration of expression (9, 10) is not possible and therefore requires the use of numerical methods.
When setting p is the minimization of (11) by selecting the value a, which provides a given value of kfm (9)? In the case where for a given p can not provide appropriate value of kfill, the change in p (increase) seek the solution set pi and ai.
Ranges of a and p depend on the shape of load profiles kfill and kmin. Modeling the load profile corresponding to given values of parameters kfill and kmin is not always possible to provide. For example, setting the parameter a <2 often does not provide specified kZ, kmin and ktmin by choosing a.
As a preliminary criterion for complete selection of options appropriate approximation of (9) proposed to use the condition
(Imax - Imin) ' e-0^ <Sk3 (12)
Here is invited sK3 =0.0005.
If condition (12) holds, then we can assume that (8) provides an approximation having an acceptable error.
The start time of minimum load graphics in the program is determined by the condition
(I " I ). e-(at)Pi " k <P (13)
V^max min/ tmin _ ktmin v1J/
The condition of precision search ktmin invited to take within sKtmin =0.001 ^ 0.01 Simulation modeling of load profiles family to determine the ranges of shape factor.
Simulation marginal production schedules by the condition of obtaining the lowest and highest values for k2f. Technique for modeling load schedule (8) with kfill (9) reduces to the problem
of finding the parameters a and p from (kp - kfill)2 ^ min. For this purpose, using quadratic interpolation functions (11) for values of a for a given p in three different locations
f(a) = a + ba + a 2 (14)
Algorithm for simulation of load profiles based on an iterative coordinate descent method and the method of quadratic interpolation. To find the minimum of the method of quadratic interpolation.
Programming. A program for simulation of load profiles (Fig. 3). The initial data inputs are: the filling factor of load profile kfill, the ratio of minimum load to maximum kmin, the relative duration of the off-peak schedule - ktmin, defined as the ratio of the length of the minimum load for the duration of the billing period. Simulation modeling of a family of load profiles by specifying parameters kfm, kmin, ktmin. In this case, determined by a and ktmin. Provides lower and upper limits of change p and pitch changes Ap. Usually for 10-40 iterations can be obtained practically acceptable approximation for a load profile using a quadratic interpolation.
Fig. 3. Flow chart of load profiles simulation.
Recommended values of a and p depending on the load shape. Below are the recommended values of a and p to approximateload profiles function of the form (8) in duration, depending on the kZ. For a given kfill and kmin and relative growth p increases the dispersion and the value of a. When a = const and increasing p, kfill decreases. An increase in a and p variance increases. To obtain relatively large kZ must specify a relatively large value of p. To obtain profiles with longer duration load minimum ktmin necessary to increase the value of a. To obtain profiles with longer duration of maximum load is also necessary to increase the value of a. Ranges of the dispersion of production schedules with the specified kfill defined by setting p of 0.5 < p < 30 and a step change in Ap 0.1 < Ap < 1, the choice of a corresponding to the set parameters and kfill and kmin.
Numerical experiment. Simulation results for the load profile with kfill = 0.4, kmin = 0.1 are shown in Table 2.
Table 2. Parameters of the simulation load profile.
Numbe Coefficient of Results of
r formula (8) simulation (9-13)
ai pi k2f ktmin ktmax
1 2.666 1.8 1.587
2 2.661 1.9 1.61 0.17 0.08
3 2.658 2 1.631 0.2 0.08
4 2.679 3 1.777 0.38 0.14
5 2.719 4 1.857 0.46 0.17
6 2.755 5 1.907 0.5 0.2
7 2.783 6 1.941 0.53 0.22
8 2.806 7 1.966 0.55 0.23
9 2.825 8 1.985 0.57 0.24
10 2.841 9 2.0 0.58 0.25
11 2.854 10 2.012 0.59 0.26
12 2.865 11 2.022 0.59 0.26
13 2.875 12 2.03 0.6 0.27
14 2.883 13 2.037 0.61 0.27
15 2.891 14 2.043 0.61 0.28
16 2.897 15 2.049 0.61 0.28
17 2.903 16 2.053 0.62 0.28
18 2.908 17 2.058 0.62 0.29
19 2.906 18 2.059 0.62 0.29
Dependence of the squared form factor, the resulting simulation of load profiles for the values of kfill = 0.4, kmin = 0.1 shows that for the same values of the kfill value k2f varies within 1.587 ^ 2.059. The average value is set to k2fsr =1.823. Limits k2f deviations from the average amount ±13%. The relative duration of minimum load varies in ktmin = 0.13 ^ 0.62.
Simulation modeling of load profiles as we obtain the limit profiles for k2f appropriate minimum, taking kmin = 0.25 * kfill, and maximize, taking kmin = 0.6 * kfill algorithm (6-14), whose results are shown in Figure 4.
* at kmin=0.6*kfill • at kmin=0.25*kfill
-Power (at kmin=0.25*kfill) -Power (at kmin=0.6*kfill)
Fig. 4. Profiles of variation squared form factor of the kZ to the possible load profiles.
Dependence of the squared form factor, the resulting simulation schedules (Table 2) and (Fig. 4) shows that for the same values of the kZ and kmin time duration of treatment with minimal impact ktmin and the corresponding values k2f vary widely. In the presence of advance information about ktmin available to assess k2f depending on three parameters: kfill, kmin and ktmin, allowing more accurate simulation.
Comparison of calculation results of simulation program schedules with the most commonly used empirical formulas.
Produced by comparing the results of the calculation k2f based on simulation graphs of electrical loads for the duration of the algorithms (3) - (5) and (6-14) (Table 3).
Table 3. The results of comparison k2f for kmin = 0.1 and p = 2 by simulation load profiles _ by function of the form (11) and (3) - (5)_
Filling Results of simulation Results of Error of simulation
coefficient, (10) discrete (5)-(7), %
k3 simulation
a k2fn k2fd (k2fg- k2fn)*100\ k2fn
0.250 5.317 2.167 2.198 1.43
0.300 3.988 1.970 1.905 -3.30
0.350 3.190 1.789 1.709 -4.47
0.400 2.658 1.631 1.582 -3.00
Produced by comparing the results of the calculation for the average k2f formula (1) and simulation plots of electrical loads for the duration of the algorithm (6-14) for load profiles, depending on the kZ and kmin, taking kmin = v * kmi. v = 0.666, 0.5, 0.25, 0.125. The calculation results k2f shown in Table 4.
Table 4. Calculation errors k2f formula (1) for small values of kfill
№ Filling coefficient , Minimum of graph, Model estimated value k2f by Error of k2f by formula (1)
kfill k formula (1) algorithm (8-13)
1 0.5 0.3333 1.143 1.236 -7.52
2 0.25 1.2 1.366 -12.15
3 0.125 1.273 1.592 -20.04
4 0.0625 1.304 1.684 -22.57
5 0.4 0.2666 1.225 1.355 -9.59
6 0.2 1.321 1.585 -16.66
7 0.1 1.45 1.823 -20.46
8 0.05 1.508 1.971 -23.49
9 0.3 0.2 1.363 1.522 -10.45
10 0.15 1.527 1.792 -14.79
11 0.075 1.754 2.213 -20.74
12 0.0375 1.86 2.457 -24.30
13 0.2 0.1333 1.64 1.909 -14.09
14 0.1 1.941 2.344 -17.19
15 0.05 2.371 2.991 -20.73
16 0.025 2.577 3.371 -23.55
Empirical formula (1) has negative systematic inaccuracy, and (3) has a positive systematic inaccuracy in k2f compared with the average values obtained by the technique (6-14). The values of the systematic inaccuracy k2f vary in the range (7 ^ 45%) depending on the kfill.
Fig. 5. Depending on the results k2f simulation load profiles (6-14) and by (1) kfill = 0.3 from kmin.
The errors increase with decreasing kfill from 0.5 in the direction of small values.
kfin = 0.5 for the values of the errors in k2f vary in the range (7 - 15%), for kfill = 0.4 in the range (10 - 17%), for kfm = 0.3 in the range (16 - 24%) and kfm = 0.2, range (22 - 45%).
Depending on the results k2f simulation load profiles (6-14) and by (1) kfill = 0.3 from kmin shown in Fig. 5.
Dependence of the error in k2f by (1) kfill = 0.3 from kmin is shown in Fig. 6.
Fig. 6. Dependence of the error in k2f by (1) kZ = 0.3 from kmin .
A comparison of shape factor family of load profiles on duration as a power function (7) and expression (8) show that, depending on the value of tmax approximation coefficients a and pload profiles take different values. k2f value varies in the range 1.631 ^ 1.856.
Approximation load profiles dependence (7) compared with (8) and algorithm (9) - (13) has a negative error of the estimated 7-30% depending on the kfill and kmin.
The reasons for the growth of systematic errors (1) for small values of the fill factor due to the use for the approximation of production schedules depending on the form (7), which in this case, the forms chart from almost zero to a maximum load.
Using simulation diagrams of electrical loads according to the algorithm (6) - (14) allows more flexible modelling k2f and meets the additional desired parameters: the duration of the minimum and maximum loads.
Thus, the use of simulation load profiles exponential dependence of the form (8) differs from the known fact that is based on close to real load profiles and, accordingly, improves the accuracy of simulation k2f.
Conclusions
1. The technique of simulation load profiles of possible schedules for the duration of the electrical loads in the form of a continuous function approximation schedules exponentially.
2. Produced by comparison of the calculated form factor k2f with the results used in practice, empirical formulas, and establishes the presence of significant systematic errors at small values of the fill factor to 30%.
3. The proposed technique for modelling the characteristics of load profiles duration as a continuous function of improving the accuracy and flexibility of modelling k2f and losses in distribution networks.
REFERENCES
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2. Klebanov L.D. Question of a technique of definition and decrease in energy losses of electric networks. H: Publishing house I LIE, 1973, - 72 p.
3. Anisimov L.P., Levin H.P., Pekelis V.G. Design procedure of energy losses in working distributive networks. Electricity, 1975, №4, p 27-30.
4. Balametov A.B., Mamedov S.Q. About definition of factor of the form at calculations of losses of the electric power in view of restrictions in electrosupply. Energetics Minsk-2002.-№2, p. 21-29.
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