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                                                       ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013



                  Loss Estimation: A Load Factor Method
                               T. Murali Krishna1, Dr. N.V. Ramana2, Dr. S. Kamakshaiah3
                                1
                                CVR College of Engineering/EEE Department, Hyderabad, India
                                                  Email: cvr.murali@gmail.com
                           2
                             JNT University College of Engineering/EEE Department, Jagityala, India
                                                   Email: nvrjntu@gmail.com
                                     3
                                       JNT University/EEE Department, Hyderabad, India
                                                 Email: skamakshaiah@yahoo.com

Abstract— This paper focuses on Power Loss estimation in             distribution companies with taxing challenges. The accurate
Electrical Sub Transmission and Distribution systems. The            estimation of electrical losses enables the supply authority
earliest empirical approaches were by Buller and Woodrow in          to determine the operating costs, with greater accuracy, and
1928, Hoebel in 1959, M.W. Gustafson from 1983 to 1993.              abate the dissipation of power.
Gustafson changed the values of the coefficients and provided
                                                                         Total system loss is the difference between the energy
them with a constant loss term. It has been observed that this
approach is not suitable for the present load scenario. In this
                                                                     sent and the energy received by the end users. Distribution
paper, its successive approach has been proposed and tested          system losses are of two types:
with real time data. It has been concluded that the relationship     · Technical losses, which arise due to the current passing
between loss factor and load factor is not as complicated as              through the network components, i.e. the conductors,
perceived but easily understandable. By using exponential                 coils and the excitation of the windings of the
curve fitting, a relationship that is very close to reality can be        transformers and others.
obtained. To verify the obtained equation, data has been             ·     Non-Technical Losses are those ones that cannot be
collected from a 33kV sub transmission line existing between              calculated beforehand. For e.g. problems with metering,
132kV/33kV Thurkayamjal substation and 33kV/11kV
                                                                          billing or collection systems, and the consumers pilfering
Hayathnagar substation, APTRANSCO, Andhra Pradesh.
                                                                          power.
Index Terms— Load factor, Loss factor, Load curve, Loss curve,           The loss factor can be used to calculate energy losses for
exponential curve fitting, Sub Transmission and Distribution.        those parts of the electric system where the current flowing
                                                                     is proportional to system load each hour, which would
                        I. INTRODUCTION                              typically be the distribution and sub transmission systems.
                                                                     Attempting to make a lead in this aspect in 1928, Buller and
    According to the fundamental laws of nature, energy can-         Woodrow proposed a relationship between load factor and
not be converted from one form to the other, unless until            loss factor, by using a completely empirical approach [1].
there is an opposition to the conversion. An analogous state-        This approach was used by Hoebel in 1959 [2]. Hoebel
ment exists for Electric Power Transmission and Distribu-            asserted that “A number of typical load curves experienced
tion. According to this statement, power cannot be transmit-         on a large system have been studied to determine this
ted from one port to the other, without losses. They arise as        relationship”. Martin. W. Gustafson, a Transmission/
power flows, to satisfy consumer demands through the net-            Substation design engineer, modified the existing relationship
work. Electrical energy is converted from various forms of           between the two factors [3]. He modified the constant
conventional and non conventional energy sources at suit-            coefficient and also added a new coefficient for losses [4],
able locations, transmitted at a high voltage over long dis-         [5]. Later on various methodologies were proposed to estimate
tance and distributed to the consumers at a medium or low            sub transmission and distribution losses [8]. Simplified
voltage. A distribution system consists of a substantial num-        models of distribution feeder components for load flow
ber of distribution transformers, feeders, and laterals, due to      calculations are proposed in [9]. An approximate loss formula
a large number of consumers spread over a wide geographi-            based on the load-flow model is proposed in [10].Simulation
cal area. Some of the input energy is dissipated in the con-         of distribution feeders with load data estimated from typical
ductors and transformers along the delivery route. Losses            customer load patterns is presented in [11]. Closed-form
increase the operating cost, estimated to add 10% to the cost        formulas for computation of losses using a three phase load-
of electricity and also approximately 25% to the cost of deliv-      flow model are proposed in [12]. Loss formulas based on
ery. These costs are levied on the consumers.                        fuzzy-c-number (FCN) clustering of losses and cluster-wise
    Studies have indicated that losses in a distribution system      fuzzy regression techniques are presented in [13]. A hybrid
contribute to a major part of losses in a power system               top/down and bottom/up approach is applied to estimate
(approximately about 70% of total losses). Therefore, the            losses in [14]. A new approach has being proposed for
distribution system loss has become more and more of                 nontechnical loss analysis for utilities using the modern
concern, because of the growth of load demand and the wide           computational technique extreme learning machine [15].
area it covers. It is important to estimate them accurately in       Difficulty in obtaining accurate data has a significant impact
order to take appropriate measures for reducing the losses.          on the accuracy of their results.
The evaluation and reduction of energy losses present the
© 2013 ACEEE                                                    9
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                                                     ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013


    Mathematics provides us curve fitting methods to                    As loss varies with the square of the load,
establish the exact relationship between two sets of data.
                                                                                  LSf = t/T + (PLD 2/PLD 1)2(1 – t/T)              (3)
The basic approaches like that of Buller and Woodrow utilized
curve fitting, to find the relationship. In any relationship                Buller and Woodrow correctly believed that the rela-
between loss factor and load factor, providing fixed limits to          tionship of load factor and loss factor on an actual electric
the values of the coefficients is important, as this helps in           system should fall between the two boundary conditions
judging if the values of the coefficients obtained are right or         viz. complete peak (t’!T) and complete off -peak (t’!0) . They
not. Before discussing the new approach to estimate energy              plotted these two extremities and applied curve fitting for the
losses by using mathematical methods, it is important to                area within the two curves, to get the relationship between
understand the basic approach i.e., Buller & Woodrow                    the two factors as shown in the Fig. 2. This work led them to
Equation. By using all the following approaches, the loss               the development of equation (4) with a constant coefficient
factor of a system can be approximated. But, the loss factor            of 0.3. The constant coefficient was derived from representa-
of a system does not give us any information of the                     tive load curves based on the available data and limited com-
performance of the system. We must use the basic loss factor            puting capability in 1928.
definition to calculate the average losses of the system.                            LSf = (LDf)2(1-X) + (LDf)X                     (4)
                                                                                          Where, LSf = Loss Factor
                     II.   BASIC APPROACHES                                                   LDf = Load Factor
                                                                                          X = Constant Coefficient
    In 1928, F.H. Buller and C.A. Woodrow, two engineers
with the General Electric Company, developed an empirical
equation (4) between Loss factor and Load factor [l]. Buller
and Woodrow assumed an arbitrary and idealized case of a
load curve, where it consisted of a peak load for a time t, and
an off-peak load for a time (T-t), where T was the complete
time period taken into account as shown in Fig. 1 [7]. As the
loss varies with square of the current, the loss varies with the
square of the load. Therefore, the loss curve too is in the
same shape as that of the load curve. The load curve is a plot
of load variation as a function of time for a definite group of
end-users. Similarly, the loss curve is a plot of loss variation
as a function of time. Load Factor (LDf) is defined as the ratio
                                                                                    Figure 2. Loss Factor versus Load Factor
of average load to the peak load over a designated period of
time. Loss Factor (LSf) is defined as the ratio of average loss             Even though, Buller and Woodrow’s approach was
to the peak loss over a designated period of time.                      considered to be useful, and it has been used, it consists of
                                                                        approximations far from reality. The load curve considered
                                                                        was an ideal one consisting of only peak and off peak as
                                                                        shown in Fig 1. The average is a function of only peak and
                                                                        off peak. But the intermediate values affect the average. In
                                                                        equation (3) it was considered that

                                                                                          PLSk = Ck PLDk [k=1,2,3…]

                                                                            With an assumption C1 = C2. This need not be right, as
                                                                        the value of the coefficient can be different at the off-peak,
                                                                        than at the peak.
                                                                            H. F. Hoebel used the original equation (4) and also
                                                                        developed equation (5) in terms of loss factor and load factor
                                                                        with an exponential coefficient [2]. The value for the exponent
                                                                        commonly used in practice was 1.6.
                                                                                     Loss Factor = (Load Factor)1.6                 (5)
                                                                            Martin. W. Gustafson developed a quadratic equation to
                                                                        determine sub transmission and distribution system losses
                                                                        [3]. He also observed that the values of 0.3 for the constant
                                                                        coefficient and 1.6 for the exponential coefficient, no longer
              Figure 1. Load Curve and Loss Curve                       appear to be appropriate and revised the coefficients as 0.08
         LDf = t/T + (PLD 2/PLD 1)(1 – t/T)                 (1)         for the constant coefficient and 1.912 for the exponent [4]. A
                                                                        constant term is added which represents no load losses [5].
        EQf = t/T + (PLS 2/PLS 1)(1 – t/T)                  (2)         He extended the same approach to determine transmission
© 2013 ACEEE                                                       10
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                                                      ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013


system losses with special considerations [6].                     and loss factors. Calculated value of the coefficient ‘X’ was
                                                                   0.6432. It is further used to calculate the loss factors from
             III. PROPOSED MATHEMATICAL APPROACH                   respective monthly load factors. The coefficients of the pro-
                                                                   posed approach ‘A’ and ‘B’ were calculated from the average
    Many processes in nature have exponentia1
                                                                   monthly load and loss factors. The calculated value of the
dependencies. The advantage of exponential equations is
                                                                   coefficients A and B were -2.2073 and 2.231. They are further
that with the right values of coefficients, any curve can be
                                                                   used to calculate the loss factors from the respective monthly
approximated in the form of an exponential equation i.e., within
                                                                   load factors. The summary of the analysis is shown in Table
the limits of the given data. In exponential curve fitting, any
                                                                   1. From the results, it can be seen that the method presented
function is approximated as
                                                                   is a new tool capable of estimating sub-transmission and
                  y = AeBx                                   (6)
                                                                   distribution system losses with greater accuracy, than what
    The advantage of such an equation is that ‘A’ decides
                                                                   was previously available. It is also observed that the values
the ‘y’ intercept of the curve, and ‘B’ gives the function its
                                                                   of the coefficients are very important in estimating losses,
curvature. Taking the logarithm of both sides of (6)
                                                                   but they cannot be generalized for any system. This is due to
               Log y = Log A + Bx                            (7)
                                                                   the fact that each system has a different load profile, which
Then, the best fit-values are
                                                                   leads to different values of coefficients. Hence, it is advised
                   n         n 2 n                                 that a distribution company, use previous data, to calculate
                   ln yi  xi   xi ln yi
                                                                   the values of the coefficients, for their respective load pro-
             A  i 1     i 12     i 1
                                         2                         files. Fitting an exponential curve to data does not need any
                         n 2
                                n
                      n  xi   xi
                       i 1     i 1
                                                            (8)
                                                                   tedious programming or simulation.

                   n            n n                                      V. CALCULATION OF LOSSES FROM THE LOAD FACTOR
                 n  xi ln yi   xi  ln y                            Using the above mentioned method, we calculated the
                  i1          i1 i1 i
              B                                                   loss factor of a system, at a particular instant. The loss at that
                       n 2 n 2
                               
                     n  xi   xi
                      i1       i1
                                                            (9)
                                                                   time can be estimated by using the basic definition of loss
                                                                   factor, viz.
    Where ‘xi’ and ‘yi’ are the sets of data, which must be fit                          AverageLoss
                                                                                LS f 
in the required curve. Using (6) to fit a curve between the two                           PeakLoss
factors.                                                           Therefore,
Let the relationship between them be
                 LSf = aebLDf                              (10)                  AverageLoss  LS f  PeakLoss
Taking the natural logarithm on both sides of (10)                    But, using the above mentioned approach, the loss factor
                                                                   was estimated as
              ln(LSf ) = ln(a) + bLDf                     (11)
                                                                   LS f  aebLD f
   This can be plotted as a straight line between the natural      Hence,
logarithm of loss factor and load factor. Therefore, the general
equation of the relationship can be shown                                                          bLD
                                                                                                          
                                                                               AverageLoss  ae f  (PeakLoss)
                                                                      Most of the electrical systems have constant values of
          ln(LSf ) = A + B (LDf )                          (12)    their peak losses. This value of the peak loss can be taken
    The values of these coefficients can be determined by          from the data is collected regularly, by the distribution
using (8) and (9). Due to its desired features of a given ‘y’      company. Therefore,
intercept value, and curvature, it can be used easily to                                                      bLD f
estimate the loss factor of a system, from a given value of a                              AverageLoss  Pe
load factor.                                                       Where,                 P  a  PeakLoss

                           IV. RESULTS                                 This relationship was used to calculate the average losses
                                                                   of the system from which data was collected. The results are
    The proposed approach was tested on a 33kV sub
                                                                   shown in Table. II.
transmission line existing between 132kV/33kV Thurkayamjal
substation and 33kV/11kV Hayathnagar substation,
                                                                                       CONCLUSIONS
APTRANSCO, Hyderabad, Andhra Pradesh. The annual load
and loss data of the test system was collected for one year        Buller and Woodrow were right, but their approach con-
and the monthly load and loss factors were evaluated by        tained too many assumptions to be considered to be close to
plotting the monthly load and loss curves. The monthly Load    the original value. Hoebel relationship had the capacity to
Curves and Loss curves are shown in Fig. 3 and Fig. 4.         get a curvature close to the original curve, but it ignored
    The constant coefficient ‘X’ of Buller and Woodrow (1)     constant losses. Every relationship between two sets of data
has been calculated from the average values of monthly load    can be expressed in terms of infinite powers of the ‘x’ coeffi
© 2013 ACEEE                                                11
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                                                       ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013




                                                    Figure 3. Monthly Load Curves
cient. Martin. W. Gustafson made use of this polynomial, but                                ACKNOWLEDGMENT
utilized only a second order equation. This was the source of
                                                                        We acknowledge the contributions of V. Prameela, Assistant
error. The new approach can be considered to be very close
                                                                    Engineer, APTRANSCO, Hyderabad in Instrumentation and the
to the original relationship, because it considers all the terms    data collection and also thank Kiran Kumar Challa, M.E (IISc Ban-
in the polynomial expansion (in the expansion of ex) includ-        galore) and P. Sai Gautham for several discussions during this work.
ing a constant term which represents total constant losses
due to large number of Distribution Transformers in the Elec-
trical Distribution System.
© 2013 ACEEE                                                     12
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                                                      ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013




                                                    Figure 4. Monthly Loss Curves

                          REFERENCES                                     [4] M.W. Gustafson, J. S. Baylor, and S. S. Mulnix, “Equivalent
                                                                             hours Loss Factor revisited,” IEEE Trans. Power Syst., vol. 3,
[1] F. H. Buller and C. A. Woodrow, “Load factor equivalent hours            no. 4, pp. 1502–1507, Nov. 1988.
    values compared,” Electr. World, Jul. 1928.                          [5] M. W. Gustafson and J. S. Baylor, “Approximating the system
[2] H. F. Hoebel, “Cost of electric distribution losses,” Electr.            losses equation,” IEEE Trans. Power Syst., vol. 4, no. 3, pp.
    Light and Power, Mar. 1959.                                              850–855, Aug. 1989 .
[3] M. W. Gustafson, “Demand, energy and marginal electric               [6] M. W. Gustafson and J. S. Baylor, “Transmission loss evaluation
    system losses,” IEEE Trans. Power App. Syst., vol. PAS-102,              for electric systems,” IEEE Trans. Power Syst., vol. 3, no. 3,
    no. 9, pp. 3189–3195, Sep. 1983.                                         Aug. 1988.

© 2013 ACEEE                                                        13
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                                                          ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013

                                                     TABLE 1. COMPARISON O F LOSS FACTORS




        TABLE II. C OMPARISON OF ACTUAL AND C ALCULATED LOSS                 “Simplified feeder modeling for load flow calculations,” IEEE
                                                                               Trans. Power Syst., vol. PWRS-2, no. 1, pp. 168–174, Feb.
                                                                               1987
                                                                          [10] R. Baldick and F. F. Wu, “Approximation formulas for the
                                                                               distribution system: The loss function and voltage
                                                                               dependence,” IEEE Trans. Power Del., vol. 6, no. 1, pp. 252–
                                                                               259, Jan. 1991
                                                                          [11] C. S. Chen, M. Y. Cho, and Y.W. Chen, “Development of
                                                                               simplified loss models for distribution system analysis,” IEEE
                                                                               Trans. Power Del., vol. 9, no. 3, pp. 1545–1551, Jul. 1994.
                                                                          [12] H.-D. Chiang, J.-C. Wang, and K. N. Miu, “Explicit loss formula,
                                                                               voltage formula and current flow formula for large scale
                                                                               unbalanced distribution systems,” IEEE Trans. Power Syst.,
                                                                               vol. 12, no. 3, pp. 1061–1067, Aug. 1997.
                                                                          [13] Y.-Y. Hong and Z.-T. Chao, “Development of energy loss
                                                                               formula for distribution systems using FCN algorithm and
                                                                               cluster-wise fuzzy regression,” IEEE Trans. Power Del., vol.
                                                                               17, no. 3, pp. 794–799, Jul. 2002.
                                                                          [14] C. A. Dortolina and R. Nadira, “The loss that is unknown is no
                                                                               loss at all: A top-down/bottom-up approach for estimating
[7] Turan Gönen, “Electric Power Distribution System Engineering”,             distribution losses,” IEEE Trans. Power Syst., vol. 20, no. 2,
     McGraw-Hill, 1986.                                                        pp. 1119–1125, May 2005.
[8] “Energy Loss Estimation in Distribution Feeders” P. S.                [15] Power Utility Nontechnical Loss Analysis with Extreme
     Nagendra Rao and Ravishankar Deekshit, IEEE Transactions                  Learning Machine Method” A. H. Nizar, Z. Y. Dong,Y. Wang,
     on Power Delivery, vol. 21, no. 3, July 2006                              IEEE Transactions On Power Systems, vol. 23, no. 3, August
[9] N. Vempati, R. R. Shoults, M. S. Chen, and L. Schwobel,                    . 2008.
© 2013 ACEEE                                                         14
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                                                         ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013


             T. Murali Krishna has received B.Tech in Electrical                            Dr.S.Kamakshaiah has received BE (Hons)
             & Electronics Engineering from Nagarjuna University,                           (Gold Medalist) at S. V. University in 1962, India.
             India in 2002 and M.Tech from National Institute of                            He received his ME (HVE) and PHD (Gold
             Technology Calicut, India in 2004. He is a research                            Medalist) from IISC, Bangalore, India. He worked
             student of JNT University, Hyderabad, India and                                as Professor and Head, Dept of EEE, Chairman of
working as an Associate Professor at CVR College of Engineering,            Electrical science at JNT University Ananthapur. He won the ‘Best
Hyderabad, India. His areas of interest are Power Systems and               Teacher Award’ for 1997-98. He has around 40 IEEE publications
Electrical Machines.                                                        to his name. He is author of 10 text books. He has guided many
              Dr. N. Venkata Ramana has received M.             Tech        people for Ph.D. His areas of interest are Electrical Networks,
              from S.V.University, India in 1991 and Ph.D. in               Electrical Machines, HVDC, GIS, and Power Systems. He presented
              Electrical Engineering from Jawaharlal Nehru                  papers at various IEEE conferences at USA and Canada.
              Technological University (J.N.T.U), India in January
              2005. He is currently Professor and Head of the
Department, EEE at J.N.T. University College of Engineering,
Jagityal, Andhra Pradesh, India. His main research interest includes
Power System Modeling and Control. He published 32 international
journals and conference papers and two text books titled “Power
System Analysis” and “Power system operation and control for
Engineering Graduate Students. He visited various universities in
USA, Canada, Singapore and Malaysia.




© 2013 ACEEE                                                           15
DOI: 01.IJEPE.4.1.1065

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Loss Estimation: A Load Factor Method

  • 1. Full Paper ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013 Loss Estimation: A Load Factor Method T. Murali Krishna1, Dr. N.V. Ramana2, Dr. S. Kamakshaiah3 1 CVR College of Engineering/EEE Department, Hyderabad, India Email: cvr.murali@gmail.com 2 JNT University College of Engineering/EEE Department, Jagityala, India Email: nvrjntu@gmail.com 3 JNT University/EEE Department, Hyderabad, India Email: skamakshaiah@yahoo.com Abstract— This paper focuses on Power Loss estimation in distribution companies with taxing challenges. The accurate Electrical Sub Transmission and Distribution systems. The estimation of electrical losses enables the supply authority earliest empirical approaches were by Buller and Woodrow in to determine the operating costs, with greater accuracy, and 1928, Hoebel in 1959, M.W. Gustafson from 1983 to 1993. abate the dissipation of power. Gustafson changed the values of the coefficients and provided Total system loss is the difference between the energy them with a constant loss term. It has been observed that this approach is not suitable for the present load scenario. In this sent and the energy received by the end users. Distribution paper, its successive approach has been proposed and tested system losses are of two types: with real time data. It has been concluded that the relationship · Technical losses, which arise due to the current passing between loss factor and load factor is not as complicated as through the network components, i.e. the conductors, perceived but easily understandable. By using exponential coils and the excitation of the windings of the curve fitting, a relationship that is very close to reality can be transformers and others. obtained. To verify the obtained equation, data has been · Non-Technical Losses are those ones that cannot be collected from a 33kV sub transmission line existing between calculated beforehand. For e.g. problems with metering, 132kV/33kV Thurkayamjal substation and 33kV/11kV billing or collection systems, and the consumers pilfering Hayathnagar substation, APTRANSCO, Andhra Pradesh. power. Index Terms— Load factor, Loss factor, Load curve, Loss curve, The loss factor can be used to calculate energy losses for exponential curve fitting, Sub Transmission and Distribution. those parts of the electric system where the current flowing is proportional to system load each hour, which would I. INTRODUCTION typically be the distribution and sub transmission systems. Attempting to make a lead in this aspect in 1928, Buller and According to the fundamental laws of nature, energy can- Woodrow proposed a relationship between load factor and not be converted from one form to the other, unless until loss factor, by using a completely empirical approach [1]. there is an opposition to the conversion. An analogous state- This approach was used by Hoebel in 1959 [2]. Hoebel ment exists for Electric Power Transmission and Distribu- asserted that “A number of typical load curves experienced tion. According to this statement, power cannot be transmit- on a large system have been studied to determine this ted from one port to the other, without losses. They arise as relationship”. Martin. W. Gustafson, a Transmission/ power flows, to satisfy consumer demands through the net- Substation design engineer, modified the existing relationship work. Electrical energy is converted from various forms of between the two factors [3]. He modified the constant conventional and non conventional energy sources at suit- coefficient and also added a new coefficient for losses [4], able locations, transmitted at a high voltage over long dis- [5]. Later on various methodologies were proposed to estimate tance and distributed to the consumers at a medium or low sub transmission and distribution losses [8]. Simplified voltage. A distribution system consists of a substantial num- models of distribution feeder components for load flow ber of distribution transformers, feeders, and laterals, due to calculations are proposed in [9]. An approximate loss formula a large number of consumers spread over a wide geographi- based on the load-flow model is proposed in [10].Simulation cal area. Some of the input energy is dissipated in the con- of distribution feeders with load data estimated from typical ductors and transformers along the delivery route. Losses customer load patterns is presented in [11]. Closed-form increase the operating cost, estimated to add 10% to the cost formulas for computation of losses using a three phase load- of electricity and also approximately 25% to the cost of deliv- flow model are proposed in [12]. Loss formulas based on ery. These costs are levied on the consumers. fuzzy-c-number (FCN) clustering of losses and cluster-wise Studies have indicated that losses in a distribution system fuzzy regression techniques are presented in [13]. A hybrid contribute to a major part of losses in a power system top/down and bottom/up approach is applied to estimate (approximately about 70% of total losses). Therefore, the losses in [14]. A new approach has being proposed for distribution system loss has become more and more of nontechnical loss analysis for utilities using the modern concern, because of the growth of load demand and the wide computational technique extreme learning machine [15]. area it covers. It is important to estimate them accurately in Difficulty in obtaining accurate data has a significant impact order to take appropriate measures for reducing the losses. on the accuracy of their results. The evaluation and reduction of energy losses present the © 2013 ACEEE 9 DOI: 01.IJEPE.4.1.1065
  • 2. Full Paper ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013 Mathematics provides us curve fitting methods to As loss varies with the square of the load, establish the exact relationship between two sets of data. LSf = t/T + (PLD 2/PLD 1)2(1 – t/T) (3) The basic approaches like that of Buller and Woodrow utilized curve fitting, to find the relationship. In any relationship Buller and Woodrow correctly believed that the rela- between loss factor and load factor, providing fixed limits to tionship of load factor and loss factor on an actual electric the values of the coefficients is important, as this helps in system should fall between the two boundary conditions judging if the values of the coefficients obtained are right or viz. complete peak (t’!T) and complete off -peak (t’!0) . They not. Before discussing the new approach to estimate energy plotted these two extremities and applied curve fitting for the losses by using mathematical methods, it is important to area within the two curves, to get the relationship between understand the basic approach i.e., Buller & Woodrow the two factors as shown in the Fig. 2. This work led them to Equation. By using all the following approaches, the loss the development of equation (4) with a constant coefficient factor of a system can be approximated. But, the loss factor of 0.3. The constant coefficient was derived from representa- of a system does not give us any information of the tive load curves based on the available data and limited com- performance of the system. We must use the basic loss factor puting capability in 1928. definition to calculate the average losses of the system. LSf = (LDf)2(1-X) + (LDf)X (4) Where, LSf = Loss Factor II. BASIC APPROACHES LDf = Load Factor X = Constant Coefficient In 1928, F.H. Buller and C.A. Woodrow, two engineers with the General Electric Company, developed an empirical equation (4) between Loss factor and Load factor [l]. Buller and Woodrow assumed an arbitrary and idealized case of a load curve, where it consisted of a peak load for a time t, and an off-peak load for a time (T-t), where T was the complete time period taken into account as shown in Fig. 1 [7]. As the loss varies with square of the current, the loss varies with the square of the load. Therefore, the loss curve too is in the same shape as that of the load curve. The load curve is a plot of load variation as a function of time for a definite group of end-users. Similarly, the loss curve is a plot of loss variation as a function of time. Load Factor (LDf) is defined as the ratio Figure 2. Loss Factor versus Load Factor of average load to the peak load over a designated period of time. Loss Factor (LSf) is defined as the ratio of average loss Even though, Buller and Woodrow’s approach was to the peak loss over a designated period of time. considered to be useful, and it has been used, it consists of approximations far from reality. The load curve considered was an ideal one consisting of only peak and off peak as shown in Fig 1. The average is a function of only peak and off peak. But the intermediate values affect the average. In equation (3) it was considered that PLSk = Ck PLDk [k=1,2,3…] With an assumption C1 = C2. This need not be right, as the value of the coefficient can be different at the off-peak, than at the peak. H. F. Hoebel used the original equation (4) and also developed equation (5) in terms of loss factor and load factor with an exponential coefficient [2]. The value for the exponent commonly used in practice was 1.6. Loss Factor = (Load Factor)1.6 (5) Martin. W. Gustafson developed a quadratic equation to determine sub transmission and distribution system losses [3]. He also observed that the values of 0.3 for the constant coefficient and 1.6 for the exponential coefficient, no longer Figure 1. Load Curve and Loss Curve appear to be appropriate and revised the coefficients as 0.08 LDf = t/T + (PLD 2/PLD 1)(1 – t/T) (1) for the constant coefficient and 1.912 for the exponent [4]. A constant term is added which represents no load losses [5]. EQf = t/T + (PLS 2/PLS 1)(1 – t/T) (2) He extended the same approach to determine transmission © 2013 ACEEE 10 DOI: 01.IJEPE.4.1.1065
  • 3. Full Paper ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013 system losses with special considerations [6]. and loss factors. Calculated value of the coefficient ‘X’ was 0.6432. It is further used to calculate the loss factors from III. PROPOSED MATHEMATICAL APPROACH respective monthly load factors. The coefficients of the pro- posed approach ‘A’ and ‘B’ were calculated from the average Many processes in nature have exponentia1 monthly load and loss factors. The calculated value of the dependencies. The advantage of exponential equations is coefficients A and B were -2.2073 and 2.231. They are further that with the right values of coefficients, any curve can be used to calculate the loss factors from the respective monthly approximated in the form of an exponential equation i.e., within load factors. The summary of the analysis is shown in Table the limits of the given data. In exponential curve fitting, any 1. From the results, it can be seen that the method presented function is approximated as is a new tool capable of estimating sub-transmission and y = AeBx (6) distribution system losses with greater accuracy, than what The advantage of such an equation is that ‘A’ decides was previously available. It is also observed that the values the ‘y’ intercept of the curve, and ‘B’ gives the function its of the coefficients are very important in estimating losses, curvature. Taking the logarithm of both sides of (6) but they cannot be generalized for any system. This is due to Log y = Log A + Bx (7) the fact that each system has a different load profile, which Then, the best fit-values are leads to different values of coefficients. Hence, it is advised n n 2 n that a distribution company, use previous data, to calculate  ln yi  xi   xi ln yi the values of the coefficients, for their respective load pro- A  i 1 i 12 i 1 2 files. Fitting an exponential curve to data does not need any n 2  n n  xi   xi i 1 i 1 (8) tedious programming or simulation. n n n V. CALCULATION OF LOSSES FROM THE LOAD FACTOR n  xi ln yi   xi  ln y Using the above mentioned method, we calculated the i1 i1 i1 i B loss factor of a system, at a particular instant. The loss at that n 2 n 2   n  xi   xi i1 i1 (9) time can be estimated by using the basic definition of loss factor, viz. Where ‘xi’ and ‘yi’ are the sets of data, which must be fit AverageLoss LS f  in the required curve. Using (6) to fit a curve between the two PeakLoss factors. Therefore, Let the relationship between them be LSf = aebLDf (10) AverageLoss  LS f  PeakLoss Taking the natural logarithm on both sides of (10) But, using the above mentioned approach, the loss factor was estimated as ln(LSf ) = ln(a) + bLDf (11) LS f  aebLD f This can be plotted as a straight line between the natural Hence, logarithm of loss factor and load factor. Therefore, the general equation of the relationship can be shown  bLD  AverageLoss  ae f  (PeakLoss) Most of the electrical systems have constant values of ln(LSf ) = A + B (LDf ) (12) their peak losses. This value of the peak loss can be taken The values of these coefficients can be determined by from the data is collected regularly, by the distribution using (8) and (9). Due to its desired features of a given ‘y’ company. Therefore, intercept value, and curvature, it can be used easily to bLD f estimate the loss factor of a system, from a given value of a AverageLoss  Pe load factor. Where, P  a  PeakLoss IV. RESULTS This relationship was used to calculate the average losses of the system from which data was collected. The results are The proposed approach was tested on a 33kV sub shown in Table. II. transmission line existing between 132kV/33kV Thurkayamjal substation and 33kV/11kV Hayathnagar substation, CONCLUSIONS APTRANSCO, Hyderabad, Andhra Pradesh. The annual load and loss data of the test system was collected for one year Buller and Woodrow were right, but their approach con- and the monthly load and loss factors were evaluated by tained too many assumptions to be considered to be close to plotting the monthly load and loss curves. The monthly Load the original value. Hoebel relationship had the capacity to Curves and Loss curves are shown in Fig. 3 and Fig. 4. get a curvature close to the original curve, but it ignored The constant coefficient ‘X’ of Buller and Woodrow (1) constant losses. Every relationship between two sets of data has been calculated from the average values of monthly load can be expressed in terms of infinite powers of the ‘x’ coeffi © 2013 ACEEE 11 DOI: 01.IJEPE.4.1.1065
  • 4. Full Paper ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013 Figure 3. Monthly Load Curves cient. Martin. W. Gustafson made use of this polynomial, but ACKNOWLEDGMENT utilized only a second order equation. This was the source of We acknowledge the contributions of V. Prameela, Assistant error. The new approach can be considered to be very close Engineer, APTRANSCO, Hyderabad in Instrumentation and the to the original relationship, because it considers all the terms data collection and also thank Kiran Kumar Challa, M.E (IISc Ban- in the polynomial expansion (in the expansion of ex) includ- galore) and P. Sai Gautham for several discussions during this work. ing a constant term which represents total constant losses due to large number of Distribution Transformers in the Elec- trical Distribution System. © 2013 ACEEE 12 DOI: 01.IJEPE.4.1.1065
  • 5. Full Paper ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013 Figure 4. Monthly Loss Curves REFERENCES [4] M.W. Gustafson, J. S. Baylor, and S. S. Mulnix, “Equivalent hours Loss Factor revisited,” IEEE Trans. Power Syst., vol. 3, [1] F. H. Buller and C. A. Woodrow, “Load factor equivalent hours no. 4, pp. 1502–1507, Nov. 1988. values compared,” Electr. World, Jul. 1928. [5] M. W. Gustafson and J. S. Baylor, “Approximating the system [2] H. F. Hoebel, “Cost of electric distribution losses,” Electr. losses equation,” IEEE Trans. Power Syst., vol. 4, no. 3, pp. Light and Power, Mar. 1959. 850–855, Aug. 1989 . [3] M. W. Gustafson, “Demand, energy and marginal electric [6] M. W. Gustafson and J. S. Baylor, “Transmission loss evaluation system losses,” IEEE Trans. Power App. Syst., vol. PAS-102, for electric systems,” IEEE Trans. Power Syst., vol. 3, no. 3, no. 9, pp. 3189–3195, Sep. 1983. Aug. 1988. © 2013 ACEEE 13 DOI: 01.IJEPE.4.1.1065
  • 6. Full Paper ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013 TABLE 1. COMPARISON O F LOSS FACTORS TABLE II. C OMPARISON OF ACTUAL AND C ALCULATED LOSS “Simplified feeder modeling for load flow calculations,” IEEE Trans. Power Syst., vol. PWRS-2, no. 1, pp. 168–174, Feb. 1987 [10] R. Baldick and F. F. Wu, “Approximation formulas for the distribution system: The loss function and voltage dependence,” IEEE Trans. Power Del., vol. 6, no. 1, pp. 252– 259, Jan. 1991 [11] C. S. Chen, M. Y. Cho, and Y.W. Chen, “Development of simplified loss models for distribution system analysis,” IEEE Trans. Power Del., vol. 9, no. 3, pp. 1545–1551, Jul. 1994. [12] H.-D. Chiang, J.-C. Wang, and K. N. Miu, “Explicit loss formula, voltage formula and current flow formula for large scale unbalanced distribution systems,” IEEE Trans. Power Syst., vol. 12, no. 3, pp. 1061–1067, Aug. 1997. [13] Y.-Y. Hong and Z.-T. Chao, “Development of energy loss formula for distribution systems using FCN algorithm and cluster-wise fuzzy regression,” IEEE Trans. Power Del., vol. 17, no. 3, pp. 794–799, Jul. 2002. [14] C. A. Dortolina and R. Nadira, “The loss that is unknown is no loss at all: A top-down/bottom-up approach for estimating [7] Turan Gönen, “Electric Power Distribution System Engineering”, distribution losses,” IEEE Trans. Power Syst., vol. 20, no. 2, McGraw-Hill, 1986. pp. 1119–1125, May 2005. [8] “Energy Loss Estimation in Distribution Feeders” P. S. [15] Power Utility Nontechnical Loss Analysis with Extreme Nagendra Rao and Ravishankar Deekshit, IEEE Transactions Learning Machine Method” A. H. Nizar, Z. Y. Dong,Y. Wang, on Power Delivery, vol. 21, no. 3, July 2006 IEEE Transactions On Power Systems, vol. 23, no. 3, August [9] N. Vempati, R. R. Shoults, M. S. Chen, and L. Schwobel, . 2008. © 2013 ACEEE 14 DOI: 01.IJEPE.4.1.1065
  • 7. Full Paper ACEEE Int. J. on Electrical and Power Engineering, Vol. 4, No. 1, Feb 2013 T. Murali Krishna has received B.Tech in Electrical Dr.S.Kamakshaiah has received BE (Hons) & Electronics Engineering from Nagarjuna University, (Gold Medalist) at S. V. University in 1962, India. India in 2002 and M.Tech from National Institute of He received his ME (HVE) and PHD (Gold Technology Calicut, India in 2004. He is a research Medalist) from IISC, Bangalore, India. He worked student of JNT University, Hyderabad, India and as Professor and Head, Dept of EEE, Chairman of working as an Associate Professor at CVR College of Engineering, Electrical science at JNT University Ananthapur. He won the ‘Best Hyderabad, India. His areas of interest are Power Systems and Teacher Award’ for 1997-98. He has around 40 IEEE publications Electrical Machines. to his name. He is author of 10 text books. He has guided many Dr. N. Venkata Ramana has received M. Tech people for Ph.D. His areas of interest are Electrical Networks, from S.V.University, India in 1991 and Ph.D. in Electrical Machines, HVDC, GIS, and Power Systems. He presented Electrical Engineering from Jawaharlal Nehru papers at various IEEE conferences at USA and Canada. Technological University (J.N.T.U), India in January 2005. He is currently Professor and Head of the Department, EEE at J.N.T. University College of Engineering, Jagityal, Andhra Pradesh, India. His main research interest includes Power System Modeling and Control. He published 32 international journals and conference papers and two text books titled “Power System Analysis” and “Power system operation and control for Engineering Graduate Students. He visited various universities in USA, Canada, Singapore and Malaysia. © 2013 ACEEE 15 DOI: 01.IJEPE.4.1.1065