COMPARING METHODS OF CALCULATING ...

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COMPARING METHODS OF CALCULATING VOLTAGE DROP IN LOW VOLTAGE FEEDERS IL L Sellick* and C. T. Gaunt** *Department ofElectrical Engineer-2.'1g, University of Cape Town. **IIKS LAW GIBB, P 0 Box 3965, Cape Town, 8000. Abstract. A generalised model of loads and networks is used to assess the accuracy of several different algorithms for calculation of voltage drop in low voltage feeders. The load models include single parameter, Normal and Beta distributions. Benchmark networks include simple and branched feeders with balanced and unbalanced three phase loading, as well as a single phase feeder. The results of the different algorithms are compared with a base case Monte Carlo simulation. The Herman Beta algorithm is found to be the most reliable. The other algorithms tested give large errors of estimate of the voltage drop. Keywords. Volt drop, load model, network model.

1.

INTRODUCTION

The sizing of most low voltage distribution feeders is constrained by the expected voltage drop and energy losses on the feeder. The future loads are not easy to define at the time of designing a distribution system for domestic consumers. However, a consistent basis is needed for making design decisions. A wide variety of methods has been used to predict voltage drop, based on load measurements taken and theory developed in several different countries. In South Africa the need for economic and effective electrification prompted a reassessment of methods of low voltage feeder design. Load measurements and analysis have led to new approaches to voltage drop calculations. This paper reviews the alternatives.

2.1.

The objectives of reticulation network voltage drop calculations are to obtain a prediction of performance which can be used to guide capital investment decisions, such as assessing the adequacy of an existing network or selecting cable routes and sizes in a planned network. These algorithms are also used to predict or assess the conditions on a real network. If the method of prediction correlates poorly with actual performance, the risks of over- or underinvestment are high. A method is sought which balances effort and risks. Various methods and their properties and assumptions have been analysed and compared to assess their validity in predicting the voltage drop experienced by consumers on a feeder.

2.2. 2.

CONCEPTS

The key concepts in this analysis are . that the objectives of a designer can be defined, the loads and network represented by a model, the model used to estimate network conditions at different stages in the future and that a certain amount of simplification is desirable. Although many of the concepts introduced here are more familiar in control theory, there are certain insights which can be gained by the application of these concepts to voltage drop calculation.

Objectives

Overall Model

The topology and loading for any specific network is unique. Since many networks need to be designed, a suitable model is required. The model is illustrated in Figure 1.

Input Load Model.

The input load model is a representation of the distribution of consumer currents at the time of maximum demand on the system. For the purposes of this investigation, only homogenous load models typical of residential consumers have been considered.

Submitted: 4/95 9/95 Accepted:

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Input Load Model

Process Method: Transfer Function

Calculated Outputs

tolerable voltage drop, thermal cable rating and economic loss. For nominal system voltages lower than 500V, the South African regulations require that the voltage supplied to the consumer shall not vary by more than ±6% of the nominal voltage for a period of more than 10 consecutive minutes [1]. Changes to the range of voltage within this regulation have been . proposed, extending the limits to ±10%.

etw ors Model N

CI

Comparator

!Constraint,

M inim um or Maximum Threshold Conditions

September 1995

Figure 1: Flow Chart for Volt Drop Calculation

In selecting a cable to be used in a feeder, the continuous load current and fault current define the thermal rating required of the cable.

Network Model.

The network model is also an input. The parameters governing the network model are allocation of consumers to phases (if this is applicable to the process method), conductor resistances, the source voltage and the network topology, if this is not incorporated into the process method.

There is an economic constraint on the electricity supplier, in the financial value of the losses due to non-zero resistances of the distribution cables.

Process Method.

The process represents the physical network and the transfer function, which is the relationship between current, impedance and voltage. It may incorporate the number of phases, the network topology (nodes and branches), assignment of loads to nodes, and a function relating output (eg in the form of a voltage) to the input (eg load currents, conductor impedances). The process depends on the type of the network. There are various types of system available to the design engineer: 1 phase, 2 wire 3 phase, 4 wire 2 phase, 3 wire In addition to the network type and topology, a voltage-drop calculation method may take into account the cable type, for example cables with a neutral impedance different from the phase impedance, as is the case with some aerial bundle conductors.

Calculated Outputs. The model is used to evaluate voltage drop or losses on the feeder. The output of any one of the methods provides either a calculated voltage or a calculated voltage drop. This voltage drop is an index of the performance of the system under consideration. This is not the actual performance of the network, because of the wide variation in load magnitude and distribution, but an index of the network's performance. Constraints. The constraints given to the design engineer will provide an indication of the acceptable threshold conditions. There are several constraints placed on the network, relating to

Threshold Conditions. There are certain limits of the calculated outputs which are acceptable, although they may not be directly the constraints. Since there is a range of inputs to the model, there will be a spread of outputs, of which the calculated output index represents one value. The acceptable or threshold index must be defined in similar terms to the output index. Once the voltage drop or current 'rating' or losses have been derived from the network model, the acceptability decision is made by comparing the output index with the threshold index. The threshold index may often be, for example, the acceptable limit of voltage drop. In general terms, however, it is only an index. 2.3.

Application

There are two groups of network solution models: Monte Carlo simulation. The network transform function is applied to randomly selected loads from a load model. The distribution of outputs indicates the range of conditions which may be expected for the load model used. Many calculations are needed for each network (or feeder), so this method is not suitable for most design work. However, it provides a useful comparison with deterministic models. A variety of parametric load distribution models or typical real distributions can be used in Monte Carlo simulations. An analytical Deterministic models. function produces a single valued output

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(index) which usually represents the conditions expected with a defined input load. The input (load model) must be a single value or parametric model of a load distribution. 2.4.

Time

The factors of load growth and cost are taken into account by making projections about future conditions. The simplest assumptions are that cost structure will remain constant and load growth can be projected during a defined period. The model is tested with different inputs and processes representing the network changes during the period of study, and the set of outputs used to evaluate the time-dependent conditions. 2.5.

Simplification

The stochastic nature of consumer loads results in a range of possible loading conditions and voltage profiles, as would be illustrated in a Monte Carlo simulation. Given that the output of the calculation process is an index, there is no single correct answer. There is however a diminishing return on further calculation effort, and hence simplifying assumptions will be made. These assumptions need to be evaluated or understood by the design engineer, in order to assess whether the simplified model is a sufficiently accurate representation of the network being analysed.

3. LOAD MODELS 3.1.

Resistance, Current, Power

The advantages of modelling the loads as resistance, current or power have been discussed elsewhere [2, 31. All of the methods considered treat the load as a stochastic current sink. Even where characteristic loads are presented as power (kVA), the analysis converts the load to a current, using the nominal voltage. Hence, all consumers will be modelled as current loads. A statistical model is obtained by sampling the load over a period of time, at regular intervals and applying a statistical model to the resulting distribution. However, not all algorithms model the load statistically. Alternatives to a statistical models are a measured load sample, or a single value model like "4 kVA / consumer". 3.2.

Different transfer functions use different statistical models, such as the Gaussian Normal distribution and the Beta distribution. The properties of the model may affect its usefulness and validity. The British and the Loss of Diversity methods of calculating voltage drop both use a Gaussian Normal distribution to model the consumer currents. The Herman Beta method uses a Beta . distribution to model the consumer current. Regardless of the load model used, groups of consumers need to be monitored over a sufficiently long period of time to establish the model. Simplicity. At the design stage of a new system, it is necessary to make assumptions about the characteristics of the consumers which will be supplied. That subject is beyond the scope of this paper. However, since there is uncertainty in the projection of these characteristics, a highly refined statistical model with many parameters is usually not justified. A model which is simple and easy to manipulate is required. A Normal distribution is described by only two parameters. A Beta distribution is described by three parameters. Due to the small number of describing parameters, both of these distributions are easy to use. Variation within the model. The load model is influenced by many factors, including socioeconomic class, maturity of the installation, climate, social patterns and tariffs. Both the magnitude and distribution change with time and load growth, and can be modified by tariff changes. In the short term (less than a year), actual values also change according to weather and the use of energy. In most cases, these variations will also be stochastic. A "conservative" upper limit might need to be assumed for design purposes. 3.3.

Projection of maximum loads

A design load is chosen, up to which the network will "perform" adequately. The design load is a projection of future conditions. The future may be 5 years or 25 years, depending on the investment philosophy. The design load may be exceeded by the actual load, or the planning criteria may be violated at a lesser load due to a change in distribution of consumers or load characteristics. In theory, under these conditions reinforcement is needed, or performance is likely to not meet the conditions adopted by the designer.

Distribution

Description in terms of a statistical model. Due to the stochastic nature of consumer currents, a statistical model is useful to represent loads.

The design load is a representative load and the analytical algorithm a representation of the expected conditions. If either the load model or process method is "conservative" the constraint

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conditions may not be exceeded until actual loads grow beyond those which the analysis attempted to depict.

Assumptions Two assumptions are made with most load models: There are no generators (current sources) on the distribution network. The power factor is unity [5]. These two assumptions are based on the fact that the consumers are residential, as opposed to commercial or industrial, and are applied for all the algorithms being considered here. The first assumption above has an important consequence: the load current drawn by any one consumer is bounded on both sides. These limits are 0 A (the consumer is not a generator), and the circuit breaker size, c b . A Normal distribution does not comply with this bounding constraint. In this respect, a Beta distribution is a more representative load model than a Normal distribution.

3.5.

namely a, 13 and c b . Some properties relating to the Beta distribution and comparisons with the Normal distribution are contained in Table 1. The mapping of parameters between the Normal and Beta distributions is obtained from the following equations. [6] Beta to Normal: Cb

o- =

a a + fl

(1) a/3

(a + ,6) 2 (a +fl+1)

.Cb

(2)

Normal to Beta:

a—

P(cbil - 1-12 - 0-2)

(3)

Cba

fl

(Cb - 1-1)(Cb11-1-12-c?)

(4)

Ca a2

In practice, u and a can be calculated for any real set of data. Assuming a value for cb, the parameters for the corresponding Beta distribution, namely a and 3, can be calculated.

Relationships between some common models 4. TRANSFER FUNCTION

The Gaussian Normal distribution is described by two parameters, i.e. ;I is the mean, and a is the standard deviation. Most single-value models are the mean or average load, for example the after diversity maximum demand (ADMD), which is the same as u in the Normal distribution. The Beta distribution has three describing parameters,

Different methods have different inputs. As a result, certain models are more sensitive to a change in the network parameters defining the network than others. Accurate models are essential for evaluating alternative technologies such as aerial bundle conductors, with neutral-to-line

Table 1: Pro p erties of the Beta distribution Property: The distribution is bounded by 0 and the circuit breaker size. A symmetrical Beta distribution where a ---- p approximates a Normal distribution. As a and 13 increase, the degree of slenderness increases; the distribution becomes more steep and less wide. The Beta distribution can be skewed in either direction For a < p, the distribution is skewed right. For a > f3, the distribution is skewed left.

For the same ratio of c b to i_t, the parameters a and p will be the same. If either a or f3 is less than 1, the distribution is unbounded, and tends towards cc.

Significance: There are no generators on the system, and there is a limit to the amount of current that a consumer can draw. Methods which use a Normal distribution to model the load can be tested against methods which use a Beta distribution, for certain conditions of the Beta distribution. With an increase in slenderness, the homogeneity of the population increases. This corresponds to a decrease in a for a Normal distribution, The Beta distribution is sufficiently versatile to be able to model a group of consumers with high load limiting (extreme left skewness) and unrestricted loads (extreme right skewness). The group ADMD [A] is less than half of the circuit breaker size, cb. The group ADMD [A] is more than half of the circuit breaker size. This represents a greater degree of load limiting than in the previous case. The transform of a Beta distribution to a Normal distribution is scaled by cb If either a or 13 is less than 1, the distribution is not representative of a real group of consumers.

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impedance ratio (NLIR [5]) # 1, small or large transformers with few or many consumers each, or load control. 4.1.

Topology

A There are many network configurations. network algorithm should be sufficiently flexible to incorporate a different number of sections from feeder to feeder, and an ability to calculate branched sections.

4.2.

Parameters

The transfer function used to process the load and network input models needs to be sensitive to the following: supply voltage the parameters defining the input models the current/impedance relationship In any real network, a neutral current will flow. This current will contribute to the voltage drop experienced by the consumer, and is affected by the resistance of the neutral conductor. The information about the line impedances and, in some cases the NLIR, is contained in the network model and used in the current/impedance function. This function may need to be sensitive to changes in temperature, which affect the cable impedances. 4.3.

Loads assignment

In a network, single-phase consumers are connected to different phases of a three-phase feeder. The assignment of consumers to phases can be defined by the design engineer, but may be modified during construction or maintenance. This configuration forms an important part of voltage drop and losses calculation. For large numbers of consumers evenly spread over the three phases, the overall system will tend towards being balanced, but a network model may be able to calculate the volt drop experienced by the consumers using the actual consumer arrangement on the feeder. This voltage drop calculation algorithm should be sensitive to: size and extent of stochastic imbalance of loads. consumer arrangement.. the loss of diversity, from the mean demand, increases as the number of consumers decreases down a feeder. Different connection philosophies may reduce the costs in different ways. For example, the approach may be to balance the number of consumers per phase as much as possible at every node, reducing

voltage drop, or connect consumers to only one phase at each node, reducing connection cost. The impact of the imbalance may be treated with an empirical factor or analysed directly in the transfer function.

5. OUTPUTS 5.1.

Sensitivity to input and network parameters

The outputs index will be sensitive to the network parameters and to the input parameters. If the accuracy of the network model is high, but the load model is not a good representation of the consumer group, then the output index will not reflect the network's performance, and similarly for an accurate load model, but a poor network model. There is a balance between calculation effort, and the risks taken by designing to a less accurate output index. 5.2.

Confidence

The calculation method yields values corresponding to voltages and currents as outputs. Deterministic algorithms yield expectation values of voltages and currents. Monte Carlo simulation yields both expectation values and bandwidths for these quantities. Since the load and network data models are both representations of real loads and networks, the real conditions may exceed or not meet the calculated values, which must be considered an index of expected performance. A confidence level can be assigned to the possibility that conditions will be within those indicated (i.e. system performs adequately). A level of 100% is not usually used, since this will lead to the network never being loaded to its maximum. The network would not be economically optimised with unnecessary capital expenditures. Methods for calculating the expected system response according to a model should indicate or be sensitive to the associated confidence level. As different confidence levels are appropriate for different consumer groups, the process method needs to be sensitive to the confidence level. 5.3.

Practical limits

Cable sizes. The mass of conductor in a feeder to "achieve" a chosen voltage drop can be calculated from the loads (factored for loss of diversity) and the distance from the source. However, cables are made in standard diameters and the variety on a single project is limited - therefore there is a practical limit to how closely one can approach the "optimum" or "minimum" conductor volume.

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There arc also practical limits of maximum and minimum conductor sizes for feeder construction. Therefore progressive refinement of algorithms results in diminishing benefits.

Statistical methods.

Ease or difficulty of reinforcement.

6.2.

When designing a reticulation system, decisions need to be made about the practical aspects of reinforcing a system when loads exceed the design capacity. Such decisions are sensitive to forecasts of load growth.

5.4.

These methods use a statistical probability and analytical derivation to calculate the expected voltage drop, within a defined confidence. Description of various methods

Six volt drop calculation methods have been investigated: Monte Carlo simulation developed by Dekenah. [7]

Decision

Models which are accurate representations of real conditions produce outputs which reflect accurately the likely conditions. Comparison of the calculated output with the threshold index yields an "Acceptable / Not acceptable" decision. If it is known that the model does not produce reliable calculated outputs, the comparator decision is "fuzzy", having to take into account other factors not incorporated in the model. The threshold indices are of less value in taking decisions, because the implications of them being exceeded are unknown.

6. SOME DIFFERENT VOLTAGE DROP CALCULATION METHODS 6.1.

September 1995

Groups of calculation methods

There are four groups of methods which have been used or described for voltage drop calculations in South Africa:

Monte Carlo simulation method.

This type of method involves randomly drawing consumers, with replacement, from a load distribution representing the group of consumers. The voltage drops are then calculated, using these consumer currents.

Single parameter methods, in which a balanced voltage drop is calculated. This value is then adjusted by means of empirical factors to cater for unbalance of loads, and diversity.

Multiple parameter methods, derived from Monte Carlo simulations, calculate the balanced voltage drop, and then adjust this value to cater for diversity and unbalance. However, the factor used for this adjustment is not calculated empirically, but has been derived from simulation.

a British method described in [8] and DT Volt Drop, a software package described in [9], both of which are single-parameter methods. the Loss of Diversity method developed by Gaunt [4,10] and the Unbalanced Voltage Drop method developed by Dekenah [5,7], both of which are multiple parameter methods derived from Monte Carlo simulations. the Herman Beta algorithm, developed by Herman [2], which uses stochastic modelling of the load. Brief descriptions and characteristics of these methods are included as Appendix A. Both the British and the Loss of Diversity methods are based on Normal load distributions. The difference lies in the method by which the factors for loss of diversity and unbalance are obtained. The British method assumes that these two factors are independent, where the Loss of Diversity method treats them as a single factor. Also, both methods solve a feeder with distributed loads in a single step. For the purposes of a single step solution, all the consumers along the feeder are treated as lumped at one point. Other methods solve feeders as cascaded sections. The Unbalanced Volt Drop method is based on real load distribution and the Herman Beta method on a Beta model of the loads. Table 2 contains a summary of the load model, transfer function and output index properties of each of the algorithms under consideration. Table 3 contains a comparison of all the factors which can be defined or modified during analysis.

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Table 2: Comparison of individual components of each method Method:

Load Model:

Transfer Function:

Output Limit:

Monte Carlo Simulation

Any defined load model

Vector analysis

Defined confidence level.

British Method

Normal distribution, characterised by ;..i. c is undefined.

Independent empirical factors for diversity and imbalance.

Confidence level not defined

Loss of Diversity Method

Normal distribution, with the variation defined by confidence, a, and s.

Dependent factors for diversity and imbalance, derived by simulation.

Defined confidence level.

Unbalanced Volt Drop Method

Any defined load model.

Derived by simulation; accommodates NLIR # 1 and load allocation.

Defined confidence level.

Beta Distribution Method

Beta distribution, defined by a., 3 and c b .

Analytical transform, accommodates NLIR # 1, and load allocation.

Defined confidence level.

Table 3: Factors accounted for by the various algorithms considered Method

ADMD

NLIR

Confidence Level

Consumer Configuration

British Method Loss of Diversity Method Unbalanced Voltage Method Beta Distribution Method

Yes Yes Yes Derived

No No Yes Yes

No Yes Yes Yes

No No Generalised Yes

7. COMPARATIVE STUDIES USING "STANDARD" NETWORKS AND LOAD MODELS 7.1.

Benchmark networks

The purpose of benchmark networks is to test the sensitivity of calculation methods to factors which are present in real networks. From the analysis of the benchmark cases, the general suitability of an algorithm or model can be assessed. More than one benchmark is desirable. The reason for this is that a network which will test all the different aspects of an algorithm will become unrealistically large. Hence, five benchmark cases have been proposed, each designed to test certain aspects of the possible network topology. In the circuit diagrams for the benchmark networks given in the following sections, the following conventions have been adopted: the line resistance for each of the 3 phases is the resistance quoted with that section of the line. Values quoted are in ohms (n).

the consumers allocated to the three phases at any node are given by the three numbers above the node. These are allocated respectively to red, white and blue. if there is an NLIR which is not equal to unity, this is stated next to the sections for which it applies. If no NLIR is quoted, it is taken as unity. The five benchmark networks are illustrated in Figure 2 and described briefly below. Network NI: A balanced network. This network is

a standard balanced network, of 6 nodes with 4 consumers at each node. These consumers have been allocated in a circular fashion, i.e. RWBRWB. This network tests the capacity to model effects of load unbalance and a neutral of size different from the phase conductor.

Network N2: An unbalanced network.

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Network Ni: A balanced network 211

121

SA R = 0,01611 n R =0,01611 n

112

R=0,01611 n

211

R = 0,01611

n

112

121

R = 0,01611 D

it=o,oisii

Network N2: An unbalanced network 222

R= 0 04296

R=0,04296

731

220

111

400

v\A,

R =0,02615

n

220

220 NLER=1,32

Network N3: Cascaded sections 201 120 012 201 120 001 Of0 0285 C) 2=0 1O 0 85 n R =0 0 85 0 R = 0 0 \9\!5 R=0,02§85 0 = 0,0,2685 0 -R C A

Network N4: Sections of different lengths iL-9\19\1A1!

211 A

121

112

B

C

(21m1112.

R=oms n

110 D

Network N5: A single-phase feeder 100 100 R= 0 0 11 n R = 0 01.11 n R= 0 0 A

B

100 200 100 11 0 R = 0 01 11 C) R = 0 01 11 o D E C

Figure 2: Benchmark networks

The nature of the simulation algorithm used as the 'base case' makes it is very difficult to solve networks which are branched. It is not possible to determine with reasonable accuracy what the voltage levels in the network will be. However, due to the manner in which the network has been set up, there are simple conditions which need to be met:

Network N3: Cascaded sections. This network will test an algorithm's ability to accurately calculate the voltage drops in a network when the most heavily loaded phase changes in successive cascaded sections. This network has been chosen to highlight the potential source of error when cascaded sections are calculated separately.

Network N4: Sections of different length. AVDG z AVDF VDE # VDF

is the largest voltage drop of all the sections AVsA

AVDF AVB.c = — AVAB AVAD

This network tests sensitivity to sections of different length. This network is balanced up to a point, with a long unbalanced spur coming off this, of a different length to the remaining sections.

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All the Network N5: Single phase feeder. consumers are connected on a single phase. This represents a case of extreme unbalance, and will test a network's ability to accurately calculate voltage drops on a single-phase feeder. 7.2.

Benchmark load models

To accompany the networks proposed above, several load models are needed. Different algorithms are based on different load models. The Herman Beta Method requires the load model to be a Beta distribution. The results (output index) cannot be compared with methods based on Normal distribution loads without establishing an equivalence between the load models (based on section 3.5).

proposed benchmark loads is possible. The difference in slenderness (or standard deviation) is clearly shown. Figure 4 contains the probability distribution functions for the benchmark Beta distribution load models on a real axis. This gives an accurate description of the demand on the system during the interval of maximum demand. This curve is useful to demonstrate the effects of varying the size of the circuit breaker for a set of consumers with the same mean, . or ADMD. The slenderness can also be seen.

Due to the assumption of a homogenous group of consumers on any one feeder, it is not possible to test more than one load model simultaneously with another on the same network. Hence more than one load model has been proposed. The results of running the different load models on the same network can be compared. These load models are described in Table 4. All the distributions have the same mean value.

104

20

40

Current [A] Full scale = 100 divisions

Figure 4: Probability Density Functions of Benchmark Load Models on a Normalised Axis

7.3.

Definition of the "base case"

Two ''base cases" have been used as references for the algorithms being tested. These are:

0.4

2

A Monte Carlo simulation method [7]. This method randomly draws, with replacement, consumers from an operating curve, and calculates the volt drops with that loading on the network. This process is repeated many times, and the result averaged for accuracy.

0

0.6

X Full scale =

100 divisions

Figure 3: Probability Density Functions of Benchmark Load Models on a Real Axis

Figure 3 contains the probability distribution functions for the benchmark load models on a normalised axis. This shows the shape of each of the proposed benchmark load models. From this graph, an easy comparison between the three Table 4: Pro p osed benchmark load models. S a ti Cb a Pt [A] [A] [A] LI

1.65

7.37

60

11

1.5

7.33

L2

2.63

6.92

40

11

2

5.50

L3

3.50

2.86

20

11

3

3.67

A perfectly homogenous group of consumers, with every consumer drawing exactly the ADMD current at the interval of maximum demand. This case is the ideal from a designer's point of view, and is equivalent to a group with s = co. This

Comments: This function is skewed right. It represents a group of consumers with effectivel y unconstrained maximum loads. This is almost a symmetrical function. It is an approximation to a Normal distribution function, and represents a group with moderate load-limiting. This function is skewed left, and represents a group with a high degree of load limiting.

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means that for calculation purposes, voltage drops can calculated without stochastic variation. However, this is not a realistic case; it is used as a reference.

Accuracy of the simulation

7.4.

Due to the nature of simulation, there is no one correct output. Given enough simulations, a range of outputs will result. The size of this range will depend on the quality of the random number generator, the number of simulation runs performed in each case, and the accuracy , of the calculation process. Tests were conducted on network N3, which is the network for cascaded sections. This network can be used, since the Monte Carlo simulation method implemented here calculates the cumulative voltage drop to a particular node, and not simply the voltage drop across an individual section. It was found that the best results were obtained using 1500 simulation runs for each network loading. This decision was arrived at by testing the same loading on a particular network 7 times. Then the maximum change in each of the respective columns of figures representing different confidence levels was calculated. Once the sum of all of these "variations" had been calculated, the number from which this total was the smallest was chosen. However, since there is still an element of inaccuracy in the simulation, each loading on the network was repeated 5 times. The results quoted are the averages of these 5 calculations. These results are shown in Table 5. This low degree of variation is indicative of repeatability, and not necessarily accuracy.

Tables of results using benchmark networks and load models

7.5.

The results of the tests run on the benchmark networks described in the previous section, using the benchmark load models are contained in Appendix B. In these tables, each network was calculated with each equivalent load model as the inputs. The benchmark networks are labelled Ni, N2, N3, N4 and N5 as in section 7.1. The benchmark load models are labelled LI, L2 and L3 as in section 7.2.

September 1995

For the feeders which are not branched, the output index shown is the voltage drop to the end of the feeder, expressed in per cent of the nominal source voltage. Network N2 has several end points, and therefore the results shown are to the ends of the four branches (C, E, F, G).

8. INTERPRETATION OF BENCHMARK ANALYSES

8.1.

Specific Methods

The mean load base case, using the ADMD as the load for all consumers is not a realistic case. It serves as a theoretical minimum bound on the voltage drop, since all other methods make allowance, in some form, for the loss of diversity from the average and/or the neutral current, both of which can cause the voltage drop to be higher in practice than indicated by average, balanced loading. The Monte Carlo simulation (base case) provides the most reliable indicator of the expected voltage conditions, taking into account the load model, the distribution of consumers across the phases, and the confidence level. In all of this analysis, a confidence of 90% has been used as the basis of comparison. The differences between the results of the British method and the DT Vdrop method using the "British factors" result from the DT Vdrop taking account of the actual consumer connections on each phase. In addition, the DT Vdrop method uses the maximum consumer current to modify the factors applicable to very small numbers of consumers, for which the British method has no equivalent procedure. For all the networks calculated using the DT Vdrop method, the results using the "DT factors" are not always an improvement on those using the "British factors", or the "AMEU factors", and are sometimes a worse estimate. The DT Vdrop program allows the choice of three-

Table 5: Variation of results for simulation

98%

95%

Confidence Level 90% 85%

80%

50%

Total Variation

0.35 0.23 0.20 0.33 0.08 0.07 0.11

0.21 0.20 0.16 0.19 0.13 0.12 0.07

0.26 0.07 0.14 0.14 0.07 0.06 0.14

0.17 0.07 0.12 0.10 0.06 0.07 0.05

0.19 0.06 0.06 0.08 0.03 0.04 0.05

1.36 0.74 0.81 0.96 0.41 0.42 0.53

N 500 750 1000 1200 1500 1800 2000

0.18 0.11 0.13 0.12 0.04 0.06 0.11

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phase or single-phase calculation. The results obtained for network N5 were identical for calculation in the three-phase mode or in the single-phase mode.

the British, Loss of Diversity and Unbalanced Volt Drop methods do not make allowance for allocation of consumer loads to specific phases, and so calculated the same volt drop in sections DE and DF.

The DT Vdrop program generally overestimated the voltage drop by more than 15% of the base case, and sometimes by more than 40%. This would lead to a significant oversizing of feeder conductors. The estimation of volt drop in the single-phase network, for which the program makes specific provision, was conservatively higher than the base case by more than 40% in most cases. The best performance for DT Vdrop occurred on network N2, which is the network with branched sections, and a large degree of unbalance.

the British and Loss of Diversity methods do not make allowance for a neutral impedance which differs from the phase impedances, so calculated the same volt drop in sections DF and DG.

The Unbalanced Volt Drop method also tended to provide conservatively high voltage drops, up to 30% above those of the base case, except for the single-phase network for which the method is unsuitable.

Comparison of the Results of Different Methods.

8.3.

All the deterministic methods met the conditions incorporated in the benchmark network N2, with the following exceptions (see section 7.1.):

8.2.

and overestimating the volt drop with L3 loads (load limiting). The method is not suitable for single-phase feeders.

A comparison of the different algorithms must take into account the large number of results derived from models specially selected to test extreme conditions. Generally, the Herman Beta algorithm is the best of all the alternatives. Only in one benchmark (N3) does it generate indices which are conservatively high. This condition arises since the Herman Beta algorithm calculates the worst voltage drop on any phase, and summates these to calculate the voltage drop to the end of a section. The network was chosen to continually change the most heavily loaded phase from section to section. The single-phase benchmark network, N5, showed that most of the algorithms are inappropriate for calculating this kind of feeder. Only the Herman Beta method provided voltage drops within +15 or -10% of the Monte Carlo base case. The British method overestimates the voltage drop in all cases other than the single-phase network. The voltage drop estimates when modelled as a single section with distributed loads were b6tween 15 and 30% higher than the base case. Calculation of cascaded separate sections resulted in much higher 'errors, even over 80%. These conservative results would cause feeder conductors to be oversized in the design stage, and operating losses to be much lower than expected from analysis. The results of the Loss of Diversity methods were mostly between +15 and -10% of the base case, but with a possibility of underestimating for LI loads

Effect of Different Load Models

It is evident from the Monte Carlo simulation that the voltage drop in feeders supplying demandlimited loads (type L3) will be 60-70% lower than may arise in feeders supplying unrestricted loads of the same ADMD, with a 90% confidence level. The Herman Beta and Loss of Diversity methods both allow for different load models and the trend of the voltage drop derived from these methods reflects the results of the Monte Carlo simulation. The change in voltage drop derived from the DT Vdrop program using different load models is not as significant as obtained from the Monte Carlo simulation or the Herman Beta and Loss of Diversity methods.

9. CONCLUSIONS The formulation of a process model for voltage drop calculations for low voltage feeders allows different algorithms to be compared on a consistent basis. As the algorithms are based on different types of load models, a set of equivalent models has been identified to use in benchmarking the algorithms. In addition, some benchmark networks have been formulated. They represent typical and extreme cases of network conditions, including a generally balanced three-phase feeder and a single-phase feeder. It is clear that the use of inappropriate algorithms can result in significant over- or under-estimation of voltage drops in low voltage feeders, resulting in significant over- or under-sizing of conductors.

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REFERENCES Of all the deterministic methods evaluated the Herman Beta method is the most general algorithm. It is able to take into account the actual allocation of consumers to phases, significant unbalance even to the extent of single-phase systems, and variation in the load model representing demand limiting. The algorithm leads to conservative estimates of volt drop when applied to cascaded sections in which the most heavily loaded phase changes at each node; however, this is an unusual condition which can be identified by inspection of the feeder. All the other methods exhibit large errors under various conditions, especially single-phase networks. The second most consistently reliable method (after the Beta Herman method) is the Loss of Diversity method, applied to each cascade section of a feeder. Not even the Herman Beta deterministic method of calculating voltage drop provides exactly the same answers as the Monte Carlo simulation. This may be due partly to variations in the random modelling. However, the deterministic algorithms are generalised solutions and cannot be tailored to meet the conditions of every case. Therefore, it must be recognised that some degree of error will exist in the output index calculated with any algorithm. It appears that a better understanding of the threshold limits for the output indices is required before further progress can be made in voltage drop calculations. The algorithms analysed in this project produce an output index of voltage drop. Acceptable limits of actual voltage drop need to be determined and the relations defined between them and an output index at a specified confidence level. In the meantime, it has been established that one method, the Herman Beta method, is more consistently reliable for voltage drop calculations than the other methods investigated. Of course, it will be necessary for system designers to adopt suitable load model parameters. Sufficient information is provided to assist in the derivation of 13-distribution models equivalent to other design parameters in common use.

10. ACKNOWLEDGEMENTS This research project was carried out with the financial support of Hill Kaplan Scott Law Gibb. The permission of the Directors to publish this paper is acknowledged.

[1]

SABS 1019, "Standard Voltages, Currents and Insulation Levels for Electricity Supply", Amendment No. 1, 19 May 1992

[2]

R. Herman, "Voltage Regulation Analysis for the Design of Low Voltage Networks Feeding Stochastic Domestic Electrical Loads", University of Stellenbosch, 1993.

[3]

R. Herman, C.T. Gaunt, "Measurement and Representation of Maximum Demand for Individual and Grouped Consumers Including Constraints", 11 th CIRED Conference, Liege, Belgium, 1991.

[4]

C.T. Gaunt, "Implications of Planning and Design Decisions in Electricity Distribution", 12 th AMEU Technical Meeting, Potchefstroom, 1988.

151 M. Dekenah, C.T. Gaunt, "Simulation of Domestic Consumer Loads and Voltage Drop on LV Distribution Feeders Based on Data Collected from Load Restricted Consumers", SAME Transactions, 1993. [6]

J.A. Rice, "Mathematical Statistics and Data Analysis", International Student Edition, California: Brooks / Cole, 1988

[7]

M. Dekenah, "Calculation of Volt Drops on Unbalanced Radial Feeders Supplying Domestic Consumers", University of Cape Town, 1991.

[8]

ACE Report No. 13, "Design of medium voltage underground networks for new housing estates", British Electricity Boards, 1966.

[9]

F. Note, "DT Vdrop 5 (Low voltage simulation package)", Eskom Distribution Technology

[10]

C.T. Gaunt, "Parameters for the design of low voltage electricity distribution to domestic consumers", Hill Kaplan Scott Inc., 1993.

APPENDIX A: DETAILED DESCRIPTIONS OF METHODS OF CALCULATING VOLTAGE DROP A.1 Monte Carlo Simulation Method This is a simulation using a load model for the calculation of voltage drops experienced by

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consumers in a network. The load model used here is a 13-distribution probability function for consumer currents. This simulation method has been used as the 'base case' against which to compare the other algorithms being tested. Monte Carlo type simulators require large numbers of simulations to be run. They arc also timeconsuming in setting up each network. Therefore, they are usually not appropriate for design or network operations monitoring. In simulation methods, there are limitations, or aspects needing special care, e.g. the randomness of the random number generator. Further, in solving a branched feeder on this particular simulator, only an approximation can be made. This method does take the following into account: neutral-to-line impedance ratio (NLIR) consumer configuration supply voltage load model parameters confidence level. This algorithm calculates the volt drop experienced for the consumers along the feeder. Although the output indices are only quoted for specific values of the confidence level, this simulator could be made to generate outputs for any desired confidence level. A.2 British Method This is an analytical algorithm, using empirical formulas to cater for unbalance, and diversity. As a result, it is quick and easy to use if implemented in a spreadsheet environment. It provides for the modelling of distributed loads along a feeder as an equivalent lumped load at the end of the feeder. In this method, the following factors are not taken into account: NLIR (this is always assumed to be unity) confidence level (empirical formulas are based on a 90% confidence level) consumer configuration (this algorithm is independent of the arrangement of the consumers at a specific node, and only takes into account the number of consumers). alternative load distribution model (a Normal distribution is assumed, with standard deviation undefined). A.3 DT Vdrop This is a computer program which calculates

voltage drops from the nominal voltage to consumers in a network across all three phases. To do this, it implements the previous British method, and an adaptation of the British method to South African conditions, known as the AMEU method. Thirdly, a modification on both of these factors is used for the DT method [9]. The empirical formulas used arc given in Appendix C. In this method, the following factors are not taken into account: confidence level (the empirical formulas used are those given for a 90% confidence level). load distribution model (only the ADMD and the maximum current size are used) However, unlike the British method, the DT method can take into account the NLIR This is implemented by the cable line impedance and neutral impedances being specified in the cable file. In the calculation of the networks for this report, all transformer impedances in the DT Vdrop method have been set to zero, as have all service cable lengths. The supply on the secondary winding of the transformer is taken as 100% of the nominal voltage. A.4 Loss of Diversity Method This method [4] was derived from Monte Carlo simulations, using load models defined as Normal distributions with different slenderness factors (s = Wo-). Tables of the combined loss of diversity (current and unbalanced voltage) factor, k,k, have been derived for different load models and confidence levels. However, if one was looking towards a computer-based implementation of this algorithm, these tables are clumsy to implement. For such a lookup table, there are 3 inputs: namely confidence level, slenderness factor and number of consumers. These inputs cannot be simultaneously used in a single lookup table, and so a series of tables would be required, or the values derived from best fit equations. This algorithm does not take into account the following: consumer configuration NLIR A.5 Unbalance Voltage Method This method modifies a base case with consumers evenly distributed across 3 phases, by applying factors which are dependent on the number of

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consumers. The consumer configuration on the feeder is not taken into account directly, but the algorithm can allow for different connection philosophies. However, the derivation of the factors of this method was based on loads being balanced as far as possible at each node on the feeder. As with the Loss of Diversity method above, the factors used in this method [5,7] are derived from Monte Carlo simulation. A more rigorous vector analysis is used to accommodate NUR and defined consumer connection patterns, compared with the Loss of Diversity method. Factors are provided for specified confidence levels. A.6 Beta Distribution Method This is a method based on statistical probabilities. The input load model is a Beta distribution. The output index calculated by an analytical algorithm is a probable condition, within a certain confidence level. This method takes into account all of the factors mentioned in this paper. These are: NUR consumer configuration

consumer load model confidence level To calculate the voltage drop across any single section of a feeder, the voltage drop across the phase with the most consumers is taken to be the voltage drop for that section. However, for adjacent sections the most heavily loaded phases might not be the same; the voltage drop on successive sections may not be cumulative. Thus, this algorithm would have the tendency to be conservative in some cases of cascaded sections. This algorithm calculates the consumer voltage, and then subtracts this from the source voltage to obtain the voltage drop. Thus the accuracy on the calculation of the consumer voltage needs to be high, since the consumer voltage is in the order of 20 times larger than the voltage drop. It should also be noted that the value of the source voltage does not have any influence on the calculated voltage drop for a section. Whether the source voltage is taken as the nominal voltage or as the voltage at the end of the preceding section, the voltage drop is the same. However, the difference will appear when the voltage drop is taken as a percentage of the voltage either at the source, or at the preceding section.

APPENDIX B: TABLES CONTAINING TEST RESULTS Table BI: Results for Network N1: A balanced network. Method Mean load (base case) Monte Carlo Simulation (base case) British method - single step British method - cascaded sections Loss of diversity - single step

Loss of diversity - cascaded sections

Unbalanced volt drop method Herman Beta Method

DT Vdrop (AMEU)

DT Vdrop (British)

DT Vdrop (DT)

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Load

Volt Drop to End of Feeder [%]

ADMD Ll L2 L3 Ll L2 L3 LI L2 L3 L1 L2 L3 LI L2 L3 LI L2 L3 LI L2 L3

2.55 4.30 3.72 3.18 4.51 5.45 3.46 3.11 2.83 4.16 3.79 3.17 4.67 4.40 3.95 3.46 4.70 4.65 4.27 6.09 6.09 6.13 4.91 4.91 4.96

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Table B2: Results for network N2: An unbalanced network Load

Method

C Mean load (base case) Monte Carlo Simulation (base case) British method - single step British method - cascaded sections Loss of diversity - single step

Loss of diversity - cascaded sections

Unbalanced volt drop method Herman Beta Method

DT Vdrop (AMIEU)

DT Vdrop (British)

DT Vdrop (DT)

ADMD LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3

_

7.58 8.96 8.60 8.31 5.67 6.34 4.44 4.10 3.76 4.72 4.30 3.85 5.32 10.03 9.42 8.80 8.57 8.48 8.13 10.74 10.74 10.83 8.83 8.83 8.91

Volt Drop to End of Feeder p/ot E F 8.22 9.77 9.31 9.01 5.67 6.34 4.44 4.10 3.76 4.72 4.30 3.85 5.32 10.81 10.15 9.50 9.48 9.39 9.04 12.13 12.13 12.22 9.74 9.74 9.83

Table B3: Results for Network N3: Cascaded sections Method Mean load (base case) Monte Carlo Simulation (base case) British method - single step British method - cascaded sections Loss of diversity - single step

Loss of diversity - cascaded sections

Unbalanced volt drop method Herman Beta Method

DT Vdrop (ANLEU)

DT Vdrop (British)

DT Vdrop (DT)

Load

Volt Drop to End of Feeder 1%)

ADMD LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3 LI L2 L3

3.47 4.99 4.22 3.46 5.32 7.21 4.39 4.06 3.63 5.30 4.75 4.20 5.44 5.75 5.39 4.73 6.09 6.04 5.09 8.87 8.52 7.39 5.60 5.83 5.39

7.58 8.96 8.60 8.31 5.67 6.34 4.44 4.10 3.76 4.72 4.30 3.85 5.32 10.03 9.42 8.80 8.57 8.48 8.13 10.74 10.74 10.83 8.83 8.83 8.91

G 7.62 9.07 8.66 8.34 5.67 6.34 4.44 4.10 3.76 4.72 4.30 3.85 5.39 10.13 9.50 8.87 8.57 8.48 8.13 10.74 10.74 10.83 8.83 8.83 8.91

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Table B4: Results for Network N4: Sections of different len ths Method Mean load (base case) Monte Carlo Simulation (base case) British method - single step British method - cascaded sections Loss of diversity - single step

Loss of diversity - cascaded sections

Unbalanced volt drop method Herman Beta Method

DT Vdrop (AMEU)

DT Vdrop (British)

DT Vdrop (DT)

Load

Volt Drop to End of Feeder [%]

ADMD L1 L2 L3 L1 L2 L3 L1 L2 L3 LI L2 L3 L1 L2 L3 LI L2 L3 L1 L2 L3

1.54 3.02 2.59 2.17 3.04 4.87 3.34 2.91 2.58 2.85 2.50 2.15 2.98 3.05 2.67 2.29 4.22 4.17 3.57 6.48 6.48 5.30 4.04 4.04 3.43

Table B5: Results for Network N5: Sin le- hase feeder Method Mean load (base case) Monte Carlo Simulation (base case) British method - single step British method - cascaded sections Loss of diversity - single step

Loss of diversity - cascaded sections

Unbalanced volt drop method Herman Beta Method

DT Vdrop (AMEU)

DT Vdrop (British)

DT Vdrop (DT)

Load

Volt Drop to End of Feeder J%]

ADMD LI L2 L3 L1 L2 L3 L1 L2 L3 L1 L2 L3 LI L2 L3 LI L2 L3 LI L2 L3

3.08 4.03 3.80 3.56 2.11 2.82 1.09 0.93 0.81 1.46 1.24 1.08 1.70 4.45 4.30 3.74 5.57 5.52 5.09 8.52 8.52 7.91 5.48 5.48 5.17

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PAPERS/REFERA TE

Comparing methods of calculating voltage drop in low voltage feeders

96

R.L. SeIlick, C. T Gaunt Various modes of dynamic operation of an axial flux permanent magnet brushless machine

112

M.M. Elmissiry, S. Chari Volgolf analise van gekoppelde mikrostrooklyne vir die berekening van kruisspraakvermindering

120

J.C. Coetzee, J. Joubert A systematic methodology for the analysis and design of controllers for nonlinear processes

125

B. Wigdorowitz, A.L. Stevens Lightning protection of MV aerial bundle conductors (ABC)

137

R. B. Lagesse, H.J. Geldenhuys, W C. van der Merwe, J. P. Reynders

REVIEW PAPERS

Power line communications : An overview

H. C. Ferreira, 0. Hooijen

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