A national laboratory of the U.S. Department of Energy

Office of Energy Efficiency & Renewable Energy

National Renewable Energy Laboratory 

Innovation for Our Energy Future 

Modeling and Verification of 

Distributed Generation and 

Voltage Regulation Equipment 

for Unbalanced Distribution 

Power Systems 

Annual Subcontract Report 

June 2007 

M.W. Davis 

DTE Energy 

Detroit, Michigan 

R. Broadwater 

Electrical Distribution Design, Inc. 

Blacksburg, Virginia

J. Hambrick 

Virginia Polytechnic Institute and State University 

Blacksburg, Virginia 

Subcontract Report 

NREL/SR-581-41885 

July 2007 

NREL is operated by Midwest Research Institute ● Battelle Contract No. DE-AC36-99-GO10337 
National Renewable Energy Laboratory

1617 Cole Boulevard, Golden, Colorado 80401-3393 

303-275-3000 • www.nrel.gov 

Operated for the U.S. Department of Energy 

Office of Energy Efficiency and Renewable Energy 

by Midwest Research Institute • Battelle 

Contract No. DE-AC36-99-GO10337

Subcontract Report 

NREL/SR-581-41885 

July 2007 

Modeling and Verification of 

Distributed Generation and 

Voltage Regulation Equipment 

for Unbalanced Distribution 

Power Systems 

Annual Subcontract Report 

June 2007 

M.W. Davis 

DTE Energy 

Detroit, Michigan 

R. Broadwater 

Electrical Distribution Design, Inc. 

Blacksburg, Virginia

J. Hambrick 

Virginia Polytechnic Institute and State University 

Blacksburg, Virginia 

NREL Technical Monitor: T. Basso 

Prepared under Subcontract No. ZAT-5-32616-06 
The National Renewable Energy Laboratory is a national laboratory of the U.S. Department of Energy (DOE) managed by 

Midwest Research Institute for the U.S. Department of Energy under Contract Number DE-AC36-99GO10337. This report was 

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Executive Summary 

Introduction

An earlier study (Davis 2003) of the limits of distributed generation (DG) penetration indicated 

that DG generators in a wide range of sizes (rated in megavolt-amperes, or MVA) can be 

interconnected to a specific distribution circuit. The size limit is dependent on the system 

voltage, the location of DG on the circuit, system protection issues, voltage regulation issues, 

and other issues related to DG and circuit characteristics. This study showed that the size of the 

DG generator can be much larger if it is allowed to actively regulate voltage rather than operate 

at a fixed unity power factor. 

Operating the DG generator at a fixed power factor has minimum effect on the traditional 

voltage regulation controls of a circuit because, as the system voltage changes, the field current 

of the DG synchronous generator is adjusted to bring the kilovolt-ampere reactance (kVAR) 

output back to zero (typically) and to maintain the desired kilowatt setpoint. Therefore, the only 

effect on system voltage is kilowatt injection at that point on the distribution circuit. However, if 

the DG synchronous generator is allowed to absorb or export VAR, voltage can be decreased or 

increased at that point on the circuit, and much larger kilowatt injections, and thus larger DG 

generators, can be installed on the circuit. 

Some of the problems associated with interconnecting DG generators to a distribution circuit are 

related to the design of the circuit and its operation; others are related to the analytical tools used 

to evaluate DG operation. Distribution circuits are designed primarily for radial, one-way flow 

of power. Distribution line voltage regulators typically are designed to regulate voltage based on 

a unidirectional flow of power. When DG generators are interconnected to the circuit, two-way 

flows can result. Most of the load served on a distribution circuit is single-phase. However, most 

of the analytical tools used to evaluate circuit performance are based on balanced three-phase 

loads and balanced three-phase line circuit impedances. 

Balanced three-phase power flow programs are used to calculate the voltage profile on the 

distribution circuit to determine whether DG generators are exceeding voltage limits. When that 

occurs, there is concern as to the accuracy of the resulting service voltages at individual singlephase loads on single-phase laterals, because only the three-phase portion of the circuit is 

modeled. American National Standards Institute (ANSI) Standard C84.1 voltage limits can be 

satisfied based on a three-phase balanced load/impedance analysis, but limits for single-phase 

loads can be exceeded. 

This is a significant concern for a utility, because liability issues arise when a customer’s 

equipment is damaged as a result of either high or low voltage on the circuit. Therefore, it is 

critically important to evaluate the effects of DG on the distribution circuit voltage profile to 

ensure that customers do not receive service voltages (voltages at the customer’s billing meter) 

outside Range A or Range B of the ANSI C84.1 standard. This can be accomplished by using 

modeling and simulation tools that recognize single-phase loads, unsymmetrical distribution 

transformer connections, and unbalanced line impedances. 

 iii
Purpose 

The purpose of the project described in this report was to (1) develop models for an actual 

distribution circuit and all DG generator types (synchronous, induction, and inverter), 

recognizing unbalanced loading and unbalanced circuit impedances; (2) validate these 

models by comparing power flow simulations and voltage profiles with actual measured 

circuit data; and (3) determine the optimum generator operating conditions (e.g., P and Q) 

which provide the greatest improvement in terms of released capacity, reduced energy losses, 

and voltage regulation. 

Project Objectives 

The overall project objectives were fivefold: (1) develop a load model for an actual 13.2 kV 

distribution circuit that represents how the real and reactive load on the circuit changes when the 

voltage is raised or lowered with capacitor switching, distribution line voltage regulator step 

changes, and load tap changing (LTC) and DG voltage regulation; (2) develop models for 

overhead and underground distribution circuit line elements, transformers, shunt capacitors, step 

voltage regulators, and all DG generator types; (3) verify that the models are correct by 

comparing simulated data for power flow with actual measured circuit data; (4) determine the 

maximum DG generator size that can be interconnected to the circuit; and (5) determine the 

optimum generator operating conditions that will result in maximum released capacity savings, 

reduced energy losses, and improved voltage regulation. This annual report addresses objectives 

2 and 3; the final report covers all five objectives. 

Project Results 

The circuit load was modeled using constant current, constant power, and a voltage-dependent 

current (VDC). The VDC model best represented the load characteristics of the circuit with a 

variance of only 2.0%. The percent variance of the constant power model, which is typically 

used, was 12.5%. 

The VDC model that best represents how the load changes with changes in voltage is: 

%ΔV

%ΔP

= 1.26 , 

and 

%ΔV

%ΔQ = 4.66 . 

During heavy load conditions, with all voltage regulation equipment turned on, the circuit tag 

end voltage improvement was 14.87 V when compared with the case in which only the LTC 

transformer regulated the voltage. When the first step regulator was turned on, the tag end 

voltage improved by 4.11 V, and when the second regulator was turned on, the voltage 

increased another 11.34 V. With the first capacitor turned on, the voltage increased only 0.97 V; 

with the second turned on, the total voltage gain was 1.75 V; and with the third turned on, 

another 1.66 V rise occurred at the tag end. It may be necessary to operate DG generators to 

absorb VAR and thus prevent high voltages during light load conditions. 

 iv
The highest unbalanced voltage, 1.52%, occurred during heavy load conditions with no LTC 

regulation. Since most synchronous generators trip when the unbalanced voltage is greater than 

3%, unbalanced three-phase power flow studies should always be conducted on the circuit to 

ensure that the unbalance does not exceed 3% at the point of interconnection (not the point of 

common coupling). Adding LTC regulation during heavy load conditions lowered the maximum 

unbalance voltage to 1.44%; adding step regulators worsened the unbalanced voltage. When all 

three capacitors were turned on, the maximum unbalanced voltage was reduced to 1.31%. 

Turning on all regulation also lowered the maximum unbalanced voltage to 1.31%. During light 

load conditions, the voltage unbalance did not exceed 1.26%. A 13-utility survey on unbalanced 

voltage reported a maximum measured voltage unbalance of 5.94%. The high imbalances 

occurred on open delta transformer connections. 

On many locations on the circuit, the current imbalance exceeded 20%. Most synchronous 

generators trip at between 10% and 20% current imbalance. Therefore, studies must be 

conducted to determine the current imbalance at the location where the DG generator might be 

sited; otherwise, it might never operate without tripping. The load imbalance at the substation 

reached 4% during heavy load conditions and 5.45% during light load conditions. 

Models were developed for the following circuit elements: 

• Line impedance 

• Line voltage drop 

• Line loss (the model was validated using unbalanced and balanced line configurations 

and unbalanced and balanced load conditions) 

• Transformers (models were developed for three-phase and single-phase conditions and 

different loading combinations) 

• Secondary and service impedance, voltage drop, and losses 

• Shunt capacitors and step regulators 

• Synchronous, induction and inverter generators. 

The models were validated by measuring power quantities throughout the circuit on the peak 

day and comparing the measured data with simulation data. The percent variances between 

measured data and simulation data for phase currents at eight different nodes throughout the 

circuit were within 5.7%, and the highest phase voltage variance was only 1.5%. 

Conclusions 

The VDC model best represents how the real and reactive components of the load change with 

changes in voltage. This model, which had an error of only 2%, should always be used in place 

of the constant power model, which had an error of 12.5%. 

Heavy load conditions produce the highest voltage unbalance, and adding voltage regulation 

equipment reduced the maximum percent voltage unbalance from 1.52% to 1.31%. Inverterbased generation was not sensitive to voltage or current unbalance, and it operated at a current 

imbalance of as much as 100%. 

 v
The models developed for the line elements, distribution equipment, and generation and the 

simulations conducted during peak load conditions compared favorably with actual phase 

currents, phase voltages, and power factors measured at eight locations throughout the circuit. 

However, it is essential to know the phasing of the loads on the circuit to obtain such a high 

degree of accuracy. The percent variance between simulated and measured data was less than 

5.7% for phase currents and did not exceed 1.5% for phase voltages. These low variance values 

indicate that the models are accurate enough to represent the actual operation of a circuit under 

unbalanced load conditions. The only measured circuit data needed to perform accurate 

simulations are phase voltages and currents at the source of the circuit and at the regulators, 

capacitors, and DG generation locations. 

Recommendations 

The project has resulted in a number of recommendations; some of the major ones are: 

• Always model distribution circuit loads with a voltage-dependent current. In addition, 

always use a validated unbalanced three-phase power flow to determine the percent 

unbalanced load and voltage during peak loads and at all the locations where DG 

generators are to be placed. Using a balanced three-phase power flow will not indicate 

unbalanced load and voltage where DG generators are to be installed, so a location could 

possibly be selected at which a synchronous DG generator would never operate because 

of circuit unbalance conditions. 

• It is advisable to conduct a three-phase unbalanced power flow study of a circuit before 

installing a synchronous DG generator, because the unit might be installed at a place on 

the circuit where voltage unbalance causes the unit to trip (thus, it will never operate at 

that location). Furthermore, the current imbalance cannot exceed 20% or the 

synchronous DG generator will trip, and there are many places on the distribution circuit 

where the current imbalance exceeds 20%. 

• Adding both LTC voltage regulation equipment and switched capacitors reduces the 

percent voltage unbalance. Adding DG generators to the circuit improves the voltage 

regulation of the circuit when they are allowed to produce and absorb VAR. Larger units 

can be installed if they are allowed to regulate voltage. 

• Future research should include developing an optimal control algorithm that controls the 

substation transformer LTC, bidirectional step regulators, switched capacitors, and DG 

units. This could be accomplished for less than $500,000, because the equipment is 

already installed on Milford Circuit 8103. 

Benefits to California 

Test data indicate that the models developed for this project accurately represent the operation 

of a distribution circuit. In addition, applying these models allows planners to select the sites on 

a circuit for distributed generation that allow the installed generators to perform as intended, 

resulting in the highest possible released capacity savings, the lowest energy losses, and the 

most improved voltage regulation. 

 vi
Table of Contents 

1.0 Introduction............................................................................................................................................1

2.0 Approach................................................................................................................................................2

3.0 Results for Model Development ..........................................................................................................3 

3.1 Three-Phase Substation Transformer Models...................................................................3 

3.2 Delta Wye-Grounded Three-Phase Substation Transformer............................................3 

3.3 Three-Phase and Single-Phase Distribution Transformer Connections for the 

Distribution Circuit...........................................................................................................5 

3.4 Distribution Transformer Impedance................................................................................8 

3.5 Distribution Transformer Voltage Drop ...........................................................................9 

3.6 Distribution Transformer Losses .....................................................................................11 

3.7 Line Impedance Model ...................................................................................................13 

3.8 Line Voltage Drop Model...............................................................................................16 

3.9 Line Losses Model Validation........................................................................................17 

3.10 Secondary and Service Impedances and Voltage Drops.................................................34 

3.11 Secondary and Service Real Losses................................................................................37 

3.12 Shunt Capacitor Models..................................................................................................38 

3.13 Step Voltage Regulator Models......................................................................................42 

3.14 Synchronous Generator Model .......................................................................................42 

3.15 Self-Excited Induction Generator Model........................................................................43 

3.16 Inverter-Based Generator Model ....................................................................................47

4.0 Results for the Design of Field Voltage Regulation and Metering Equipment Used to Verify 

the Models ...........................................................................................................................................50 

4.1 Voltage Regulation Equipment..........................................................................................50 

4.2 Major System Protection Equipment.................................................................................52 

4.3 Metering Equipment and Accuracy of Measurements.......................................................53

5.0 Results for Distributed Generation Control Strategies Used in Field Verification......................55 

5.1 1000 kW Synchronous Generator......................................................................................55 

5.2 400 kW High-Speed Generator and Inverter .....................................................................55 

5.3 400 kW Self-Excited Induction Generator ........................................................................55 

5.4 Voltage Regulation Simulations and Field Verification Strategy 17.................................55

6.0 Results for Field Verification of Models ...........................................................................................62 

6.1 Circuit and Generation Measured Data.............................................................................62 

6.2 Bidirectional Voltage Regulator Measured Data..............................................................64 

6.3 Circuit Equipment (Capacitors) and Customer Measured Data .......................................64 

6.4 Percent Variance Between Simulated and Field Measured Data......................................64 

6.5 Summary of Variance Results ...........................................................................................70

7.0 Conclusions and Recommendations................................................................................................72 

Conclusions..............................................................................................................................72 

Recommendations....................................................................................................................73 

Benefits to California...............................................................................................................73

8.0 References...........................................................................................................................................74 

 vii
List of Figures 

Figure 1. Delta wye-grounded three-phase transformer........................................................ 3 

Figure 2. Four-wire wye-grounded overhead distribution line with multigrounded 

neutral .................................................................................................................. 14 

Figure 3. Flat line spacing configuration............................................................................. 14 

Figure 4. Configuration, phase, and neutral spacings; phase and neutral sizes; and 

conductor types.................................................................................................... 18 

Figure 5. Line transposition and load.................................................................................. 19 

Figure 6. Total line losses versus load imbalance for each line configuration ................... 29 

Figure 7. Percent losses versus load imbalance for each line configuration....................... 29 

Figure 8. Voltage imbalance versus load imbalance for each line configuration ............... 30 

Figure 9. Sequence currents versus load imbalance for each line configuration ................ 30 

Figure 10. Voltage imbalance versus current imbalance for equilateral spacing line........... 31 

Figure 11. Voltage imbalance versus current imbalance for nontransposed line.................. 31 

Figure 12. Voltage imbalance versus current imbalance for transposed line........................ 32 

Figure 13. Sequence currents versus current imbalance for equilateral spacing line............ 32 

Figure 14. Sequence currents versus current imbalance for nontransposed line................... 33 

Figure 15. Sequence currents versus current substance for transposed line ......................... 33 

Figure 16. Distribution transformer servicing secondaries and services............................... 35 

Figure 17. Distribution service drop...................................................................................... 36 

Figure 18. Steady-state synchronous machine model ........................................................... 42 

Figure 19. Two-phase primitive machine to be interconnected to an RLC load................... 43 

Figure 20. Stator direct axis with an R load added.................................................................. 45 

Figure 21. RLC load RL, LL, and CL connected to the self-excited capacitor CS.................. 46 

Figure 22. One line diagram of a 400 kW inverter-based generator and prime mover......... 48 

Figure 23. Voltage pullback curves....................................................................................... 49 

Figure 24. Inverse time-current characteristic....................................................................... 49 

Figure 25. Milford Circuit DC 8103 ..................................................................................... 51 

Figure 26. Milford Substation one line diagram ................................................................... 52 

Figure 27. DG control strategies for 1000 kW synchronous machine .................................. 56 

Figure 28. DG control strategies for 400 kW high-speed generator and inverter in 

current mode ........................................................................................................ 57 

Figure 29. DG control strategies for 400 kW self-excited induction generator.................... 58 

Figure 30. Daily circuit load profile and time stamps........................................................... 63 

 viii
List of Tables 

Table 1. Distribution Transformer No-Load (Core Losses) and Load (Copper Losses)... 10 

Table 2. Evaluation of Equilateral Spacing Kilowatt Losses............................................. 23 

Table 3. Evaluation of Nontransposed kW Losses............................................................ 25 

Table 4. Evaluation of Transposed kW Losses.................................................................. 27 

Table 5. The Effects of Circuit Spacing and Unbalanced Load on Percent kW Losses.... 34 

Table 6. Summary of Transformer, Secondary, and Service Voltage Drops..................... 37 

Table 7. Comparison of Peak Day Real Losses................................................................. 38 

Table 8. Voltage Regulation Equipment............................................................................ 50 

Table 9. Major System Protection Equipment................................................................... 52 

Table 10. Measurement Locations and Data Collection...................................................... 53 

Table 11. Accuracy of 3720 ACM....................................................................................... 54 

Table 12. Matrix of Voltage Regulation Simulations and Control Strategy 17 for 

Field Verification................................................................................................. 59 

Table 13. Test Dates and Data Collection Periods............................................................... 62 

Table 14. Circuit and Generation Measured Data................................................................ 65 

Table 15. Bidirectional Voltage Regulator Measured Data................................................. 66 

Table 16. Circuit Equipment and Customer Measured Data ............................................... 67 

Table 17. Field Verification Data – 7/17/06 – DR Generation On...................................... 68 

Table 18. Field Verification Data – 7/29/06 – DR Generation On...................................... 69 

Table 19. Field Verification Data – 7/31/-6 – DR Generation Off...................................... 69 

Table 20. Percent Variance Between Actual and Simulated Currents, Voltages, and 

Power Factors....................................................................................................... 71 

 ix
 x
1.0 Introduction 

An earlier study (Davis 2003) reported that the largest distributed generation (DG) power 

generator that can be installed on a distribution circuit (or DC) depends on (1) the circuit 

system voltage, (2) the location of the DG generator on the circuit, (3) the circuit 

configuration and characteristics, (4) system protection issues, (5) voltage regulation issues, 

and (6) DG characteristics. Larger DG units can be installed on circuits with higher system 

voltages and at substations with less effect on system protection and voltage regulation issues. 

In addition, installing DG generators that actively regulate voltage allows units of much larger 

sizes to be interconnected to a distribution circuit. This is so because high voltage conditions 

during light load conditions can be curtailed when DG units are absorbing volt-amperes 

reactive (VAR). 

Earlier analyses were all evaluated using three-phase balanced power flows on distribution 

circuits with balanced loads. However, most distribution circuits have unbalanced load 

conditions, unbalanced line impedances, and unbalanced voltage conditions. In addition, there 

is no assurance that the results of studies in which balanced power flows are assumed actually 

represent what is happening on single-phase laterals of a three-phase circuit. The single-phase 

portion may serve upwards of 90% of the load on the circuit. Even if voltage limits are 

satisfied on the three-phase portion of the circuit, this does not mean that there are voltage 

limit violations on the single-phase loads. Installing DG generators on the circuit can add 

another level of complexity to an analysis of a distribution system, but it can also provide the 

added benefits of released capacity, lower energy losses, and improved voltage regulation. 

The objectives of the project described in this report are as follows: 

1. Select a distribution circuit and install DG generators to reduce overload and improve 

voltage regulation. 

2. Develop models and run simulations to calculate voltage profiles on this distribution 

circuit with load tap changers (LTC), step regulators, capacitors, and DG units all 

regulating the voltage. 

3. Install metering on the circuit and DG installations and compare actual measured data 

with data from the simulations. 

4. Determine the maximum size of DG units and the optimum DG operating conditions that 

will provide the greatest released capacity, the lowest energy losses, and the most 

improved voltage regulation. 

 1
2.0 Approach 

The approach to this project involves carrying out all of the following activities: 

1. Develop a load model of a selected distribution circuit that represents how the circuit load 

changes with changes in voltage. 

2. Develop circuit line elements and circuit equipment models. 

3. Develop DG models for synchronous, induction, and inverter generation. 

4. Model an entire 13.2 kV distribution circuit with all three-phase and single-phase loads 

connected to the correct phases of the distribution circuit. 

5. Install a 1000 kW synchronous generator on the circuit, and install the appropriate 

metering equipment throughout the circuit and where the DG unit is installed to measure 

the power quantities of the circuit, including single-phase loads. 

6. Validate the models by conducting a multitude of power flow simulations and comparing 

the data from these simulations to measured data for an actual circuit on the circuit peak 

day (with the DG unit both on and off). 

7. Report the variance between the simulated data and actual measured power data. 

8. Determine the largest DG units that can be installed on the circuit without violating 

voltage, thermal, and reverse power criteria. 

9. Determine the optimum DG operating conditions that result in high released capacity, low 

energy losses, and improved voltage regulation. 

 2
3.0 Results for Model Development 

This section describes the development of models for distributed generation and distribution 

circuit voltage regulation equipment for unbalanced power systems. Voltage regulation 

models were developed for the substation LTC transformer, bidirectional step regulator, 

capacitors, each of the DC transformer connections, and different loading combinations. 

Examples are given of how to calculate transformer impedances from test data, transformer 

voltage drop, and transformer losses. Models for line impedance, voltage drop, and losses as 

well as secondary and service drop impedances, voltage drop, and losses were also developed. 

Also included are models for a shunt capacitor for a wye-grounded connection and for 

synchronous, induction, and inverter-based generators. 

3.1 Three-Phase Substation Transformer Models 

Three-phase transformers are used at a distribution substation to transform the voltage from 

the subtransmission system (or, in some cases, the transmission system) to distribution circuit 

voltage levels. These three-phase transformers can feature high- or low-voltage fixed taps and 

LTC or under-load tap changing. The typical distribution circuit is a four-wire wyemultigrounded system fed from a delta wye-grounded three-phase substation transformer. 

However, there are also three-wire ungrounded delta distribution circuits, and they are 

normally fed from delta-delta three-phase substation transformers. The three-phase and 

single-phase connected transformers on the circuit feed the customer loads, which are threephase or various combinations of single-phase and three-phase. In addition, single-phase 

transformers on the circuit feed single-phase loads. 

3.2 Delta Wye-Grounded Three-Phase Substation Transformer 

The delta wye-grounded transformer connection is shown in Figure 1. 

g

Vtc

Vtb

Vta

+

+

+

-

-

-

+

Vca

-

zt 

zt 

zt 

Vab

+ 

-

Vbc

+

-

+

-

-

VAB

VCA

-

+ 

+ 

+ 

VBC

Figure 1. Delta wye-grounded three-phase transformer 

 3
From Figure 1, the primary side line-to-line voltages can be written as a function of the ideal 

secondary side voltages as: 

⎥

⎥

⎥

⎦

⎤

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⎣

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⋅

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=

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t

t

t

CA

BC

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v

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v

00nn-00

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V

V

V

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where the turns ratio nt is defined as 

LNLS

LLHS t

V

V= n . Equation 3.2 

The ideal secondary voltage can be written as a function of the secondary side line-to-neutral 

voltage, as follows: 

Vt

abc = VLGabc + Ztabc Iabc , Equation 3.3 

where 

⎥

⎥

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⎤

⎢

⎢

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=

t

t

t

t

Z00

0Z0

00Z

Z . Equation 3.4 

The primary line-to-line voltage as a function of the secondary line-to-ground voltage, and the 

secondary current Iabc is 

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00

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0 0

 . Equation 3.5 

To make the transformer model consistent with other system models and thus easier to 

calculate, the secondary side line-to-ground voltage should be expressed as a function of the 

primary side line-to-neutral voltage. This can be done using the theory of symmetrical 

components and algebraic manipulation. The resulting expression is as follows: 

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 4
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⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

c

b

a

t

t

t

c

b

a

t

t

t

C

B

A

I

I

I

z

z

z

V

V

V

n

n

n

V

V

V

00

00

00

00

00

0 0

101

110

011

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

c

b

a

t

t

t

c

b

a

C

B

A

t I

I

I

z

z

z

V

V

V

V

V

V

n

00

00

00

101

110

011

010

001

100

1

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

−

−

−

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

c

b

a

t

t

t

C

B

A

t

c

b

a

I

I

I

z

z

z

V

V

V

n

V

V

V

00

00

00

110

011

101

1 . Equation 3.6 

3.3 Three-Phase and Single-Phase Distribution Transformer Connections for 

the Distribution Circuit 

The following equations determine the voltage drop on the wye-grounded primary system 

feeding different transformer connections and three-phase and single-phase loading on these 

transformers. The factor 0.9 IL in some of the equations below represents the portion of the 

neutral current in the neutral conductor, the remainder of which is the current in the earth. 

Wye-Grounded Primary System Transformer Connections and Voltage Regulation 

Case I 

R NA V = IL [R Cos θ + X Sin θ] + 0.9 IL [RN Cos (θ – 60o

) + XN Sin (θ – 60o)], Equation 3.7 

R CN V = IL [R Cos θ + X Sin θ] + 0.9 IL [RN Cos (θ + 60o

) + XN (θ +60o)]. Equation 3.8 

C

X IL A

N

IL

IL

a

c

b

B

R

SΦ

Equal Single-Phase

R XN Loads N SΦ

R X

 5
Case II 

R X AIL

N

C

RN XN I

L

R X I

L

a

c

b 3Φ

3

B

R NA V = IL [R Cos (θ – 30o

) + X Sin (θ – 30o)] + 0.9 √3 IL [RN Cos (θ – 60o) + XN Sin (θ 

– 60o

)] . 

 Equation 3.9 

R CN V = IL [R Cos (θ + 30o

) + X Sin (θ + 30o)] + 0.9 √3 IL [RN Cos (θ + 60o) + XN Sin (θ 

 + 60o

)] . Equation 3.10 

Case III

SΦ

a

c

N

b

R X I

L A

C

B

R X 2 IL

R X I

L

R NA V = IL [R Cos (θ – 60o

) + X Sin θ – 60o)] . Equation 3.11 

R BN V = 2 IL (R Cos θ + X Sin θ) . Equation 3.12 

R CN V = IL [R Cos (θ + 60o

) + X Sin (θ + 60o)] . Equation 3.13 

Case IV 

3Φ

R X I

L A

R X IL

B

C

R X IL

N

a

c

b

R AN V = IL (R Cos θ + X Sin θ) on L-N Base Voltage . Equation 3.14 

R BN V = IL (R Cos θ + X Sin θ) on L-N Base Voltage . Equation 3.15 

R CN V = IL (R Cos θ + X Sin θ) on L-N Base Voltage . Equation 3.16 

Case V 

 6
R X I

L A

C

B

R X IL

R X I

L

3Φ

a

c

b

R NA V = IL (R Cos θ + X Sin θ) . Equation 3.17 

R NB V = IL (R Cos θ + X Sin θ) . Equation 3.18 

R NC V = IL (R Cos θ + X Sin θ) . Equation 3.19 

Case VI 

R X I

L A

RN

XN N I

L

C B

a

SΦ

N

R NA V = IL [R + RN) Cos θ + (X + XN) Sin θ] . Equation 3.20 

Case VII 

SΦ

a

b

L-L Voltage

Base

R X I

L A

RN XN

R X

C

R X I

L B

N

R BA V = 2 IL (R Cos θ + X Sin θ) . Equation 3.21 

R NA V = IL [R Cos (θ – 30o

) + X Sin (θ – 30o)] . Equation 3.22 

R BN V = IL [R Cos (θ + 30o

) + X Sin (θ + 30o)] . Equation 3.23 

 7
Case VIII 

SΦ

c

a

b L-L Voltage

Base

R X 2 IL A

RN XN

R X I

L

c

R X I

L B

N

R NA V = 2 IL (R Cos θ + X Sin θ) . Equation 3.24 

R BN V = IL [R Cos (θ + 60o

) + X Sin (θ + 60o)] . Equation 3.25 

R CN V = IL [R Cos (θ - 60o

) + X Sin (θ - 60o)] . Equation 3.26 

Case IX 

3 Φ Φ

NA 3Φ

VR + NA SΦ = IL 3Φ [R Cos (θ3Φ -30o

) + X Sin (θ3Φ -30o)] + 0.9 √3 IL 3Φ [RN Cos (θ3Φ 

– 60o

) + XN Sin (θ3Φ – 60o)] + IL SΦ [R Cos θ SΦ + X Sin θ SΦ] + 0.9 IL SΦ 

 [RN Cos (θSΦ – 60o) + XN Sin (θSΦ – 60o)] . Equation 3.27 

VR CN 3Φ = IL 3Φ [R Cos (θ3Φ +30o

) + X Sin (θ3Φ +30o)] + 0.9 √3 IL 3Φ [RN Cos (θ3Φ + 60o) 

 + XN Sin (θ3Φ + 60o

)] . Equation 3.28 

VRCA 3Φ + CA SΦ = IL 3Φ [R Cos (θ +60o

) + X Sin (θ +60o) + IL 3Φ [R Cos (θ -60o) + X Sin (θ 

 -60o

)] + IL SΦ [R Cos (θ - 30o) + X Sin (θ - 30o)] . Equation 3.29 

3.4 Distribution Transformer Impedance 

The percent resistance and percent reactance of transformers can be determined from load 

losses (in watts [W]) and the percent impedance. 

The percent Z is defined as 

,X %R % Z% 2 2 += Equation 3.30 

 8
and percent R is determined from 

. 10 kVA

Load (W) Losses R% = Equation 3.31 

Using the 25 kVA transformer load losses in Table 1, then 

59.1

(25) (10)

W 397 R% = = . 

From Equation 3.3, the %R of Equation 3.31, and the %Z of Table 1, %X is determined as 

,R % Z%X % 2 2 −=

 

2.03 )59.1((2.58)X % 2 2 = =− . 

3.5 Distribution Transformer Voltage Drop 

The voltage drop can be determined when the R and X values, the load current, and the power 

factor are known. The actual R and X values (in ohms) are found from the %R and %X values 

as follows (showing the voltage drop for a 25 kVA transformer): 

 Base Voltage = 240 V . 

 Base kVA = 25 kVA . 

 Base Current = A 104.17 

240

25,000 = . 

 Base Z = 2.30 Ω=

104.17

240 . 

 9
Table 1. Distribution Transformer No-Load (Core Losses) and Load (Copper Losses) 

Type kVA Phase Sec. Volt Pri. Volt 

NoLoad 

Losses 

(W) 

Load 

Losses 

(W) 

Tot. 

Losses 

(W) % Z % Ie

OH 15 S 120/240 4800/7620 34 280 314 2.58 0.51 

OH 25 S 120/240 4800/7620 43 397 440 2.58 0.27 

OH 50 S 120/240 4800/7620 103 564 667 1.97 0.76 

OH 100 S 120/240 4800/7620 165 1150 1315 2.10 0.21 

OH 167 S 120/240 4800/7620 267 1749 2016 2.37 0.33 

PAD 75 3 120/208 4800x13200/7620 283 836 1119 2.43 0.78 

PAD 150 3 120/208 13200/7620 328 2026 2354 2.37 0.34 

PAD 300 3 120/208 13200/7620 639 3198 3837 2.50 0.55 

PAD 500 3 120/208 4800x13200/7620 1140 4085 5225 3.45 0.46 

PAD 1000 3 480Y/277 4160 1160 7601 8761 5.51 0.23 

PAD 1500 3 480Y/277 4800x13200 1516 10294 11810 6.04 0.17 

PAD 2000 3 480Y/277 4800x13200 1894 12933 14827 5.75 0.15 

ISO 333 S 4800/8320 7620/13200 416 2937 3353 3.43 0.35 

PAD 50 S 120/240 7620 107 675 782 1.92 0.30 

PAD 100 S 120/240 7620 173 1074 1247 2.34 0.26 

PAD 167 S 120/240 4800x13200/7620 231 1466 1697 2.72 0.20 

Actual R ohms = 1.59% x 2.30 Ω 

 = 0.0366 Ω . 

 Actual X ohms = 2.0% x 2.30 Ω

 = 0.0460 Ω . 

 Actual Z ohms = 2.58% x 2.30 Ω

 = 0.0593 Ω . 

Assuming a power factor of 0.90 (25.84o

), a load current of 1.40 times rated kVA, and a 

voltage of 105% of rated voltage, then the current would be 

 I = A 138.9 1.40x 

1.05

A) (104.17 = . 

The approximate voltage drop is then 

∆ V = IR Cos θ + I X Sin θ Equation 3.32 

 = (138.9) [(0.0366) (0.90) + (0.0460) (0.4358)] 

 = (138.9) [0.03294 + 0.0200] . 

 10
∆ V = 7.36 V @ 240 V. 

 Equation 3.33 

∆ V @ 120 V = 3.68 V 

The voltage drop can be reduced if a 50 kVA transformer is substituted for the 25 kVA 

transformer. The following shows the voltage drop for a 50 kVA transformer: 

 Base Current = 208.34 A . 

 Base Z = 1.15 Ω=

208.34

240 . 

 Actual R ohms = 1.13% x 1.15 Ω

 = 0.0130 Ω . 

 Actual X ohms = 1.62% x 1.15 Ω

 = 0.0187 Ω . 

 Actual Z ohms = 1.97% x 1.15 Ω

 = 0.0227 Ω . 

Using a 0.90 power factor, 1.40 times rated current, and 105% voltage, the current is 

 I = A 277.8 1.40x 

1.05

208.34 = , 

and 

 

∆ V = IR Cos θ + I X Sin θ

 

 = (277.8) [(0.0130) (0.90) + (0.0187) (0.4358)] 

 = (277.8) [0.0117 + 0.00815]. 

∆ V = 5.51 V @ 240 V 

 Equation 3.34 

 ∆ V @ 120 V = 2.75 V 

3.6 Distribution Transformer Losses 

The distribution transformer losses consist of load losses and no-load losses. 

3.6.1 Load Losses 

The load losses are proportional to the square of the load current. The test data shown in 

Table 1 are given at rated load current. The load losses at loads other than rated load are: 

 11
Load Losses = load), ratedat losses (load I

I

 2

rated

load ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ Equation 3.35 

 = . load) ratedat losses (load kVA

kVA kV

kV

 2

rated

actual

actual

rated ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ Equation 3.36 

For the 25 kVA transformer, the load losses at rated load are 397 W. But at peak load, the 

load may be as high as 140% to 200% of rated load. The load losses at 105% of rated voltage 

and 140% of rated load are from Equation 3.36. 

 Load Losses = (397) 25

(1.40) (25) x

(1.05) (7620)

7620

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

 = (1.78) (397) 

 = 706 W . 

Percent Load Losses = 

θ) (Cos )kVA (10 (1.4)

(W) LossLoad x100

θ) (Cos )(VA (1.4)

(W) LossLoad

rated rated

= . 

 Equation 3.37 

For the 25 kVA transformer, the percent load losses can be calculated as follows: 

 Percent Load Losses = 2.24% 

(0.90) (25) (10) (1.4)

706 = . 

3.6.2 No-Load Losses 

The no-load losses are the core, dielectric, and copper losses caused by the excitation current, 

but the core losses due to hysteresis and eddy current losses are the most significant ones. 

From Table 1, the no-load losses for the 25 kVA transformer amount to 43 W. For 

operation above rated voltage, the no-load losses must be corrected. The correction factors are 

1.15 at 105% of rated voltage and 1.30 at 110% of rated voltage. At 105% voltage, 

No-Load Losses = (Rated No-Load Losses) (Voltage Correction) Equation 3.38 

No-Load Losses = (43 W) (1.15) = 49.5 W . 

3.6.3 Total Losses 

Adding the no-load losses corrected for voltage and load losses corrected for voltage and 

current gives the total percent losses of 2.40% for the 25 kVA transformer. 

Total Percent Losses = 2.40% 

(0.90) (25) (10) (1.4)

49.5 706 = + . Equation 3.39 

 12
 

For the 50 kVA transformer, the percent load losses are 

 Load Losses = (564) 50

(1.40) (50) x

(1.05) (7620)

7620

2

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ 

 = 1003 W . 

Percent Load Losses = 1.59% 

(0.90) (50) (10) (1.4)

1003 = . 

The percent no-load losses are calculated as follows: 

No-Load Losses = (103) (1.15) = 118.5 W, 

and the total percent losses are then 

Total Percent Losses = 1.78% 

(0.90) (50) (10) (1.4)

5.1181003 = + . Equation 3.40 

3.7 Line Impedance Model 

A four-wire wye grounded overhead line distribution circuit consisting of three-phase 

conductors and one neutral may be represented as a 4x4 matrix. Using Figure 2, the resultant 

4x4 matrix using Kirckhoff’s voltage law can be expressed as: 

 

S

VA LVA 

S

VB LVB 

S

VC LVC 

S

VN LVN 

ZA ZAB ZAC ZAN I

A

ZBA ZB ZBC ZBN IB

ZCA ZCB ZC ZCN IC

ZNA ZNB ZNC ZN I

N

= + Equation 3.41 

 13
ZA

S VA

S VB

S VC

S VN

ZB

ZC

ZN

L VA

L VB

L VC

L VN

ZBA

ZCB

ZNC

ZCA

ZNB ZNA

Figure 2. Four-wire wye-grounded overhead distribution line with multigrounded neutral 

For a three-conductor cable with one neutral, the impedance matrix is 

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

=

NA NB NC N

CA CB C CN

BA B BC BN

A AB AC AN

ZZZZ

ZZZZ

ZZZZ

ZZZZ

Z . 

For the flat line configuration used in the Milford circuit in California, the line spacing is: 

N 

B C 

5.25’ 

4.00’ 

7.33’ 

2.42’ A 

Figure 3. Flat line spacing configuration 

 14
The characteristics of the phase and neutral conductors are: 

A B C N 

Conductor 636 Al 636 Al 636 Al 1/0 ACSR

GMR (ft) 0.0294 0.0294 0.0294 0.006 

Resistance (Ω/mi) 0.146 0.146 0.146 1.153 

The diagonal and off-diagonal elements of the impedance matrix (Kersting 2004) are 

calculated using the following formulas. 

zii = ri + 0.09530 + j0.12134 [ln(1/GMRi) + 7.93402] Ω/mi, Equation 3.42 

zij = 0.09530 + j0.12134 [ln(1/Dij) + 7.93402] Ω/mi, Equation 3.43 

where 

ri = resistance of the conductor, in Ω/mi 

Dij = distance, in feet, from conductor i to conductor j 

 GMRi = geometric mean radius of the conductor, in feet 

 0.09530 = equivalent resistance of earth (100 Ω-m) 

 7.93402 = equivalent mutual inductive reactance between a conductor and earth 

 at 60 Hz . 

Note that the diagonal terms account for the self-inductance and resistance of the line and also 

contain expressions that take into account the effects of the earth return path. The off-diagonal 

elements account for the mutual inductance of the lines, and again take into account the 

effects of the earth return path. 

For the flat line model of Figure 3, the impedance matrix is as follows: 

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

+ + + +

+ + + +

+ + + +

+ + + +

=

0.0953 j0.70949 0.0953 j0.78573 0.0953 j0.76267 1.2483 j1.3907

0.0953 j0.72101 0.0953 j0.76963 0.2413 j1.3907 0.0953 j0.76267

0.0953 j0.8554 0.2413 j1.3907 0.0953 j0.76963 0.0953 j0.78573

0.2413 j1.3907 0.0953 j0.85548 0.0953 j0.72101 0.0953 j0.70949

 Z . Equation 3.44 

The Kron reduction technique allows the 4x4 matrix above to be reduced to a more applicable 

3x3 matrix, ZABC. This is accomplished in the following manner: 

Equation 3.45 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+ + +

+ + +

+ + +

= =

0.0953 j0.72101 0.0953 j0.76963 0.2413 j1.3907

0.0953 j0.8554 0.2413 j1.3907 0.0953 j0.76963

0.2413 j1.3907 0.0953 j0.85548 0.0953 j0.72101

CZ CB Z CA Z

BC Z BZ BA Z

AC Z AB Z AZ

ij Z . 

 

 15
 16

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+

+

+

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

0.0953 j0.76267

0.0953 j0.78573

0.0953 j0.70949

z

z

z

 Z

CN

BN

AN

in . Equation 3.46 

[ ] [ ] 0.0953 j0.70949 0.0953 j0.78573 0.0953 j0.76267 NC Z NB Z NA nj Z = Z += + + .

 Equation 3.47 

 NN 1.2483z Z +== j1.5835 . Equation 3.48 

 E

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

= =

+ + +

+ + +

+ + +

0.0953 j0.7210 0.0953 j0.76963 0.2413 j1.3907

0.0953 j0.85548 0.2413 j1.3907 0.0953 j0.76963

0.3404 j1.1566 0.0953 j0.85548 0.0953 j0.7210

njinij ABC ZZZ-Z Z

1-

N 

 . quation 3.49

For a nontransposed line with a length of 6.25 mi (33,000 ft), the impedance matrix becomes 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

+ + +

+ + +

+ + +

=

1.2750 j2.9420 1.3689 j3.0904 2.2530 j7.0188

1.3100 j3.7384 2.3108 j6.9235 1.3689 j3.0904

2.1275 j7.2289 1.3100 j3.7384 1.2750 j2.9420

ABC Z . Equation 3.50 

3.8 Line Voltage Drop Model 

The voltage drop for a given four-wire line segment is calculated using the impedance and 

matrix for that particular line segment. 

ABC LS ABC = IZ V - V . Equation 3.51 

 ΔV = ZABC IABC . Equation 3.52

 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

⋅

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

Δ

Δ

Δ

C

B

A

CA CB C

BA B BC

A AB AC

C

B

A

I

I

I

zzz

zzz

zzz

V

V

V

 . Equation 3.53 

The effects of the admittance matrix have been neglected because of the relatively short line 

length (6.25 mi). 
3.9 Line Losses Model Validation 

To validate the line losses model, three line configurations and balanced and unbalanced load 

conditions were considered. These line configurations are shown in Figure 4. Figure 4(a) is a 

balanced impedance triangular or equilateral configuration with a high B phase. The 

geometric mean distance (GMD) of the phase spacings shown in Figure 4(a) is 53 in. (4.42 ft), 

which is the same GMD for the flat configuration of Figure 3. Figure 4(b) is the flat 

nontransposed line, and Figure 4(c) is the flat transposed line. The dimensions between 

phases, neutral, and ground are identical in Figure 4(b) and Figure 4(c), but the flat transposed 

line of Figure 4(c) has two transpositions. 

Figure 5 shows the flat transposed line that consists of three line segments of 11,000 ft each 

for a total line length of 33,000 ft. The source is at node 0, and the two transpositions are 

shown at node 1 and node 2. The lump load of 9000 kVA at unity power factor is located at 

the end of the line at node 3. 

 17
88"

29" 59"

22'

N

(b) Flat Non-transposed

96" 63"

Earth

N = 1/0 ACSR

Φ = 636 Al

Φ Φ Φ

X-arm

Note: Neutral to ground distance = 22' - 63" = 201" (16.75')

63"

N

N = 1/0 ACSR

(a) Equilateral Spacing

Φ = 636 Al

53"

53

53

"

”

88"

29" 59"

22'

N

(c) Flat Transposed

96" 63"

Earth

N = 1/0 ACSR

Φ = 636 Al

Φ Φ Φ

X-arm

Note: Neutral to ground distance = 22' - 63" = 201" (16.75')

A B C

A B C

A

B

C

GMD = 53”

GMD = 53”

GMD = 53”

88"

29" 59"

22'

N

(b) Flat Non-transposed

96" 63"

Earth

N = 1/0 ACSR

Φ = 636 Al

Φ Φ Φ

X-arm

Note: Neutral to ground distance = 22' - 63" = 201" (16.75')

63"

N

N = 1/0 ACSR

(a) Equilateral Spacing

Φ = 636 Al

53"

53

53

"

”

88"

29" 59"

22'

N

(c) Flat Transposed

96" 63"

Earth

N = 1/0 ACSR

Φ = 636 Al

Φ Φ Φ

X-arm

Note: Neutral to ground distance = 22' - 63" = 201" (16.75')

A B C

A B C

A

B

C

88"

29" 59"

22'

N

(b) Flat Non-transposed

96" 63"

Earth

N = 1/0 ACSR

Φ = 636 Al

Φ Φ Φ

X-arm

Note: Neutral to ground distance = 22' - 63" = 201" (16.75')

88"

29" 59"

22'

N

(b) Flat Non-transposed

96" 63"

Earth

N = 1/0 ACSR

Φ = 636 Al

Φ Φ Φ

X-arm

Note: Neutral to ground distance = 22' - 63" = 201" (16.75')

63"

N

N = 1/0 ACSR

(a) Equilateral Spacing

Φ = 636 Al

53"

53

53

"

”

88"

29" 59"

22'

N

(c) Flat Transposed

96" 63"

Earth

63"

N

N = 1/0 ACSR

(a) Equilateral Spacing

Φ = 636 Al

53"

53

53

"

”

88"

29" 59"

22'

N

(c) Flat Transposed

96" 63"

Earth

N = 1/0 ACSR

Φ = 636 Al

Φ Φ Φ

X-arm

Note: Neutral to ground distance = 22' - 63" = 201" (16.75')

A B C

A B C

A

B

C

GMD = 53”

GMD = 53”

GMD = 53”

Figure 4. Configuration, phase, and neutral spacings; 

phase and neutral sizes; and conductor types 

 18
A

C

B

Flat Transposed Line and Load Characteristics

Segment #1 Segment #2 Segment #3

11,000 ft. 11,000 ft. 11,000 ft.

Bus

33,000 ft.

Line

Load

Node 0 Node 1 Node 2 Node 3

9000 kVA

 = 0o

P.F. = 1.0

θ

C

B

A

B

A

C

Transpositions

Figure 5. Line transposition and load 

This validation shows that, with balanced line impedance and balanced load, the kilowatt losses 

are equal in each phase, and the total kilowatt losses are the lowest. In addition, an example is 

given to explain that, even though the kilowatt losses per phase might not be correct because of 

the application of the Kron reduction process, which reduces a 4x4 matrix to a 3x3 matrix, the 

total losses are, in fact, correct. Table 2, Table 3, and Table 4 and the associated graphs provide 

a summary of balanced and unbalanced loading for the equilateral line spacing, nontransposed 

flat line spacing, and transposed flat line spacing configurations. 

A balanced load of 3000 kW per phase, modeled using constant impedance and computed 

using a nominal voltage of 7.62 kV, is attached at the end of the nontransposed line at node 3. 

The load impedance matrix is 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

19.3500

019.350

0019.35

Zload , , Equation 3.54

where 

 

19.35

3 MVA

kV) (7.62 Z

2

l i oadi = = . Equation 3.55 

Attaching the load to the end of the above nontransposed line yields 

 Equation 3.56 S load ABC ABC += I )Z(ZV . 

 19
For a 13.2 kV line, the above load, and the 6.25-mi nontransposed line, solving for IABC gives 

S

-1

C AB l ABC V)Z(ZI oad += . Equation 3.57 

A 

371.3 108.4

130.2-377.8

10.2-358.4

I ABC

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

∠

∠

∠

= . Equation 3.58 

Note from the data for a nontransposed line in Table 3 at node 3, the magnitudes of phase 

currents are IA = 356.2, IB = 375.9, and IC = 372.0. The data are based on the voltage-dependent 

current source model rather than the constant impedance model data in the matrix above. 

The load end voltage V load is, therefore, 

ABC load S ABC = IZ-V V . 

kV 

108.419.7

130.2-31.7

10.2-6.94

Vload

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

∠

∠

∠

= . Equation 3.59 

The magnitudes of the phase voltages in kilovolts from Table 3 at node 3 are VA = 6.96, VB = 

7.31, and VC = 7.24, respectively. The power loss in the line is 

kW 

102.8

71.6

200.5

loss in out ] Re[S- ]Re[S P

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

= = . Equation 3.60 

The total loss for the line is 374.9 kW. This compares to ΦA = 194.35 kW, ΦB = 72.47 kW, 

and ΦC = 84.29 kW, for a total of 351.11 kW for the voltage-dependent current model. If the 

balanced load impedance matrix is replaced with an unbalanced load impedance matrix, the 

loss results may appear incorrect. 

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

=

25.8100

025.810

0012.90

Zload . Equation 3.61 

The impedance matrix above corresponds to a heavily loaded A phase (4500 kW) relative to 

the B and C phases (2250 kW). The last line of Table 3 at node 0 shows the voltage 

magnitudes in kilovolts of VA = 7.620, VB = 7.620, and VC = 7.620. Applying these balanced 

voltages and unbalanced loads for each phase, then 

 20
 , 12.90 10x 4500

)7620( Z 3

2

A = = 

and , 25.81 10x 2250

)7620( Z 3

2

B = =

 . 25.81 

 10x 2250

)7620( Z 3

2

C = =

Repeating the procedure above to find the line losses yields 

kW 

57.0

164.4-

516.0

l oss in out ] Re[S- ]Re[S P

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

= = . Equation 3.62 

The total losses for the line amount to 408.6 kW. The phase losses from the spreadsheet data 

for the nontransposed case are 131.8, 390.5, and -117.4, corresponding to a heavily loaded B 

phase (4500 kW) in the simulation case, or a total of 404.96 kW. It is obvious in this 

unbalanced case that the phase loss calculations do not represent the actual losses in each 

phase; however, the total line loss is correct. The inclusion of the effects of the earth and 

neutral in the 3x3 impedance matrix is the reason for the dubious phase loss numbers. To 

obtain correct losses for the individual phases, the I2

R losses should be calculated. First, the 

earth and neutral currents should be found. 

A 157.560.6IZ I ABC NtN ∠== , Equation 3.63 

where 

ABC nj Nt 1- Z -ZIZ N

= . Equation 3.64 

 

The earth current is found as follows: 

E NCBA 127.9)III-(II ∠=+++= A 126.4 . Equation 3.65 

Next, the I2

R losses for each phase, earth, and neutral are calculated: 

A kW 209.1RI P 2lossA A == Equation 3.66 

B kW 88.4RI P 2lossB B == Equation 3.67 

C kW 0.57RI P 2lossC C == Equation 3.68 

E kW 7.9RI P 2lossE E == Equation 3.69 

N kW 26.4RI P 2lossN N == . Equation 3.70 

 21
When the losses are summed from the I2

R calculations, the result is 408.6. This exactly 

matches the total line loss calculated previously. There is a slight difference between the 

amount for total losses in the Table 3 spreadsheet data, 404.96 kW, and the total losses 

amount of 408.6 kW shown above, because the spreadsheet data are for a voltage-dependent 

current model and the total losses above are based on a constant impedance model. 

 22
Table 2. Evaluation of Equilateral Spacing Kilowatt Losses 

ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC

3000.00 3000.00 3000.00 117.17 117.02 116.89 351.08 515.76 515.74 515.68 1547.18 2758.79 2759.06 2759.28 8277.13

3150.00 2925.00 2925.00 143.85 94.42 113.39 351.66 568.41 507.63 472.37 1548.41 2856.88 2713.97 2703.31 8274.16

3300.00 2850.00 2850.00 171.35 72.44 109.62 353.41 622.80 498.63 430.76 1552.19 2951.53 2666.94 2646.96 8265.43

3450.00 2775.00 2775.00 199.93 51.43 104.92 356.28 677.99 489.86 390.41 1558.26 3041.81 2619.49 2589.48 8250.78

3600.00 2700.00 2700.00 228.92 31.27 100.01 360.20 734.23 480.24 351.94 1566.41 3128.22 2570.23 2531.83 8230.28

3750.00 2625.00 2625.00 258.85 11.66 94.75 365.26 792.19 470.22 314.54 1576.95 3211.81 2519.64 2472.90 8204.35

3900.00 2550.00 2550.00 289.23 -6.96 89.03 371.30 850.71 459.93 278.75 1589.39 3291.28 2468.02 2413.32 8172.62

4050.00 2475.00 2475.00 320.05 -24.56 82.79 378.28 909.51 449.56 244.49 1603.56 3366.43 2415.58 2352.90 8134.91

4200.00 2400.00 2400.00 351.17 -41.40 76.48 386.25 969.41 438.57 211.70 1619.68 3438.52 2361.61 2291.72 8091.85

4350.00 2325.00 2325.00 382.50 -57.46 70.17 395.21 1030.43 426.96 180.38 1637.77 3507.79 2306.09 2229.77 8043.65

4500.00 2250.00 2250.00 414.10 -72.43 63.39 405.06 1091.39 415.48 150.55 1657.42 3572.73 2249.97 2166.85 7989.55

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.62 7.62 7.62 0 7620.91 0.11 0.0% 368.22 368.25 368.28 0.02 368.24 0.01 0.0% 0.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 382.16 362.23 360.04 6.44 368.12 8.39 2.3% 4.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 395.74 355.95 351.85 12.85 367.78 16.63 4.5% 8.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 408.87 349.61 343.58 19.24 367.2 24.7 6.7% 11.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 421.56 343.04 335.36 25.53 366.38 32.61 8.9% 15.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 433.99 336.27 327.06 31.87 365.35 40.37 11.0% 19.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 445.97 329.36 318.74 38.14 364.09 47.97 13.2% 22.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 457.5 322.35 310.36 44.35 362.59 55.41 15.3% 26.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 468.72 315.12 301.96 50.54 360.88 62.69 17.4% 30.0%

7.62 7.62 7.62 0 7620.91 0.11 0.0% 479.63 307.68 293.51 56.72 358.97 69.83 19.5% 0.33

7.62 7.62 7.62 0 7620.91 0.11 0.0% 490.12 300.16 284.99 62.84 356.83 76.81 21.5% 0.37

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.44 7.44 7.44 0.1 7444.34 0.08 0.0% 368.22 368.25 368.28 0.02 368.24 0.01 0.0% 0.0%

7.42 7.46 7.45 27.28 7444.34 10.95 0.1% 382.16 362.23 360.04 6.44 368.12 8.39 2.3% 4.0%

7.39 7.48 7.46 54.31 7444.22 21.65 0.3% 395.74 355.95 351.85 12.85 367.78 16.63 4.5% 8.0%

7.36 7.5 7.47 81.21 7444.01 32.15 0.4% 408.87 349.61 343.58 19.24 367.2 24.7 6.7% 11.0%

7.33 7.52 7.48 107.71 7443.74 42.43 0.6% 421.56 343.04 335.36 25.53 366.38 32.61 8.9% 15.0%

7.3 7.54 7.49 134.43 7443.41 52.52 0.7% 433.99 336.27 327.06 31.87 365.35 40.37 11.0% 19.0%

7.27 7.56 7.5 160.86 7443.01 62.4 0.8% 445.97 329.36 318.74 38.14 364.09 47.97 13.2% 22.0%

7.25 7.58 7.51 187.04 7442.55 72.06 1.0% 457.5 322.35 310.36 44.35 362.59 55.41 15.3% 26.0%

7.22 7.59 7.52 213.13 7442.05 81.53 1.1% 468.72 315.12 301.96 50.54 360.88 62.69 17.4% 30.0%

7.19 7.61 7.53 239.13 7441.52 90.81 1.2% 479.63 307.68 293.51 56.72 358.97 69.83 19.5% 0.33

7.16 7.63 7.54 264.96 7440.9 99.88 1.3% 490.12 300.16 284.99 62.84 356.83 76.81 21.5% 0.37

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.29 7.29 7.3 0.19 7294.81 0.15 0.0% 368.23 368.27 368.29 0.02 368.26 0.01 0.0% 0.0%

7.24 7.33 7.31 54.55 7294.79 21.95 0.3% 382.18 362.24 360.05 6.44 368.13 8.39 2.3% 3.0%

7.19 7.37 7.33 108.61 7294.48 43.37 0.6% 395.76 355.97 351.86 12.85 367.79 16.63 4.5% 7.0%

7.13 7.41 7.34 162.42 7293.96 64.36 0.9% 408.89 349.63 343.59 19.24 367.21 24.7 6.7% 10.0%

7.08 7.45 7.36 215.42 7293.28 84.91 1.2% 421.58 343.05 335.37 25.52 366.39 32.61 8.9% 13.0%

7.03 7.48 7.37 268.87 7292.42 105.1 1.4% 434.01 336.28 327.07 31.87 365.37 40.37 11.0% 17.0%

6.98 7.52 7.39 321.72 7291.4 124.86 1.7% 445.99 329.38 318.74 38.14 364.11 47.97 13.2% 20.0%

6.92 7.56 7.41 374.07 7290.21 144.19 2.0% 457.52 322.36 310.37 44.35 362.61 55.41 15.3% 23.0%

6.87 7.6 7.42 426.25 7288.9 163.12 2.2% 468.74 315.14 301.96 50.54 360.9 62.69 17.4% 26.0%

6.82 7.63 7.44 478.26 7287.49 181.69 2.5% 479.65 307.7 293.52 56.71 358.99 69.83 19.5% 0.29

6.77 7.67 7.46 529.91 7285.88 199.83 2.7% 490.14 300.18 284.99 62.84 356.84 76.81 21.5% 0.32

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.17 7.17 7.17 0.29 7174.03 0.24 0.0% 368.24 368.27 368.3 0.02 368.26 0.01 0.0% 0.0%

7.1 7.23 7.19 81.83 7173.95 32.96 0.5% 382.19 362.25 360.06 6.44 368.14 8.39 2.3% 3.0%

7.02 7.29 7.21 162.92 7173.39 65.08 0.9% 395.77 355.97 351.86 12.85 367.8 16.63 4.5% 6.0%

6.95 7.35 7.23 243.63 7172.45 96.57 1.3% 408.9 349.63 343.59 19.24 367.22 24.7 6.7% 9.0%

6.88 7.4 7.25 323.12 7171.2 127.4 1.8% 421.59 343.06 335.38 25.52 366.4 32.61 8.9% 12.0%

6.8 7.46 7.27 403.29 7169.63 157.69 2.2% 434.02 336.29 327.07 31.87 365.37 40.37 11.0% 15.0%

6.73 7.51 7.29 482.57 7167.75 187.32 2.6% 446 329.38 318.75 38.14 364.11 47.97 13.2% 17.0%

6.66 7.57 7.31 561.09 7165.55 216.31 3.0% 457.53 322.37 310.37 44.35 362.62 55.4 15.3% 20.0%

6.59 7.63 7.34 639.36 7163.11 244.72 3.4% 468.75 315.14 301.96 50.54 360.91 62.69 17.4% 23.0%

6.52 7.68 7.36 717.38 7160.47 272.57 3.8% 479.67 307.7 293.52 56.71 358.99 69.83 19.5% 0.26

6.44 7.74 7.38 794.85 7157.47 299.78 4.2% 490.16 300.18 284.99 62.84 356.85 76.81 21.5% 0.28

Node 3

Sequence Currents Imbalance

V2/V1

Voltage Sequence Voltages (V) Current

Imbalance

Node 2

Sequence Currents Imbalance

Voltage

Voltage Sequence Voltages (V)

Sequence Voltages (V)

V2/V1

Current

V2/V1

Current

Sequence Voltages (V)

V2/V1

Current

Spot Load (kW) kW Losses kVar Losses kW Flow

Voltage

Node 0

Total kW 

Losses

Total kVar 

Losses Total kW Flow

Sequence Currents Imbalance

Node 1

Sequence Currents

 23
ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC

513.61 513.62 513.49 1540.72 2641.63 2642.04 2642.39 7926.06 -2.15 -2.12 -2.20 -6.47

566.39 505.01 469.93 1541.33 2713.03 2619.55 2589.92 7922.50 -2.03 -2.62 -2.44 -7.09

620.16 496.25 428.61 1545.02 2780.18 2594.50 2537.33 7912.01 -2.64 -2.38 -2.15 -7.17

675.99 487.17 388.05 1551.21 2841.88 2568.05 2484.56 7894.49 -2.00 -2.70 -2.36 -7.06

732.12 478.01 349.25 1559.38 2899.30 2538.95 2431.82 7870.07 -2.11 -2.23 -2.69 -7.03

789.55 467.89 312.28 1569.72 2952.96 2507.99 2378.15 7839.10 -2.65 -2.33 -2.26 -7.24

847.92 457.62 276.52 1582.06 3002.05 2474.98 2324.28 7801.31 -2.78 -2.31 -2.23 -7.32

907.62 447.15 241.84 1596.61 3046.38 2440.15 2270.11 7756.64 -1.89 -2.41 -2.65 -6.95

967.68 436.16 208.98 1612.82 3087.35 2403.01 2215.24 7705.60 -1.73 -2.42 -2.72 -6.87

1027.84 424.65 177.97 1630.46 3125.30 2363.54 2159.60 7648.44 -2.58 -2.31 -2.41 -7.30

1089.67 412.92 147.93 1650.52 3158.63 2322.40 2103.46 7584.49 -1.73 -2.56 -2.61 -6.90

kW Native Load Total kW 

Load

kVar Native Load Total kVar 

Load

kVar Flow Total kVar 

Flow

Plot Information:

0.0% 351.08 4.2% 0.0% 0.0% 0.0% 0.02 368.24 0.01

2.3% 351.66 4.3% 0.1% 0.3% 0.5% 6.44 368.12 8.39

4.5% 353.41 4.3% 0.3% 0.6% 0.9% 12.85 367.78 16.63

6.7% 356.28 4.3% 0.4% 0.9% 1.3% 19.24 367.20 24.70

8.9% 360.20 4.4% 0.6% 1.2% 1.8% 25.53 366.38 32.61

11.0% 365.26 4.5% 0.7% 1.4% 2.2% 31.87 365.35 40.37

13.2% 371.30 4.5% 0.8% 1.7% 2.6% 38.14 364.09 47.97

15.3% 378.28 4.7% 1.0% 2.0% 3.0% 44.35 362.59 55.41

17.4% 386.25 4.8% 1.1% 2.2% 3.4% 50.54 360.88 62.69

19.5% 395.21 4.9% 1.2% 2.5% 3.8% 56.72 358.97 69.83

21.5% 405.06 5.1% 1.3% 2.7% 4.2% 62.84 356.83 76.81

I0 I1 I2

kW 

Losses / 

kW Flow

Node 0 

I2/I1

 Total kW 

Losses

Node 1 

V1/V2

Node 2 

V1/V2

Node 3 

V1/V2

Note: “PF” in Table 2, Table 3, and Table 4 is the percent power flow imbalance (kVA).

 24
Table 3. Evaluation of Nontransposed kW Losses 

ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC

3000.00 3000.00 3000.00 194.35 72.47 84.29 351.11 468.69 504.08 571.12 1543.89 2673.63 2820.78 2777.24 8271.65

2925.00 3150.00 2925.00 191.64 101.18 59.82 352.64 421.45 558.63 560.65 1540.73 2617.99 2918.11 2734.33 8270.43

2850.00 3300.00 2850.00 187.85 130.92 36.35 355.12 376.15 614.45 549.88 1540.48 2561.96 3010.91 2690.10 8262.97

2775.00 3450.00 2775.00 183.15 161.60 13.77 358.52 332.63 671.65 538.71 1542.99 2505.32 3099.54 2644.44 8249.30

2700.00 3600.00 2700.00 177.65 193.08 -7.92 362.81 290.90 730.13 527.13 1548.16 2448.06 3184.09 2597.33 8229.48

2625.00 3750.00 2625.00 171.40 225.19 -28.66 367.93 250.97 789.76 515.15 1555.88 2390.15 3264.58 2548.82 8203.55

2550.00 3900.00 2550.00 164.51 257.80 -48.44 373.87 212.83 850.44 502.77 1566.04 2331.59 3341.09 2498.88 8171.56

2475.00 4050.00 2475.00 157.03 290.77 -67.24 380.56 176.49 912.05 490.01 1578.55 2272.37 3413.67 2447.55 8133.59

2400.00 4200.00 2400.00 149.06 323.97 -85.02 388.01 141.95 974.51 476.88 1593.34 2212.46 3482.4 2394.83 8089.69

2325.00 4350.00 2325.00 140.67 357.27 -101.76 396.18 109.20 1037.72 463.38 1610.30 2151.87 3547.33 2340.72 8039.92

2250.00 4500.00 2250.00 131.84 390.48 -117.36 404.96 78.33 1101.20 449.63 1629.16 2090.66 3608.08 2285.35 7984.09

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.6200 7.6200 7.6200 0 7620.91 0.11 0.0% 356.11 375.94 371.99 3.24 367.98 9.3 2.5% 0.03

7.62 7.62 7.62 0 7620.91 0.11 0.0% 347.9 389.79 366.19 9.06 367.9 15.93 4.3% 0.06

7.62 7.62 7.6200 0 7620.91 0.11 0.0% 339.73 403.15 360.22 15.38 367.57 23.48 6.4% 0.1

7.62 7.62 7.6200 0 7620.91 0.11 0.0% 331.58 416.08 354.06 21.72 367.01 31.17 8.5% 0.13

7.62 7.62 7.6200 0 7620.91 0.11 0.0% 323.45 428.57 347.7 28.05 366.2 38.8 10.6% 0.17

7.62 7.62 7.6200 0 7620.91 0.11 0.0% 315.32 440.64 341.15 34.34 365.14 46.31 12.7% 0.2

7.62 7.62 7.6200 0 7620.91 0.11 0.0% 307.19 452.3 334.41 40.59 363.85 53.68 14.8% 0.24

7.62 7.62 7.6200 0 7620.91 0.11 0.0% 299.04 463.56 327.47 46.8 362.32 60.9 16.8% 0.28

7.62 7.62 7.62 0 7620.91 0.11 0.0% 290.89 474.41 320.35 52.97 360.56 67.97 18.9% 0.31

7.62 7.62 7.62 0 7620.91 0.11 0.0% 282.71 484.88 313.04 59.1 358.57 74.88 20.9% 0.35

7.6200 7.62 7.62 0 7620.91 0.11 0.0% 274.51 494.92 305.56 65.15 356.34 81.62 22.9% 0.38

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.38 7.49 7.46 17.54 7442.93 56.3 0.8% 356.11 375.94 371.99 3.24 367.98 9.3 2.5% 0.03

7.39 7.46 7.48 42.54 7442.11 46.62 0.6% 347.9 389.79 366.19 9.06 367.9 15.93 4.3% 0.06

7.39 7.43 7.5 71.6 7441.23 37.83 0.5% 339.73 403.15 360.22 15.38 367.57 23.48 6.4% 0.1

7.4 7.4 7.52 101.14 7440.28 30.53 0.4% 331.58 416.08 354.06 21.72 367.01 31.17 8.5% 0.13

7.41 7.37 7.54 130.7 7439.27 25.77 0.3% 323.45 428.57 347.7 28.05 366.2 38.8 10.6% 0.17

7.42 7.34 7.57 160.16 7438.22 24.81 0.3% 315.32 440.64 341.15 34.34 365.14 46.31 12.7% 0.2

7.43 7.3 7.59 189.47 7437.11 27.84 0.4% 307.19 452.3 334.41 40.59 363.85 53.68 14.8% 0.24

7.43 7.27 7.61 218.59 7435.97 33.64 0.5% 299.04 463.56 327.47 46.8 362.32 60.9 16.8% 0.28

7.44 7.24 7.63 247.53 7434.78 40.92 0.6% 290.89 474.41 320.35 52.97 360.56 67.97 18.9% 0.31

7.45 7.21 7.65 276.27 7433.56 48.93 0.7% 282.71 484.88 313.04 59.1 358.57 74.88 20.9% 0.35

7.46 7.18 7.67 304.7 7432.31 57.25 0.8% 274.51 494.92 305.56 65.15 356.34 81.62 22.9% 0.38

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.16 7.39 7.33 35.08 7291.9 112.7 1.5% 356.14 375.94 372 3.24 367.99 9.29 2.5% 0.04

7.17 7.33 7.38 85.08 7290.1 93.32 1.3% 347.93 389.8 366.2 9.06 367.91 15.91 4.3% 0.06

7.18 7.27 7.42 143.2 7288.13 75.74 1.0% 339.76 403.16 360.23 15.37 367.59 23.47 6.4% 0.09

7.19 7.21 7.46 202.27 7285.99 61.12 0.8% 331.61 416.08 354.07 21.72 367.02 31.16 8.5% 0.13

7.21 7.15 7.5 261.4 7283.7 51.56 0.7% 323.47 428.58 347.71 28.04 366.21 38.79 10.6% 0.16

7.22 7.09 7.54 320.31 7281.26 49.59 0.7% 315.34 440.65 341.16 34.34 365.16 46.3 12.7% 0.19

7.24 7.03 7.59 378.92 7278.69 55.61 0.8% 307.21 452.31 334.41 40.59 363.87 53.67 14.7% 0.22

7.25 6.98 7.63 437.17 7276 67.19 0.9% 299.07 463.57 327.48 46.8 362.34 60.89 16.8% 0.25

7.27 6.92 7.67 495.05 7273.19 81.75 1.1% 290.91 474.43 320.36 52.97 360.58 67.96 18.8% 0.28

7.28 6.86 7.71 552.53 7270.27 97.75 1.3% 282.73 484.9 313.05 59.09 358.59 74.87 20.9% 0.31

7.3 6.8 7.75 609.38 7267.27 114.39 1.6% 274.52 494.94 305.57 65.15 356.36 81.61 22.9% 0.34

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

6.96 7.31 7.24 52.62 7169.53 169.1 2.4% 356.17 375.94 372 3.24 368 9.27 2.5% 0.05

6.97 7.23 7.3 127.61 7166.6 140.03 2.0% 347.95 389.79 366.2 9.05 367.92 15.9 4.3% 0.07

6.99 7.14 7.37 214.79 7163.34 113.65 1.6% 339.78 403.16 360.23 15.37 367.6 23.47 6.4% 0.09

7 7.06 7.43 303.4 7159.76 91.71 1.3% 331.63 416.08 354.07 21.71 367.03 31.15 8.5% 0.12

7.02 6.98 7.49 392.08 7155.89 77.36 1.1% 323.49 428.58 347.71 28.04 366.22 38.78 10.6% 0.15

7.04 6.9 7.55 480.44 7151.74 74.38 1.0% 315.36 440.65 341.16 34.33 365.17 46.29 12.7% 0.18

7.05 6.82 7.62 568.36 7147.34 83.38 1.2% 307.23 452.32 334.41 40.58 363.88 53.66 14.7% 0.2

7.07 6.74 7.68 655.73 7142.68 100.74 1.4% 299.08 463.57 327.48 46.8 362.35 60.89 16.8% 0.23

7.09 6.66 7.74 742.55 7137.8 122.56 1.7% 290.92 474.43 320.36 52.96 360.59 67.95 18.8% 0.25

7.11 6.58 7.8 828.77 7132.71 146.57 2.1% 282.74 484.9 313.05 59.09 358.6 74.86 20.9% 0.28

7.14 6.5 7.86 914.04 7127.42 171.52 2.4% 274.54 494.94 305.57 65.14 356.37 81.6 22.9% 0.31

Current

Sequence Voltages (V)

V2/V1

Current

Sequence Voltages (V)

V2/V1

Current

Node 2

Voltage Sequence Currents Imbalance

Spot Load (kW) kW Losses kVar Losses kW Flow

Node 1

Sequence Currents

Voltage

Node 0

Total kW 

Losses

Total kVar 

Losses Total kW Flow

Sequence Voltages (V)

Imbalance

V2/V1

Current Imbalance

Voltage

Sequence Currents

Node 3

Voltage Sequence Voltages (V) Sequence Currents Imbalance

V2/V1

 25
ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC

465.92 501.64 569.08 1536.64 2479.28 2748.31 2692.95 7920.54 -2.78 -2.43 -2.04 -7.25

419.06 555.98 558.44 1533.48 2426.36 2816.93 2674.51 7917.80 -2.39 -2.66 -2.21 -7.26

373.75 611.81 547.65 1533.21 2374.11 2879.99 2653.75 7907.85 -2.40 -2.63 -2.23 -7.26

330.22 669.05 536.47 1535.74 2322.17 2937.94 2630.67 7890.78 -2.41 -2.60 -2.24 -7.25

288.48 727.55 524.87 1540.90 2270.41 2991.02 2605.25 7866.68 -2.42 -2.58 -2.26 -7.26

248.53 787.22 512.87 1548.62 2218.75 3039.39 2577.47 7835.61 -2.44 -2.54 -2.28 -7.26

210.38 847.92 500.48 1558.78 2167.09 3083.29 2547.33 7797.71 -2.45 -2.52 -2.30 -7.27

174.03 909.56 487.70 1571.29 2115.34 3122.90 2514.79 7753.03 -2.46 -2.48 -2.31 -7.25

139.48 972.05 474.55 1586.08 2063.40 3158.43 2479.85 7701.68 -2.47 -2.47 -2.33 -7.27

106.72 1035.28 461.04 1603.04 2011.20 3190.07 2442.48 7643.75 -2.48 -2.44 -2.35 -7.27

75.60 1099.26 447.30 1622.16 1958.82 3217.60 2402.71 7579.13 -2.73 -1.94 -2.32 -6.99

Total kW 

Load

kVar Native Load Total kVar 

Load

kVar Flow Total kVar 

Flow

kW Native Load

Plot Information:

2.5% 351.11 4.2% 0.8% 1.5% 2.4% 3.24 367.98 9.30

4.3% 352.64 4.3% 0.6% 1.3% 2.0% 9.06 367.90 15.93

6.4% 355.12 4.3% 0.5% 1.0% 1.6% 15.38 367.57 23.48

8.5% 358.52 4.3% 0.4% 0.8% 1.3% 21.72 367.01 31.17

10.6% 362.81 4.4% 0.3% 0.7% 1.1% 28.05 366.20 38.80

12.7% 367.93 4.5% 0.3% 0.7% 1.0% 34.34 365.14 46.31

14.8% 373.87 4.6% 0.4% 0.8% 1.2% 40.59 363.85 53.68

16.8% 380.56 4.7% 0.5% 0.9% 1.4% 46.80 362.32 60.90

18.9% 388.01 4.8% 0.6% 1.1% 1.7% 52.97 360.56 67.97

20.9% 396.18 4.9% 0.7% 1.3% 2.1% 59.10 358.57 74.88

22.9% 404.96 5.1% 0.8% 1.6% 2.4% 65.15 356.34 81.62

I0 I1

 Total kW 

Losses

Node 1 

V1/V2

Node 2 

V1/V2

Node 0 

I2/I1

Node 3 

V2/V1

kW 

Losses / 

kW Flow I2

Note: “PF” in Table 2, Table 3, and Table 4 is the percent power flow imbalance (kVA).

 26
Table 4. Evaluation of Transposed kW Losses 

ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC

3000.00 3000.00 3000.00 117.05 117.04 117.06 351.15 515.80 515.77 515.79 1547.36 2758.93 2758.99 2759.00 8276.92

3150.00 2925.00 2925.00 144.46 91.77 115.41 351.64 570.85 508.10 469.82 1548.77 2854.89 2716.82 2702.11 8273.82

3300.00 2850.00 2850.00 172.80 67.30 112.99 353.09 627.40 500.00 425.46 1552.86 2946.78 2673.22 2644.53 8264.53

3450.00 2775.00 2775.00 201.93 43.65 109.86 355.44 685.35 491.46 382.72 1559.53 3034.65 2628.21 2586.23 8249.09

3600.00 2700.00 2700.00 231.72 20.86 106.09 358.67 744.55 482.47 341.61 1568.63 3118.54 2581.78 2527.21 8227.53

3750.00 2625.00 2625.00 262.04 -1.04 101.75 362.75 804.88 473.04 302.13 1580.05 3198.50 2533.96 2467.46 8199.92

3900.00 2550.00 2550.00 292.76 -22.01 96.92 367.67 866.23 463.17 264.30 1593.70 3274.59 2484.74 2406.97 8166.30

4050.00 2475.00 2475.00 323.76 -42.03 91.64 373.37 928.50 452.87 228.11 1609.48 3346.87 2434.14 2345.73 8126.74

4200.00 2400.00 2400.00 354.91 -61.08 85.99 379.82 991.58 442.13 193.57 1627.28 3415.40 2382.16 2283.73 8081.29

4350.00 2325.00 2325.00 386.12 -79.12 80.03 387.03 1055.39 430.97 160.70 1647.06 3480.26 2328.81 2220.98 8030.05

4500.00 2250.00 2250.00 417.26 -96.12 73.83 394.97 1119.83 419.41 129.49 1668.73 3541.49 2274.10 2157.48 7973.07

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.62 7.62 7.62 0 7620.91 0.11 0.0% 368.23 368.24 368.24 0 368.23 0 0.0% 0

7.62 7.62 7.62 0 7620.91 0.11 0.0% 381.96 362.61 359.83 6.43 368.11 8.38 2.3% 0.04

7.62 7.62 7.62 0 7620.91 0.11 0.0% 395.27 356.79 351.42 12.82 367.74 16.59 4.5% 0.07

7.62 7.62 7.62 0 7620.91 0.11 0.0% 408.16 350.78 343 19.18 367.13 24.63 6.7% 0.11

7.62 7.62 7.62 0 7620.91 0.11 0.0% 420.63 344.57 334.58 25.5 366.28 32.5 8.9% 0.15

7.62 7.62 7.62 0 7620.91 0.11 0.0% 432.71 338.18 326.15 31.78 365.19 40.21 11.0% 0.18

7.62 7.62 7.62 0 7620.91 0.11 0.0% 444.38 331.59 317.7 38.03 363.86 47.74 13.1% 0.22

7.62 7.62 7.62 0 7620.91 0.11 0.0% 455.67 324.82 309.22 44.25 362.29 55.12 15.2% 0.25

7.62 7.62 7.62 0 7620.91 0.11 0.0% 466.58 317.85 300.71 50.42 360.49 62.33 17.3% 0.29

7.62 7.62 7.62 0 7620.91 0.11 0.0% 477.12 310.7 292.17 56.55 358.47 69.38 19.4% 0.33

7.62 7.62 7.62 0 7620.91 0.11 0.0% 487.29 303.37 283.59 62.65 356.22 76.27 21.4% 0.36

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.36 7.5 7.47 25.23 7444.34 60.26 0.8% 368.23 368.24 368.24 0 368.23 0 0.0% 0

7.33 7.53 7.48 53.13 7445.92 64.24 0.9% 381.96 362.61 359.83 6.43 368.11 8.38 2.3% 0.04

7.31 7.55 7.49 81.93 7447.38 69.72 0.9% 395.27 356.79 351.42 12.82 367.74 16.59 4.5% 0.07

7.28 7.57 7.5 110.8 7448.72 76.26 1.0% 408.16 350.78 343 19.18 367.13 24.63 6.7% 0.11

7.25 7.6 7.5 139.58 7449.95 83.52 1.1% 420.63 344.57 334.58 25.5 366.28 32.5 8.9% 0.15

7.22 7.62 7.51 168.23 7451.06 91.24 1.2% 432.71 338.18 326.15 31.78 365.19 40.21 11.0% 0.18

7.19 7.64 7.52 196.72 7452.07 99.22 1.3% 444.38 331.59 317.7 38.03 363.86 47.74 13.1% 0.22

7.17 7.67 7.53 225.04 7452.99 107.34 1.4% 455.67 324.82 309.22 44.25 362.29 55.12 15.2% 0.25

7.14 7.69 7.54 253.18 7453.81 115.52 1.5% 466.58 317.85 300.71 50.42 360.49 62.33 17.3% 0.29

7.11 7.71 7.55 281.14 7454.54 123.67 1.7% 477.12 310.7 292.17 56.55 358.47 69.38 19.4% 0.33

7.08 7.73 7.56 308.91 7455.2 131.77 1.8% 487.29 303.37 283.59 62.65 356.22 76.27 21.4% 0.36

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.27 7.38 7.24 25.24 7294.81 60.31 0.8% 368.26 368.24 368.25 0 368.24 0.02 0.0% 0.01

7.21 7.42 7.25 81.75 7295.27 47.15 0.6% 381.99 362.62 359.83 6.43 368.12 8.39 2.3% 0.02

7.16 7.47 7.26 140.47 7295.49 42.66 0.6% 395.3 356.8 351.42 12.82 367.76 16.6 4.5% 0.05

7.1 7.51 7.28 199.18 7295.5 48.7 0.7% 408.19 350.79 343.01 19.18 367.15 24.64 6.7% 0.09

7.05 7.55 7.29 257.64 7295.3 61.82 0.8% 420.67 344.58 334.59 25.5 366.29 32.52 8.9% 0.12

6.99 7.6 7.31 315.82 7294.91 78.2 1.1% 432.74 338.18 326.15 31.79 365.2 40.22 11.0% 0.15

6.93 7.64 7.32 373.65 7294.34 95.91 1.3% 444.42 331.6 317.7 38.03 363.87 47.76 13.1% 0.18

6.88 7.68 7.34 431.15 7293.6 114.1 1.6% 455.71 324.82 309.22 44.25 362.31 55.13 15.2% 0.21

6.82 7.73 7.35 488.3 7292.7 132.4 1.8% 466.62 317.86 300.71 50.42 360.51 62.34 17.3% 0.24

6.77 7.77 7.37 545.08 7291.65 150.59 2.1% 477.16 310.71 292.16 56.55 358.48 69.39 19.4% 0.27

6.72 7.81 7.38 601.49 7290.47 168.59 2.3% 487.34 303.37 283.58 62.65 356.24 76.29 21.4% 0.3

ΦA ΦB ΦC V0 V1 V2 ΦA ΦB ΦC I0 I1 I2 I2/I1 PF

7.17 7.17 7.17 0.02 7174.03 0.06 0.0% 368.26 368.24 368.27 0 368.25 0.02 0.0% 0.01

7.1 7.24 7.19 88.72 7173.81 32.68 0.5% 381.99 362.62 359.85 6.42 368.13 8.39 2.3% 0.03

7.02 7.3 7.2 176.95 7173.18 64.7 0.9% 395.3 356.8 351.44 12.81 367.76 16.6 4.5% 0.05

6.94 7.37 7.22 264.74 7172.14 96.06 1.3% 408.19 350.79 343.03 19.17 367.15 24.64 6.7% 0.08

6.86 7.43 7.24 352.03 7170.74 126.76 1.8% 420.67 344.58 334.61 25.49 366.3 32.51 8.9% 0.11

6.79 7.5 7.25 438.83 7168.98 156.81 2.2% 432.75 338.18 326.17 31.78 365.21 40.21 11.0% 0.14

6.71 7.56 7.27 525.1 7166.88 186.21 2.6% 444.42 331.6 317.71 38.03 363.88 47.75 13.1% 0.16

6.63 7.62 7.29 610.86 7164.47 214.97 3.0% 455.72 324.82 309.23 44.24 362.31 55.13 15.2% 0.19

6.56 7.69 7.31 696.09 7161.77 243.1 3.4% 466.63 317.86 300.72 50.41 360.52 62.34 17.3% 0.22

6.48 7.75 7.33 780.78 7158.78 270.59 3.8% 477.17 310.71 292.18 56.55 358.49 69.39 19.4% 0.24

6.41 7.81 7.35 864.92 7155.53 297.47 4.2% 487.34 303.37 283.59 62.64 356.24 76.28 21.4% 0.27

Sequence Currents Imbalance

Node 1

Sequence Currents

Voltage

Node 0

Total kW 

Losses

Total kVar 

Losses Total kW Flow

Spot Load (MW) kW Losses kVar Losses kW Flow

Sequence Voltages (V)

V2/V1

Current

V2/V1

Current

Sequence Voltages (V)

V2/V1

Current

Voltage Sequence Voltages (V)

Imbalance

Node 2

Sequence Currents Imbalance

Voltage

Node 3

Voltage Sequence Voltages (V) Sequence Currents Imbalance

V2/V1

Current

 27
ΦA ΦB ΦC ΦA ΦB ΦC ΦA ΦB ΦC

513.41 513.32 513.39 1540.12 2641.88 2641.94 2641.95 7925.77 -2.40 -2.45 -2.40 -7.25

568.47 505.63 467.41 1541.51 2710.43 2625.05 2586.70 7922.18 -2.37 -2.47 -2.41 -7.25

625.05 497.51 423.04 1545.60 2773.98 2605.92 2531.54 7911.44 -2.35 -2.48 -2.43 -7.26

683.03 488.95 380.29 1552.27 2832.72 2584.56 2476.37 7893.65 -2.32 -2.51 -2.43 -7.26

742.25 479.94 339.17 1561.36 2886.82 2560.92 2421.12 7868.86 -2.30 -2.52 -2.44 -7.26

802.62 470.50 299.68 1572.80 2936.46 2534.99 2365.71 7837.16 -2.27 -2.55 -2.45 -7.27

863.99 460.61 261.83 1586.43 2981.83 2506.75 2310.05 7798.63 -2.24 -2.56 -2.47 -7.27

926.29 450.29 225.63 1602.21 3023.12 2476.17 2254.09 7753.38 -2.22 -2.58 -2.47 -7.27

989.39 439.53 191.09 1620.01 3060.49 2443.24 2197.74 7701.47 -2.19 -2.60 -2.49 -7.28

1053.22 428.35 158.21 1639.78 3094.14 2407.93 2140.95 7643.02 -2.17 -2.62 -2.49 -7.28

1117.68 416.77 126.99 1661.44 3124.24 2370.22 2083.65 7578.11 -2.15 -2.64 -2.50 -7.29

Total kW 

Load

kVar Native Load Total kVar 

Load

kVar Flow Total kVar 

Flow

kW Native Load

Plot Information:

0.0% 351.15 4.2% 0.8% 0.8% 0.0% 0.00 368.23 0.00

2.3% 351.64 4.3% 0.9% 0.6% 0.5% 6.43 368.11 8.38

4.5% 353.09 4.3% 0.9% 0.6% 0.9% 12.82 367.74 16.59

6.7% 355.44 4.3% 1.0% 0.7% 1.3% 19.18 367.13 24.63

8.9% 358.67 4.4% 1.1% 0.8% 1.8% 25.50 366.28 32.50

11.0% 362.75 4.4% 1.2% 1.1% 2.2% 31.78 365.19 40.21

13.1% 367.67 4.5% 1.3% 1.3% 2.6% 38.03 363.86 47.74

15.2% 373.37 4.6% 1.4% 1.6% 3.0% 44.25 362.29 55.12

17.3% 379.82 4.7% 1.5% 1.8% 3.4% 50.42 360.49 62.33

19.4% 387.03 4.8% 1.7% 2.1% 3.8% 56.55 358.47 69.38

21.4% 394.97 5.0% 1.8% 2.3% 4.2% 62.65 356.22 76.27

kW 

Losses / 

kW Flow

Node 0 

I2/I1

 Total kW 

Losses

Node 1 

V1/V2

Node 2 

V1/V2

Node 3 

V1/V2 I0 I1 I2

Note: “PF” in Table 2, Table 3, and Table 4 is the percent power flow imbalance (kVA).

 28
Total kW Loss

340.00

350.00

360.00

370.00

380.00

390.00

400.00

410.00

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0%

Node 0 I2/I1

kW Loss

Transposed Total kW Loss Nontransposed Total kW Loss Equal Spacing Total kW Loss

Figure 6. Total line losses versus load imbalance for each line configuration 

Total kW Loss / kW Flow

4.2%

4.3%

4.4%

4.5%

4.6%

4.7%

4.8%

4.9%

5.0%

5.1%

5.2%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Node 0 I2/I1

kW Loss / kW Flow

Transposed % kW Loss Nontransposed % kW Loss Equal Spacing % kW Loss

Figure 7. Percent losses versus load imbalance for each line configuration 

 29
Voltage Imbalance vs Current Imbalance

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Node 0 I2/I1

V2/V1

Node 1 V. Imbalance, NT Node 2 V. Imbalance, NT Node 3 V. Imbalance, NT

Node 1 V. Imbalance, T Node 2 V. Imbalance, T Node 3 V. Imbalance, T

Node 1 V. Imbalance, ES Node 2 V. Imbalance, ES Node 3 V. Imbalance, ES

Figure 8. Voltage imbalance versus load imbalance for each line configuration 

Sequence Currents vs Current Imbalance

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0%

Node 0 I2/I1

Amps

I0, NT I1, NT I2, NT

I0, T I1, T I2, T

I0, ES I1, ES I2, ES

Figure 9. Sequence currents versus load imbalance for each line configuration 

 30
Voltage Imbalance vs Current Imbalance

Equilateral Spacing Line

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 18.0% 20.0%

Node 0 I2/I1

V2/V1

Node 1 Voltage Imbalance Node 2 Voltage Imbalance Node 3 Voltage Imbalance

Figure 10. Voltage imbalance versus current imbalance for equilateral spacing line 

Voltage Imbalance vs Current Imbalance

Nontransposed Line

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Node 0 I2/I1

V2/V1

Node 1 Voltage Imbalance Node 2 Voltage Imbalance Node 3 Voltage Imbalance

Figure 11. Voltage imbalance versus current imbalance for nontransposed line 

 31
Sequence Currents vs Current Imbalance

Transposed Line

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Node 0 I2/I1

Amps

I0 I1 I2

Figure 12. Voltage imbalance versus current imbalance for transposed line 

Sequence Currents vs Current Imbalance

Equilateral Spaced Line

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Node 0 I2/I1

Amps

I0 I1 I2

Figure 13. Sequence currents versus current imbalance for equilateral spacing line 

 32
Sequence Currents vs Current Imbalance

Nontransposed Line

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Node 0 I2/I1

Amps

I0 I1 I2

Figure 14. Sequence currents versus current imbalance for nontransposed line 

Sequence Currents vs Current Imbalance

Transposed Line

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

Node 0 I2/I1

Amps

I0 I1 I2

Figure 15. Sequence currents versus current substance for transposed line 

 33
Table 5. The Effects of Circuit Spacing and Unbalanced Load on Percent kW Losses 

Total

ΦA ΦB ΦC ΦA ΦB ΦC

ΦI's 368.22 368.25 368.28 490.12 300.16 284.99

kW 

Losses 117.17 117.02 116.89 351.08 414.10 -72.43 63.39 405.06

% Losses 4.2% 5.10%

Native 

Load kW 7926 7584

ΦI's 356.11 375.94 371.99 274.51 494.92 305.56

kW 

Losses 194.35 72.47 84.29 351.11 131.84 390.48 -117.36 404.96

% Losses 4.2% 5.10%

Native 

Load kW 7920 7579

ΦI's 368.23 368.24 368.24 487.29 303.37 283.59

kW 

Losses 117.05 117.04 117.06 351.15 417.26 -96.12 73.83 394.97

% Losses 4.2% 5.0%

Native 

Load kW 7925 7578

Notes (1) Balanced loads result is less losses

(2) kW losses are greater for the non-transposed line than for transposed

 because % I2/I1 is greater for the non-transposed.

Balanced Load

I2/I1 = 0%

Unbalanced Load

I2/I1 = 21.4%

Flat Transposed

I2/I1 = 21.5%

Balanced Load

I2/I1 = 2.5%

Unbalanced Load

I2/I1 = 22.9%

Flat Non-Transposed

Balanced Load Total Unbalanced Load

I2/I1 = 0%

Equilateral Spacing

 (3) Unbalanced load is 4500 kW on Phase A, 2250 kW on Phase B, and 2250 kW on Phase 

C. Balanced load is 3000 kW on each of the three-phases. 

Table 5 presents a summary of the effects of circuit spacing and unbalanced load on percent 

losses. For the balanced load of 3000 kW on each phase (total load = 9,000 kW), the percent 

real losses, 4.2%, are the same for the equilateral spacing line, the flat nontransposed line, and 

the flat transposed line. For the unbalanced loads of 4500 kW on Phase A, 2250 kW on Phase 

B, and 2250 kW on Phase C, the percent losses are 5.10% for the equilateral and flat 

nontransposed configurations and 5.0% for the flat transposed configuration. 

3.10 Secondary and Service Impedances and Voltage Drops 

Secondary and service impedances and voltage drops can be represented as follows. 

 34
11.66 kVA 11.66 kVA 11.66 kVA

11.66 kVA 11.66 kVA 11.66 kVA

#1 #2 #3

#4 #5 #6

#0 #0 #0

100'

100'

100'

100'

100'

100'

C C C

C C C

B 150' #0

150'

#0 150' B

150'

48.63A

48.63A

97.25A I

sec = 97.25A

Load =

70 kVA

50 kVA

transf.

#0

150'

#0 #0

Figure 16. Distribution transformer servicing secondaries and services 

ransformer Size = 50 kVA 

m = maximum coincident total demand on the transformer 

m = 70 kVA of load and losses 

os θ = 0.90 

sec. = 

T

D

D

C

. A 97.25

V 240

kVA 46.68

2

1

V 240

kVA) (11.66 2kVA 70

2

1 =⎥

⎦

⎤ ⎢

⎣

⎡ =⎥

⎦

⎤ ⎢

⎣

⎡ − I

Iservice = 1⁄2 (97.25 A) = 48.63 A . 

 35
R

R

I

R = 43.76A

I

R = 43.76A

Load =

48.63A 25.84o

Figure 17. Distribution service drop 

The R and X values for the #0 triplex secondary and service are 

 R = 0.211 Ω/1000 ft 

 X = 0.031 Ω/1000 ft. 

The voltage drops in secondary B are 

∆ V = 2 (I R Cos θ + I X Sin θ) Equation 3.71 

 = 2 [97.25 (0.211 Ω/1000 ft x 150 ft) 0.90 + 97.25 (0.031 Ω/1000 ft x 150 ft) .436] 

 = 2 (2.770 + 0.197). 

∆ V = 5.93 V @ 240 V or 2.97 V @ 120 V base , Equation 3.72 

and the drops in the services C are 

∆ V = 2 [48.63 (0.211 Ω/1000 ft x 100 ft) 0.90 + 48.63 (0.031 Ω/1000 ft x 100 ft) 0.436] 

 = 2 (0.920 + 0.067) 

∆ V = 1.97 V @ 240 V or 0.99 V @ 120 V base . Equation 3.73 

The voltage drop for customer #2 and customer #5 is only 0.99 V, but for customers #1, #3, 

#4, and #6, it is 3.96 V (2.97 + 0.99). 

 36
Table 6. Summary of Transformer, Secondary, and Service Voltage Drops

25 kVA 

N = 3 

No Secondary 

50 kVA 

N = 2 N = 4 

No Secondary Secondary 

Transformer 3.68 2.75 2.75 

Secondary -- -- 2.97 

Service 0.99 0.99 0.99

Total 4.67 3.74 6.71 

3.11 Secondary and Service Real Losses 

Service drop real losses are determined as follows. 

The resistance of #0 aluminum service drops for 100 ft is 

R service = 2 (0.211 Ω/1000 ft x 100 ft) = 0.0422 Ω . Equation 3.74 

The resistance is doubled because of the forward and return path of the current. 

I

2

service R service = (48.63)2

 (0.0422) = 99.80 W per service drop . Equation 3.75 

Service Drop Total Real Losses = (6) (99.80 W) = 598.8 W, Equation 3.76 

Secondary real losses are determined as follows. 

R secondary = 2 (0.211 Ω/1000 ft x 150 ft) = 0.0633 Ω , Equation 3.77 

I

2

sec. R secondary = (97.25)2

 (0.0633) = 598.7 W per secondary. Equation 3.78 

Secondary Total Real Losses = 2 (598.7 W) = 1197.3 W. Equation 3.79 

Total Service and Secondary Real Losses = 598.8 + 1197.3, Equation 3.80 

 = 1796.1 W. 

Percent Real Losses = 2.9% 100x 

(0.90) kVA) (50 (1.4)

kW 1.796 = . Equation 3.81 

Note that we must add kW losses for each circuit element and divide by total kW flow to 

obtain the total circuit percent losses; percent losses cannot be added for each element. 

 37
The peak day (July 17, 2006) real power losses for the Milford substation transformer, the 

Milford Distribution Circuit 8103 primary, overhead and padmount transformers, and 

secondary and services are summarized in Table 7. 

The percent real losses shown in the first column are for the 4.8 kV Hickory Distribution 

Circuit (Davis, Krupa, and Diedzic 1983). (See also Note 1 in Table 7.) These are actual 

measured losses and total 5.2%, excluding the substation transformer. The next two columns 

represent the percent real losses for Milford DC 8103. The second column depicts calculated 

losses based on the average peak loading of the transformers and services, and a lumped load 

representing the peak circuit load placed at the midpoint to calculate the primary losses. The 

third column shows the calculated losses determined from the simulation of the circuit for the 

peak load hour on July 17, 2006. 

Table 7. Comparison of Peak Day Real Losses 

Percent Losses 

Transmission 

Subtransmission 

-- 

-- 

3.5 

4.5 

-- 

-- 

Hickory DC Milford DC 

8103 

Milford DC 

8103 

Substation Transformer 0.70 0.729 0.702 

Primary 

O.H. & UG Transformers 

Secondary & Services 

3.1(1)

1.3(1)

0.8(1)

3.82(2)

1.22(2)

0.95(4)

3.59(3)

1.50(3)

 0.89(4)

 0.31(5)

Total Circuit Losses 5.2 5.99 5.98(4) 5.4(5)

Total System Losses 13.9 14.72 14.68 14.10 

Notes: 

(1) Data [3] are taken from “The Economics of Load Management on the Design and Operation 

of Distribution System,” Murray W. Davis, Theodore J. Krupa, and M. J. Diedzic (March 

1983). 

(2) Approximate, based on average loading of transformers and services and lump load placed 

at midpoint for the primary losses. 

(3) Distribution Engineering Workstation. 

(4) #4 MXAT Services used. 

(5) #0 MXAT Services used. 

3.12 Shunt Capacitor Models 

When capacitors are connected in parallel, kVAR = kVAR1 + kVAR2 + kVAR3 + ..., 

and when connected in series, 

. 

kVAR

1

kVAR

1

kVAR

1

1 kVAR

1 2 3

+++ •••

=

The capacitive reactance (in ohms) is 

 38
 XC = 2 π 10 x C 

1

6- f , Equation 3.82 

where 1 μf is 2653 Ω at 60 Hz, or 

XC = 

 C

2653 , where C is 1 μf.

From Equation 3.82, C is then 

C = 2 π X 

10

C

6

f . Equation 3.83 

Now, XC can be defined in terms of voltage and VAR: 

kVAR = kV IX, where IX = 

XC

V . 

Since kV = 

1000

V , then 

kVAR = 

 X

1000 kV

C

2

, and 

kVAR

1000 kV X

2

C = . Equation 3.84 

The capacitance C in microfarads can be determined in terms of voltage and VAR by 

substituting Equation 3.82 into Equation 3.84 and solving for C: 

 2 kV πf

kVAR 1000 C 2 = . Equation 3.85 

If the capacitance C and voltage are known, then kVAR can be found from Equation 3.85: 

1000

 2 kV π kVAR

2 C f = . Equation 3.86 

Equations 3.84, 3.85, and 3.86 are the most commonly used. Two examples follow. 

The ohms per phase for a 300 kVAR, three-phase 4.8 kV delta-connected capacitor bank is 

 39
 kVAR/Φ = = 3

kVAR 300 100 kVAR . 

The line-to-line voltage applied across each capacitor is 4.8 kV. Applying Equation 3.84, 

XC = 

100

(1000) 4.82

= 230.4 Ω . 

The capacitance from Equation 3.85 is 

 11.51μf (377) (4.8)

(100) (1000) C 2 = = . 

The ohms per phase for a 600 kVAR, three-phase 13.2 kV wye-connected capacitor bank is 

 kVAR/Φ = 

3

600 = 200 kVAR. 

The voltage across each capacitor is ; 

3

kV 13.2 therefore, using Equation 3.84 results in 

, 290.3 200

1000 (7.620) X

2

C = Ω=

and the capacitance is 

 14.9 μf (377) (7.620)

(200) (1000) C 2 = = . 

Capacitor models can be developed using Equation 3.84, which is repeated below. 

kVAR

1000 kV X

2

C = . 

The susceptance BC is the reciprocal of XC; therefore, BC from Equation 3.84 becomes 

1000 kV

kVAR

X

1 B 2

C

C == , Equation 3.87 

where kVAR is the kVAR per phase, and kV is the applied voltage across the capacitor. For a 

wye-connected capacitor bank, 

 40
1000 kV

kVAR

X

1 B 2

C LN

C == , Equation 3.88 

and for a delta-connected capacitor bank, 

1000 kV

kVAR

X

1 B 2

C LL

C == . Equation 3.89 

The line currents for the wye-connected bank are written from Equation 3.88: 

A A = CAN X jIV 

==

A

AN

AC X j

V I VAN j BA Equation 3.90 

B CI = VBN j BB Equation 3.91 

C CI = VCN j BC. Equation 3.92 

For the delta-connected bank, the phase currents are 

AB CI = VAB j BAB Equation 3.93 

BC CI = VBC j BBC Equation 3.94 

CA CI = VCA j BCA , and Equation 3.95 

the line currents are 

CA C AB C A C −= III

 Equation 3.96 

AB C BC C B C −= III

 Equation 3.97 

BC C CA C C C −= III

 . Equation 3.98 

 41
In matrix form, Equations 3.96, 3.97, and 3.98 can be written as 

 1 0 -1 

A CIAB CI 

B CI = -1 1 0 . Equation 3.99 BC CI

C CI 0 -1 1 CA CI 

3.13 Step Voltage Regulator Models 

Both Type A and Type B step voltage regulators are modeled. The relationship between the 

source voltage VS and load voltage VL for the single-phase step voltage regulator is: 

VS = arVL , Equation 3.100 

where ar is defined as 

Tap Steps of #

 Range% Total 1ar = m × Equation 3.101 

with the following sign convention: 

Type A Type B

Raise + - 

Lower - + 

The relationship between source current and load current for the single-phase step voltage 

regulator is 

L

r

S I

a

1

=I . Equation 5.102 

3.14 Synchronous Generator Model 

The Distribution Engineering Workstation uses two models to represent the synchronous 

machine, one for power flow and another for transient analysis. Since this study does not 

involve transient analysis, only the power flow model will be considered. The following is a 

graphical representation of the power flow synchronous machine model. 

P,Q Source Z 

Controller 

Figure 18. Steady-state synchronous machine model 

 42
As shown in Figure 18, the synchronous machine model used in power flow consists of a real 

and reactive power source along with the impedance Z (in this case, the synchronous 

impedance). The controller takes measurements from machine terminals—specifically, 

voltage magnitude and power factor—and allows the machine to operate in two different 

modes: constant P with power factor control, or constant P and constant Q. 

The model takes into account the minimum and maximum generation limits of the machine, 

and the controller adjusts the P,Q source based on the control mode. For example, if the 

machine is in constant P with power factor control mode, the controller will maintain real 

rated power while varying the reactive power within the machine limits in an attempt to hold 

constant the power factor at the point of measurement. 

3.15 Self-Excited Induction Generator Model 

A self-excited induction generator (SEIG) can be represented as a two-phase primitive 

machine with stator windings fixed and rotor windings rotating. The currents ids and iqs are the 

stator currents, and idr, and iqr are the rotor currents in the direct and quadrature axes, 

respectively. Figure 19 shows the mechanical angular speed of the rotor asω= dθ/dt.

A

q

C

d

B

C

ω

Rotor

iqr iqs C

idr

ids

Figure 19. Two-phase primitive machine to be interconnected to an RLC load 

A SEIG with capacitor added is represented in Equation 3.103. The voltage drop across the 

capacitor for the direct axis is vCd, and for the quadrature axis, it is vCq. The term 1/pC 

represents this voltage drop, since 

 43
,idt C

1

v ∫ = 

 p = d/dt , and 

dt 

p

1 t

∫ = . 

= 

vds R1 + L1 p + pC

1

0 Mp 0 ids

vqs 0 R1 + L1 p + pC

1

0 Mp iqs

vdr Mp ωM R2 + L2 p L ω 2 idr

vqr -ω M Mp -ω L2 R2 + L2 p iqr

. 

 

Equation 3.103 

Equation 3.103 represents the no-load condition, and the turns ratio is assumed to be unity; 

therefore, if needed, the ratio must be included when referring the rotor parameters to the stator. 

The mutual inductance M varies with the relative position between the stator and rotor. R1 and 

R2 are the stator and rotor resistances, and L1 and L2 are their inductances. The additional 

voltage drop due to the mutual flux is 

φm = Mi, and 

v = 

dt

idM

dt

di M

dt

 d φm += . 

 = Mpi + ipM Equation 3.104 

There is a nonlinear relationship between the magnetizing reactance and the magnetizing current; 

therefore, the mutual inductance M varies continuously. The term Mpi represents the current 

variation due to the stator and the term ipM is the variation due to the rotation of the rotor. 

3.15.1 Resistive Load 

When a resistive load R is added in parallel with the self-excitation capacitor C, then the 

voltage across this R load is the same as the voltage across the capacitor. When referring to 

the direct axis stator of Figure 19, we see that the addition of the load resistor changes the 

equivalent circuit to that of Figure 20. 

 44
Figure 20. Stator direct axis with an R load added 

Since v load d = vCd = R i load d , and Equation 3.105 

iCd = C = dt

 vd Cd C pvCd , then 

iCd = CpRi load d . 

Because 

ids = iCd + i load d , therefore 

ids = RCpi load d + i load d , and 

 = i load d (RCpi + 1) . Equation 3.106 

From Equation 3.105, 

i load d =

1RCp

ids

+

 . Equation 3.107 

Substituting Equation 3.107 into Equation 3.105 results in 

v load d = 

1RCp

Rids

+

 . Equation 3.108 

The quadrature load voltage is obtained in a similar manner as 

v load q = 

1RCp

Riqs

+

 . Equation 3.109 

Now, Equation 3.103 can be rewritten with the addition of the resistive load R from Equations 

3.108 and 3.109 as

 45
 

vds R1 + L1 p + 1 RCp

R

+

0 Mp 0 ids

vqs 0 R1 + L1 p + 1 Cp R

R

+

0 Mp iqs

 vdr Mp ωM R2 + L2 p ωL2 idr

 vqr -ωM Mp -ωL2 R2 + L2 p iqr

= 

 Equation 3.110 

 

3.15.2 Resistance-Inductance-Capacitance Load 

When a resistance and an inductance load RL and LL is added in parallel with the selfexcitation capacitor Cs or a combination of this excitation capacitor and other capacitors CL

on the circuit in parallel, then C in the following equations considers this combination of 

capacitors (see Figure 21). 

SEIG CS CL

RL

LL

Figure 21. RLC load RL, LL, and CL connected to the self-excited capacitor CS

Since v load d = Ri load d from Equation 3.105, then v load d with load inductance added is 

v load d = Ri load d + L pi load d , Equation 3.111 

 

and 

iCd = Cpv load d = (RCp + LCp2

) i load d . Equation 3.112 

Now, from Equation 3.112, 

Cd d = ii s + i load d = (RCp + LCp2

) i load d + i load d , Equation 3.113 

and 

 46
i load d = 

1 LCpRCp

i 

2

ds

++

. Equation 3.114 

From Equation 3.111, 

v load d = Ri load d + Lpi load d , and 

v load d = (R + Lp) i load d . Equation 3.115 

Substituting i load d from Equation 3.114 into Equation 3.115 results in 

 v load d = . i 

1 LCpRCp

 Lp R

2 ds ++

+ Equation 3.116 

The quadrature load voltage would become 

 v load q = . i 

1 LCpRCp

 Lp R

2 qs ++

+ Equation 3.117 

Equation 3.103 can be rewritten with the addition of resistance load R, inductive load L, and 

capacitive load C using Equation 3.117. 

Equation 3.118 is the impedance matrix for a single self-excited induction generator serving 

an RLC load. Note that the variation in the magnetizing reactance due to the magnetizing 

current must be included and corrected, as shown in the development of Equation 3.104. 

= 

vds

 R1 + L1 p + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++

+

1 2p LC RCp

 Lp R 0 Mp 

0 

ids

vqs

0 R1 + L1 p + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

++

+

1 2p LC RCp

 Lp R 0 

Mp 

iqs

vdr Mp ω M R2 + L2

p 

ωL2 idr

vqr -ω M Mp - L ω 2 R2 + L2 p iqr

 Equation 3.118 

3.16 Inverter-Based Generator Model 

The 400 kW, 500 kVA inverter-based generator consists of a high-speed a-c generator driven 

by a 440 kW twin spool gas turbine engine. Figure 22 is a simplified one-line diagram of the 

400 kW, 60 Hz generator and inverter. 

 47
Gas

Turbine

Engine Controller

High Speed

Generator Generator

Bus

Step up Transformer

( optional )

Load

Bus Main Circuit

Breaker Rectifier CT Inverter

=

Figure 22. One line diagram of a 400 kW inverter-based generator and prime mover 

The unit has seamless transfer capability where it can be operating in a current source mode 

parallel to the external power supply with only 10% load and switch without outage to deliver 

power in the voltage source mode to an islanded load. This generating unit has a continuous 

overload capability of 110% of rated load at any power factor from -0.8 to 0.8, and it can 

follow load steps of 25% up or down while maintaining frequency to less than +/-0.1 Hz. The 

voltage total harmonic distortion is less than 2%, and it can handle 100% load imbalance. 

During short circuits, it can deliver 200% of rated current for about 8 seconds. 

The inverter model is represented by the voltage pullback curves given in Figure 23. The unit 

can deliver 1.10 per unit current (602 A root-mean-square [rms]) at 1.0 per unit rated voltage 

(277 V rms) and 2.0 per unit current at 0.56 per unit voltage. The inverse time current 

characteristic is given in Figure 24. 

 48
Figure 5.22 Voltage Pullback Curves

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

Current (P.U.), 602 Arms = 1.0 P.U.

Voltage (P.U.), 277 Vrms = 1.0 P.U.

y = -0.0477x3 + 0.5556x2 - 1.8574x + 2.4336

R2

 = 0.9998

Current Limit

kVA Limit = 550 kVA

P.F. < 0.8

Figure 23. Voltage pullback curves 

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Current (P.U.)

log(Time (secs))

Data

Curve Fit

log(y)=175.265*x6

-1715.528*x5+6957.537*x4-

14963.291*x3

+17999.078*x2-11485.441*x+3041.135

1000

10000

100

10

Time (secs)

Figure 24. Inverse time-current characteristic 

 49
4.0 Results for the Design of Field Voltage Regulation and 

Metering Equipment Used to Verify the Models 

This section defines the equipment installed on the Milford DC 8103 for controlling voltage 

and the metering equipment used to validate the models developed in Section 3.0. 

4.1 Voltage Regulation Equipment 

Figure 25 shows the locations of the voltage regulation equipment on the Milford Circuit DC 

8103. Table 8 describes this equipment. 

Table 8. Voltage Regulation Equipment 

Location Description 

A. 10 MVA delta-wye transformer #1 

Z = 7.02%, 41.57 kV/13.8 kV, 0, + 2.5, + 5.0 high side setting = 0, low side setting = 0, 

LTC + 16 steps + 10%, aN = 3 (see Figure 26) 

B. 3 – 167 kVA single-phase step voltage regulators, 32 steps 

D. 1000 kW synchronous generator, 1050 kW power prime mover, rated output = 1312 kVA 

@ P.F. = 0.8, 1800 rpm, 480 V, 60 Hz, voltage regulation + 5%. 

F. (1) 900 kVAR 3 Φ capacitor Y-connected, neutral grounded 

G. (2) 900 kVAR 3 Φ capacitor Y-connected, neutral grounded 

H. (3) 1200 kVAR 3 Φ capacitor Y-connected, neutral grounded 

 

 50
MT3Φ

MTSΦ

Figure 7.1 Milford Circuit DC 8103

M

Note (1):

PCC

M

Metering

Permanent

Installations

Note (2):

MT

Metering

Temporary

Installations

MTSΦ MTSΦ

VA, VB, VC

I

A, IB, IC

P, Q

ROC

(SOC)

M 3Φ

J. K.

I.

L.

80K 125K

100K

65K

40K

43199

43244

43052 C.

400 kW Inverter

Generator

(Study)

3 - S

Regulators

(Study)

Φ

E.

P6

40K

80K 40K 65K

111V

400 kW Induction

Generator (Study)

Far End

DR

(Study)

900 kVAr

43090

PTS

43095 100K G.

2

25K

65K

65K

40K

40K

40K

25K

65K

Midpoint DR

(actual site)

1000 kW

Sync. Generator D.

3 - S

Regulators

1

R

M3Φ

M3Φ

R-1 125K 40K 65K R-3

140K

900 kVAr

1

ZX

F..

MT3Φ 65K

L4

3Φ 3Φ 3Φ

3Φ

2Φ

3 Φ

43283

65K

2Φ Φ

A.

P5 Y, Z

80K

2

Near End

DR (Study)

recloser

L

2Φ

65K

2Φ 40K

Φ 65K

ZX

2Φ

80K 2Φ

80K

80K

80K

65K

P1

M3Φ

S

DTECH

(SOC)

1000 kW

Synchronous Machine

M3Φ

P

24 19

18

27

6

28

26

Note (3): Nodes where low voltage, high voltage, highest unbalance voltage, highest unbalanced load and nodes where

actual values exceed pre-set values (criteria violations) may be different than the nodes indicated above.

Note (4): The LTC voltage regulator is gang operated using the C phase as reference.

1

21 22

20 16

15

14

0

7

4

17

5

MT3Φ

H.

3

1200 kVAr 13

9

8

2

25

11

23

10

3

3-S reclosers

move

Regulation

Path

17

14

12

B.

8

3

11

Note (5): The "P" reclosure data is sent to midpoint DR consisting of IA, IB, IC, VCN, VBN, VCN, kWA, kWB, kWC,

 kVArA, kVArB, kVArC

MTSΦ

MT3Φ

Phase C

Phase B

Phase A

Figure 25. Milford Circuit DC 8103 

 51
Tie Line

8623

Proud

Trunk Line

3107

Page

CAP 2

12 MVAr

N.O.

Tie Line

2819

Cody - Proud

41.57 kV

Tie Line

6117

Placid Tie Line

6147

Placid

Section

Breaker

Tie Line

5831

"A" Cable "B" Cable

DC 8103 DC 8066

13.8 kV 13.2 kV

PTS 43283

#1 #2 0% Fixed

10 MVA

+10%

0% Fixed

41.57/13.2 kV

N.O.

5% Fixed

10 MVA

+10%

0% Fixed 41.57/13.8 kV

Figure 26. Milford Substation one line diagram 

4.2 Major System Protection Equipment 

The major system protection equipment shown in Table 9 consists of three-phase and singlephase reclosers and single-phase sectionalizers. 

Table 9. Major System Protection Equipment 

Location Description of Major System Protection Equipment 

1. R 3 – 3 Φ reclosers – 280A vacuum V4L, 2A, 2D 

2. P 1 – 3 Φ recloser – 680A vacuum VWE, R, C 

3. P5 3 – S Φ reclosers – 280A vacuum V4L, 2A, 2D 

4. P1 3 – S Φ reclosers – 280A vacuum V4L, 2A, 2D 

5. S 3 – S Φ reclosers – 140A vacuum V4L, 1A, 3D 

6. L 3 – S Φ reclosers – 140A vacuum V4L, 1A, 3D 

7. L4 3 – S Φ sectionalizers – 140A hydraulic 

8. P6 3 – S Φ reclosers – 140A V4L, 2A, 2D 

 52
Table 10. Measurement Locations and Data Collection 

Permanent Metering Locations 

Temporary Metering Locations 

A. Substation M3 Φ VAN, VBN, VCN IA, IB, IC P, Q, P.F. 

B. Regulators (1) M3 Φ VAN, VBN, VCN IA, IB, IC P, Q, P.F. 

D. 1000 kW Sync. DR M3 Φ VAN, VBN, VCN IA, IB, IC P, Q, P.F. 

Temporary Metering Locations 

F. 900 kVAR capacitor MT 3 Φ VAN, VBN, VCN IA, IB, IC

G. 900 kVAR capacitor MT 3 Φ VAN, VBN, VCN IA, IB, IC

H. 1200 kVAR capacitor MT 3 Φ VAN, VBN, VCN IA, IB, IC

I. 3Φ circuit tag end MT 3 Φ IA, IB, IC

K.L. Single-phase tag ends MT S Φ VAN VBN

4.3 Metering Equipment and Accuracy of Measurements 

4.3.1 Substation Metering 

The substation metering consists of the ION 7600 Power Measurements Meter, which 

measures the three-phase voltages, the three line currents, and the phase power factors and 

calculates kW per phase and total kW, kVAR per phase and total kVAR, kVA per phase and 

total kVA, and the unbalanced voltage and unbalanced current. The integration period is 15 

minutes. The accuracy of the metered data is 0.1% for voltage, 0.1% for current, +0.01 Hz for 

frequency, and 0.5% for power factor. The kW, kVAR, and kVA accuracy is Class 0.2. 

4.3.2 Synchronous Generator Metering 

The generator metering is a 3720 ACM meter, which measures the three-phase voltages, the 

three line currents, three-phase kW, three-phase kVAR, three-phase kVA, power factor, 

frequency, and circuit load, including generation. The data are 30-second samples integrated 

over 5 minutes. The data are 5-minute samples averaged over 15 minutes. The accuracy of the 

measured data is given in Table 11. 

 53
Table 11. Accuracy of 3720 ACM 

Parameter Accuracy 

Current 0.2% 

kW 0.4% 

kVAR 0.4% 

kVA 0.4% 

Voltage 0.2% 

Power Factor 1.0% 

Frequency 0.05 Hz 

4.3.3 Capacitor Location Metering 

The metering at the capacitor locations (nodes F, G, and H) is the Line Tracker LT40 from 

Grid Sense. These metering devices measure the three-phase voltages and three line currents. 

Voltage is sampled at 600 samples per second; current is sampled at 1200 samples per second. 

The accuracy of the current measurement is +5%. The integration period is 15 minutes. 

4.3.4 Voltage Regulator Location Metering 

The CL-6 control of the voltage regulators at location B measures the load and source phase 

voltages, line currents, phase kW, phase kVAR, phase kVA, and phase power factor. The data 

are integrated over 15 minutes, and the accuracy is Class 1 metering, or 1.0% for all power 

quantities (i.e., voltage, current, kW, kVAR, and kVA). 

4.3.5 Single-Phase Voltage Customer Metering 

The Rustrak was used to measure voltage and current. The accuracy is 0.25% for voltage 

and 0.5% for current. Each stated accuracy is in percent of reading. The integration period 

is 15 minutes. 

 54
5.0 Results for Distributed Generation Control Strategies 

Used in Field Verification 

This section defines the generation equipment voltage regulation capability and the control 

strategies used to regulate the voltage on the Milford Circuit DC 8103. 

5.1 1000 kW Synchronous Generator 

The voltage regulation control capabilities for the 1000 kW synchronous generator located at 

the midpoint on the circuit node D are given in Figure 27.

5.2 400 kW High-Speed Generator and Inverter 

The voltage regulation control capability for the 400 kW high-speed generator and inverter is 

shown in Figure 28. 

5.3 400 kW Self-Excited Induction Generator 

The voltage regulation control capability for the 400 kW self-excited induction generator is 

displayed in Figure 29. 

5.4 Voltage Regulation Simulations and Field Verification Strategy 17 

The voltage regulation simulations that were conducted and Strategy 17, which was used for 

field verification, are outlined in Table 12. 

 55
Xd

1

1

2 3

4

5

kWs

6

Synchronous

Condensor

7

8

If Rated

If Lead

I

 Rated

a

I.

Fixed P.F.

Fixed P.F.

IV.

246 kW

1050

kW

max.

984 kW

1230 kVA

Operating Points

(a) Points and is rated kW and rated kVAR 

(b) Point rated kVA and rated lagging P.F.

2

 

(c) Point ading P.F. le

(d) Point maximum kW 

(e) Point minimum kW and maximum kVAR (synchronous condensor) 

(f) Points and fixed power factor lagging and leading 

8

1

3

4

5

6

7

Figure 27. DG control strategies for 1000 kW synchronous machine 

 56
III

I

IV

+Q

Rated P.F. = 0.8 lag (producing

VAR = +Q = +240 kVAR

Operating

Quadrants

I. and IV.

Pmax = 400 kW

Rated P.F. = 0.8 lead (absorbing

VAR = -Q = -240 kVAR)

Rated kW = 320 kW

Rated kVA = 400 kVA

II

-Q

Note (4): DR Control Strategies

(a) Peak shave, P fixed

(b) P.F. = unity, P variable

(c) + P.F. constant, P variable

(d) + Q R volt. reg.) P = 0 (VA

(e) P + jQ (optimizing) - see synchronous machine

(f) Frequency Dithering (anti-islanding)

Note (5): (f) not included in model

Figure 28. DG control strategies for 400 kW high-speed 

generator and inverter in current mode 

 57
Rated kVA = 470.59 kVA

switched

capacitor

"off"

normal

switched capacitor 2

Rated P.F. = .85

"on" (producing VAR)

= +Q = +247.91 kVAR

switched

capacitor 1

"on"

P.F. = .85

P

max

 = 400 kW

(absorbing VAR = -Q

= -247.91 kVAR)

+Q

-Q

(a) Peak shave, P fixed

(b) P.F. = unity, P fixed

(c)+

(d)

 P.F. constant at P fixed

+ Q fixed, P fixed same as (c)

Note (4) Induction machine with inverter is the same as Inverter where the speed is

varied to create variable dc output after the rectifier and the ac current output

from the inverter are phase shifted (switching) to produce a variable P.F. with

respect to the system voltage.

Figure 29. DG control strategies for 400 kW self-excited induction generator 

 58
Table 12. Matrix of Voltage Regulation Simulations 

and Control Strategy 17 for Field Verification 

Line 

Regulator Capacitors

DR (Synchronous Machine) 

Locations (1)

(a)Peak shave, P fixed

(b) P.F.= Unity, P variable

(c) + P.F. Constant, Variable P & 

Q

(d) + Q (Vars Volt reg) P Min

(e) P+ jQ (Optimizing)

(f) Fixed P.F. Anti-island

#1 #2 #1 #2 #3 Near End

Midpoint 

(Actual 

Site) Far End

1 I.A. (a) (d) (f) x x

2 I.B. (d) (f) x x x

3 I.C. (d) (f) x x x x

4 I.C '. (d) (f) x x x x x

5 I.D (d) (f) x x x x

6 I.D ' (d) (f) x x x x x

7 I.D." (d) (f) x x x x x x

8 I.E. (d) (f) x x x x x x x x

9 II.A. x xx (b) (d) (e)

10 II.B. x xx (b) (d) (e)

11 II.C. x xx (b) (d) (e)

12 II.D. Repeat IIA., IIB., IIC. turn on LTC

13 II.E. Repeat IIA., IIB., IIC. turn on LTC, and Regulator (1)

14 II.F. Repeat IIA., IIB., IIC. turn on LTC, and Regulator (1) and (2)

15 II.G. Repeat IIA., IIB., IIC. turn on LTC, and Regulator (1) and (2) and Cap. (1)

16 II.H. Repeat IIA., IIB., IIC. turn on LTC, and Regulator (1) and (2) and Cap. (1), (2)

17 II.I. Repeat IIA., IIB., IIC. turn on LTC, and Regulator (1) and (2) and Cap. (1), (2), (3)

18

19

20

21

22

23

24

25

Note (1): DR Control Strategies

(e.) DR is operated with variable P and Q and is used to optimize to a specific set of criteria such as minimize 

real losses, minimize reactive losses, regulate voltage and maximize released capacity.

(c.) DR is operated at a fixed power factor either lead or lag (for example + 0.8 P.F.) with variable P and Q to 

not violate voltage criteria.

(a.) The DR is operated at fixed real power output and normally Is scheduled to run during peak periods when 

the cost of DR generation is lower than host utility cost. The Q is zero at the PCC. 984 kW rated

(d.) The DR is operated as a synchronous condenser with minimum watts and variable VArs to regulate 

voltage. 246 kW (25% of rated)

(b.) The DR is operated at unity P.F. with variable power output. 492 kW (50%) to 1050 kW max. LTC + 16 steps & neutral

Test NumbersReference. SectionLight Load Day 24 Hourly quantities

Primary 

Voltage 

Spread

(a)87%

(b) 92%

(c)93%

(d) 95%

(e) 98%

(f) 105%

Peak Load Day 24 Hourly quantities

 59
Line 

Regulator Capacitors

DR (Induction Generator) 

Locations 

(a)Peak shave, P fixed

(b) P.F.= Unity, P fixed

(c) + P.F. Constant, P fixed

(d) + Q fixed, P fixed

same as (c)

#1 #2 #1 #2 #3 Near End

Midpoint 

(Actual 

Site) Far End

1 (a) (d) (f)

2 (d) (f)

3 (d) (f)

4 (d) (f)

5 (d) (f)

6 (d) (f)

7 (d) (f)

8 (d) (f)

9 II.A. x x x (b) (d)

10 II.B. x x x (b) (d)

11 II.C. x x x (b) (d)

12 II.D. Repeat II.A., II.B., II.C. turn on LTC

13 II.E. Repeat II.A., II.B., II.C. turn on LTC and Regualator (1)

14 II.F. Repeat II.A., II.B., II.C. turn on LTC and Regulators (1) & (2)

15 II.G. Repeat II.A., II.B., II.C. turn on LTC, Reg. (1) & (2) and Cap (1)

16 II.H. Repeat II.A., II.B., II.C. turn on LTC, Reg. (1) & (2) and Cap (1) and (2)

17 II.I. Repeat II.A., II.B., II.C. turn on LTC, Reg (1) & (2) and Cap (1), (2) & (3)

18 III. Repeat synchronous and inverter generation

19 IV. Repeat synchronous, inverter and induction generation

20

21

22

23

24

25

26

27

28

29

Light Load Day 24 Hourly quantitiesLTC + 16 steps & neutral

Test NumbersReference. Section

Primary 

Voltage 

Spread

(a)87%

(b) 92%

(c)93%

(d) 95%

(e) 98%

(f) 105%

Peak Load Day 24 Hourly quantities

 60
Line 

Regulator Capacitors

DR (Induction Generator) 

Locations 

(a)Peak shave, P fixed

(b) P.F.= Unity, P fixed

(c) + P.F. Constant, P fixed

(d) + Q fixed, P fixed

same as (c)

#1 #2 #1 #2 #3 Near End

Midpoint 

(Actual 

Site) Far End

1 (a) (d) (f)

2 (d) (f)

3 (d) (f)

4 (d) (f)

5 (d) (f)

6 (d) (f)

7 (d) (f)

8 (d) (f)

9 II.A. x x x (b) (d)

10 II.B. x x x (b) (d)

11 II.C. x x x (b) (d)

12 II.D. Repeat II.A., II.B., II.C. turn on LTC

13 II.E. Repeat II.A., II.B., II.C. turn on LTC and Regualator (1)

14 II.F. Repeat II.A., II.B., II.C. turn on LTC and Regulators (1) & (2)

15 II.G. Repeat II.A., II.B., II.C. turn on LTC, Reg. (1) & (2) and Cap (1)

16 II.H. Repeat II.A., II.B., II.C. turn on LTC, Reg. (1) & (2) and Cap (1) and (2)

17 II.I. Repeat II.A., II.B., II.C. turn on LTC, Reg (1) & (2) and Cap (1), (2) & (3)

18 III. Repeat synchronous and inverter generation

19 IV. Repeat synchronous, inverter and induction generation

20

21

22

23

24

25

26

27

28

29

Light Load Day 24 Hourly quantitiesLTC + 16 steps & neutral

Test NumbersReference. Section

Primary 

Voltage 

Spread

(a)87%

(b) 92%

(c)93%

(d) 95%

(e) 98%

(f) 105%

Peak Load Day 24 Hourly quantities

 61
6.0 Results for Field Verification of Models 

This section describes the verification of the models using field test data. The voltage 

regulation models were developed in Section 3.0 for the LTC transformer at the substation, 

the bidirectional step regulators, the capacitors, all the distribution circuit transformer 

connections, and the line sections. In addition, models for synchronous, induction, and 

inverter-type generators were given. 

Field verifications were conducted using the metering equipment defined in Section 4.0, and 

DG generation control for the 1000 kW synchronous generator was described in Section 5.0. 

The control strategy used was Strategy 17, in which all three capacitors (1, 2, and 3), the 

regulator (1), and the substation LTC would be turned on. 

Field verification studies were conducted for the circuit peak day July 17, 2006, at 17:43 

hours; on July 29, 2006, at 17:11 hours when the 1000 kW DG was running; and on July 31, 

2006, at 17:56 hours when the DG was turned off. 

6.1 Circuit and Generation Measured Data 

The nodes where measured data were taken on the circuit and node D (node 10), where the 

generation unit is interconnected to the circuit, were shown in Figure 25. Test dates and data 

collection periods are presented in Table 13. 

Table 13. Test Dates and Data Collection Periods 

Date 

Circuit 

Start 

Circuit 

Peak 

Circuit 

End 

Generation Start 

(postgeneration 

start)

Generation End 

(pregeneration 

off)

7/17/06 

7/29/06 

7/31/06 

8/02/06 

10:42 

00:11 

00:11 

00:11 

17:43 

17:11 

17:56 

17:56 

23:58 

23:56 

23:56 

23:56 

17:28 

17:11 

10:26 

21:23 

20:16 

12:16 

generator 

There is a time delay between the initiation of the “start” generator command and the time 

when the unit is synchronized and carrying load, and a time delay between the initiation of the 

“stop” generator command and the time when the generator is not carrying load. Therefore, 

the times given in the Table 13 are the postgeneration start (the time the unit is carrying load) 

and the pregeneration off (the time the unit is still carrying load but has received the “stop” 

generator command). 

 62
Illustrative Example of Generation Run Time and

Distribution Circuit Time Stamps for Simulation Studies

and Field Verification

Load, MVA

Light Load Pre-generation

Start

Post generation

Start

Pre-generation

Off

Post generation

Off 24 hr

Circuit Peak

Generator

Run

0 hr

Figure 30. Daily circuit load profile and time stamps 

These times are illustrated in Figure 30. As shown in Table 13, there were four days during 

the summer when tests were conducted and four days when measured data were collected. 

The first two days, July 17 and 29, 2006, the 1000 kW DG was turned on to offset the circuit 

peak load, and on July 31, this DG was turned off. This was done to determine whether there 

was a difference in the variance between simulation and field measured data when the DG 

was running and when the DG was not running. 

Simulations and comparisons with field measured data were not conducted on the last test 

day, August 2, 2006, because the load of the South Branch of the Milford Circuit DC 8103 

had been cut over to a circuit fed from the Page Substation on August 1 at 19:00 hours. This 

removed approximately 2 MVA load (there was a 15.92 MVA peak before the cutover vs. a 

13.98 MVA peak after it) from the Milford Circuit DC 8103. See Table 14. 

The time of circuit peak data and generation data are given in Table 14 for the four test days. 

The measured data consist of : 

1. Circuit peak kVA 

2. Circuit phase currents at Substation node 1 

3. Generation line currents at node 10 

4. Generation line-to-line voltages at node 10 

5. Generation three-phase power factor at node 10 

6. Generation three-phase kW, kVAR, and kVA at node 10. 

 63
 64

6.2 Bidirectional Voltage Regulator Measured Data 

The bidirectional voltage regulator measured data for the circuit peak time of each test day are 

given in Table 15 and listed below for each phase regulator at node 9. 

1. Load current 

2. Load power factor 

3. kW load 

4. kVA load 

5. kVAR load 

6. Voltage 120 V base 

6.3 Circuit Equipment (Capacitors) and Customer Measured Data 

The measured data at the time of circuit peak for each of the three capacitors located at nodes 

6, 12, and 13 on the respective test days are given in Table 16, and they consist of the phase 

currents and phase voltages. Also given in the table are the customer-measured data for node 

I, consisting of Ia and Ic phase currents; node L, the VC phase voltage; and node K, the VA

phase voltage. 

6.4 Percent Variance Between Simulated and Field Measured Data 

Table 17 and Table 18 compare simulated data and field measured data for the two test days 

when the DG generation unit is running. Table 19 shows similar data when the DG is turned 

off. The field data for the phase currents, phase voltages, and phase power factors of the 

circuit, generator, and the VR1 bidirectional regulator are given for node 1, node 10, and node 

9, respectively, on the left side of the tables. The corresponding simulation data are shown on 

the right side of these tables. 

The measured phase currents and phase voltages for the three capacitors and their respective 

simulation data are also given in the tables. However, the simulated voltage data have been 

reduced by 3.67 V to reduce the measured primary voltage to secondary voltage values. Finally, 

customer-measured current and voltage data and simulation data for nodes I and L are listed. 

Table 20 shows the percent variance between actual and simulated current, voltage, and 

power factor for each node and summarizes the difference between measured and simulated 

values. For three-phase quantities, the percent variance is calculated as the absolute value of 

the sum of the differences between the actual and simulated values divided by three and then 

dividing by the average of the actual three-phase values. The result is multiplied by 100 to 

represent the percent difference using the average of the actuals as the base. For single-phase 

quantities, the percent variance is the absolute value of the difference divided by the actual as 

the base and multiplying by 100. 
Table 14. Circuit and Generation Measured Data 

Circuit Start: 7/17/2006 10:42 End: 7/17/2006 23:58

Generator Start: 7/17/2006 17:28 End: 7/17/2006 21:23Date Time Circuit Load Including Generation

5 Minute Integrated

3 phase kVA Ia Ib Ic 3 phase KW 3phase Kvar 3 phase Kva power factor Ia Ib Ic Va Vb 

Circuit Start 7/17/2006 10:42 10,101.50 384.00 419.00 442.00 - - - - - - - - - 

- - 

Circuit Peak 7/17/2006 17:43 15,257.22 565.00 651.00 637.00 433.00 120.50 449.45 503.60 474.10 467.60 490.00 490

0.96

Circuit Start: 7/29/2006 00:11 End: 7/29/2006 23:56

Generator Start: 7/29/2006 17:11 End: 7/29/2006 20:16

Date Time Circuit Load Including Generation

5 Minute Integrated

3 phase kVA Ia Ib Ic 3 phase KW 3phase Kvar 3 phase Kva power factor Ia Ib Ic Va Vb 

Circuit Start 7/29/2006 0:11 7,596.38 270.00 319.00 332.00 - - - - - - - - - 

- - 

Circuit Peak 7/29/2006 17:11 14,670.96 562.00 635.00 615.00 166.19 33.75 167.00 222.00 193.00 188.00 0.94 489.00 489

 

Circuit Start: 7/31/2006 00:11 End: 7/31/2006 23:56

Generator Start: 7/31/2006 16:11 End: 7/31/2006 17:11Date Time Circuit Load Including Generation

5 Minute Integrated

3 phase kVA Ia Ib Ic 3 phase KW 3phase Kvar 3 phase Kva power factor Ia Ib Ic Va Vb 

Circuit Start 7/31/2006 0:11 6,947.16 248.00 301.00 304.00 - - - - - - - - - 

Circuit Peak 7/31/2006 17:56 15,917.42 604.00 696.00 678.00 - - - - - - - - - 

Circuit Start: 8/2/2006 00:11 End: 8/2/2006 23:56

Generator Start: 8/2/2006 10:26 End: 8/2/2006 12:16Date Time Circuit Load Including Generation

5 Minute Integrated

3 phase kVA Ia Ib Ic 3 phase KW 3phase Kvar 3 phase Kva power factor Ia Ib Ic Va Vb 

Circuit Start 8/2/2006 0:11 10,241.28 428.00 408.00 448.00 - - - - - - - - - 

- - 

Generator 8/2/2006 12:16 11,277.14 475.00 433.00 469.00 409.00 112.00 424.06 551.50 500.00 447.00 0.97 489.00 489

 

Circuit Peak 8/2/2006 17:56 13,976.61 601.00 557.00 611.00 - - - - - - - - - 

August 1st moved south branch to Page Substation at 19:00

15 minute Integrated Line to L

Node 10 Generation Data

Line Current

Node 1 Circuit Phase Currents

Node 10 Generation Data

Line Current

Node 1 Circuit Phase Currents

Line to L

Node 10 Generation Data

15 Minute Integrated Line Current

Node 1 Circuit Phase Currents

15 Minute Integrated Line to L

Node 1 Circuit Phase Currents

Line to L

Node 10 Generation Data

15 Minute Integrated Line Current

 65
Table 15. Bidirectional Voltage Regulator Measured Data 

Circuit Start: 7/17/2006 10:42 End: 7/17/2006 23:58Generator Start: 7/17/2006 17:28 End: 7/17/2006 21:23Date Time

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva 

Circuit Start 7/17/2006 10:42 132.20 1,039.00 0.98 1,021.00 190.00 7,857.51 126.90 123.72 0.99 998.00 984.00 169.00 7,868.23 120.50 123.89 946.00 0.98 

Circuit Peak 7/17/2006 17:43 282.60 2,216.00 0.94 2,088.00 7841.79 123.48 742.00 255.70 2,009.00 0.95 1,913.00 7855.34 123.69 612.00 238.00 1,877.00 0.94 

Circuit Start: 7/29/2006 00:11 End: 7/29/2006 23:56Generator Start: 7/29/2006 17:11 End: 7/29/2006 20:16Date Time

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Load 

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Load 

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva 

Circuit Start 7/29/2006 0:11 157.80 1,248.00 0.89 1,110.00 7,908.95 124.53 570.00 138.80 1,096.00 991.00 0.90 7,896.23 468.00 124.33 143.40 1,133.00 

0.88

Circuit Peak 7/29/2006 17:11 312.30 2,464.00 0.93 2,286.00 7887.81 124.20 917.00 296.60 2,331.00 2,190.00 798.00 0.94 7859.16 123.75 279.70 2,195.00 

0.93

Circuit Start: 7/31/2006 00:11 End: 7/31/2006 23:56Generator Start: 7/31/2006 16:11 End: 7/31/2006 17:11Date Time

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Load 

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Load 

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva 

I

Circuit Start 7/31/2006 0:11 271.10 2,139.00 0.89 1,903.00 977.00 7,891.57 256.90 124.26 2,029.00 0.90 1,829.00 7,898.27 878.00 241.10 124.37 1,892.00 0.89 

Circuit Peak 7/31/2006 17:56 369.00 2,882.00 0.94 2,695.00 1,021.00 7811.26 123.00 355.00 2,813.00 0.94 2,640.00 7924.79 124.78 971.00 314.00 2,486.00 0.93 

Circuit Start: 8/2/2006 00:11 End: 8/2/2006 23:56Generator Start: 8/2/2006 10:26 End: 8/2/2006 12:16Date Time

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Load 

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva Load KW Load Kvar

Load 

Voltage

Load 

Voltage 

120V Base

Load 

Current

Load Power 

Factor Load Kva 

Circuit Start 8/2/2006 0:11 116.00 917.00 0.91 833.00 384.00 7,907.76 124.52 113.00 895.00 0.91 818.00 7,914.87 361.00 124.63 107.30 845.00 0.89 

Generator 8/2/2006 12:16 170.10 1,334.00 0.96 1,287.00 352.00 7,844.62 154.10 123.52 1,210.00 0.98 1,185.00 7,854.76 246.00 149.40 123.68 1,167.00 0.97 

Circuit Peak 8/2/2006 17:56 259.70 2,059.00 0.93 1,924.00 734.00 7,929.52 247.60 124.86 1,950.00 0.95 1,856.00 7,873.08 595.00 227.90 123.97 1,794.00 0.94 

August 1st moved south branch to Page Substation at 19:00

East Regulator Phase A North Regulator Phase B South R

North Regulator Phase B

East Regulator Phase A North Regulator Phase B South R

East Regulator Phase A North Regulator Phase B South R

East Regulator Phase A South R

 66
Table 16. Circuit Equipment and Customer Measured Data 

Circuit Start: 7/17/2006 10:42 End: 7/17/2006 23:58Generator Start: 7/17/2006 17:28 End: 7/17/2006 21:23Date Time

Ia Ib Ic Va Vb Vc Ia Ib Ic Va Vb Vc Ia Ib Ic 

Circuit Start 7/17/2006 10:42 33.50 62.41 121.40 8.37 120.90 121.55 33.36 11.00 12.35 121.85 121.20 120.90 44.01 21.49 42.7

 

Circuit Peak 7/17/2006 17:43 48.57 108.20 118.25 13.00 117.75 121.08 51.96 14.00 23.83 120.65 118.40 118.65 65.84 27.09 53.7

 

Date Time Node K Node L

Ia Ib Ic Va Vc

Circuit Start 7/17/2006 10:42 32.00 - 19.00 120.05 -

Circuit Peak 7/17/2006 17:43 46.00 - 31.00 117.60 -

Circuit Start: 7/29/2006 00:11 End: 7/29/2006 23:56

Generator Start: 7/29/2006 17:11 End: 7/29/2006 20:16Date Time

Ia Ib Ic Va Vb Vc Ia Ib Ic Va Vb Vc Ia Ib Ic 

Circuit Start 7/29/2006 0:11 23.82 47.09 8.78 24.69 5.42 10.78 27.23 12.53 31.5

Circuit Peak 7/29/2006 17:11 50.45 103.25 10.34 54.29 13.02 24.72 50.05 18.57 47.9

Circuit Start: 7/31/2006 00:11 End: 7/31/2006 23:56Generator Start: 7/31/2006 16:11 End: 7/31/2006 17:11Date Time

Ia Ib Ic Va Vb Vc Ia Ib Ic Va Vb Vc Ia Ib Ic 

Circuit Start 7/31/2006 0:11 22.12 44.79 7.74 21.23 5.81 10.16 30.38 14.65 31.8

Circuit Peak 7/31/2006 17:56 51.53 108.84 11.78 49.30 14.93 23.96 62.18 27.37 54.0

Circuit Start: 8/2/2006 00:11 End: 8/2/2006 23:56

Generator Start: 8/2/2006 10:26 End: 8/2/2006 12:16Date Time

Ia Ib Ic Va Vb Vc Ia Ib Ic Va Vb Vc Ia Ib Ic 

Circuit Start 8/2/2006 0:11 34.58 77.96 10.80 41.65 9.95 18.72 47.89 23.05 52.0

Generator 8/2/2006 12:16 37.82 86.89 12.02 42.15 13.98 18.46 52.82 23.95 50.0

Circuit Peak 8/2/2006 17:56 52.56 109.82 13.40 51.40 15.32 24.46 52.27 20.09 55.1

Date Time Node K Node L Notes

Va Vc (1): August 1st moved south branch to Page Substation at 19:00

Circuit Start 8/2/2006 0:11 - (2): Voltage Data At Capacitor Locations Not Available from July 26 until August 22

(3): All Capacitor Phase Voltages Are Measured On The Secondary Side At The Customer's Meter

Generator 8/2/2006 12:16 119.15 - (4): All Capacitor Phase Currents Are Measured On The Primary

Ci i P k 8/2/2006 17 56 118 25

Phase Voltages Phase VoltagesPhase Voltages 

Node F/6 Capacitor 1 (900 Kvar) Node H/13 Cap

Phase Currents

Phase Currents Phase Currents

Node G/12 Capacitor 2 (900 Kvar)

Node F/6 Capacitor 1 (900 Kvar) Node H/13 Cap

Phase Currents Phase Currents Phase Voltages

Node G/12 Capacitor 2 (900 Kvar)

Phase Currents

Phase Currents

Node G/12 Capacitor 2 (900 Kvar)

Phase Currents Phase Voltages Phase Currents Phase VoltagesPhase Currents

Node F/6 Capacitor 1 (900 Kvar) Node H/13 Cap

Node I

Node G/12 Capacitor 2 (900 Kvar)

Phase Currents Phase Voltages

Node H/13 Cap

Phase Voltages Phase Currents 

Node F/6 Capacitor 1 (900 Kvar)

 67
Table 17. Field Verification Data – 7/17/06 – DR Generation On 

Location Node

A B C

A B C

A B C

A B C

A B 

Start of Circuit 1 565.00 651.00 637.00 126.00 126.00 126.00 0.95 0.95 0.95 554.54 637.77 619.97 125.86 125.80 

2.2% 0.3% 5.7%

Generator 10 18.31 17.24 17.00 122.50 122.50 120.25 0.96 0.96 0.96 17.49 17.25 17.36 121.68 123.38 

2.3% 1.1% 0.0%

VR 1 9 282.60 255.70 238.00 123.48 123.69 124.19 0.94 0.95 0.94 274.07 246.07 227.56 123.60 123.82 

3.7% 0.2% 2.4%

Cap 1* 6 48.57 108.20 13.00 118.25 117.75 121.08 48.03 104.47 13.77 121.25 121.14 

3.0% 0.7% 117.58 117.47 

Cap 2* 12 51.96 14.00 23.83 120.65 118.40 118.65 52.18 14.87 24.53 120.51 123.09 

2.0% 1.5% 116.84 119.42 

Cap 3* 13 65.84 27.09 53.70 117.05 116.93 118.15 64.44 27.92 50.18 119.04 121.44 

3.9% 1.2% 115.37 117.77 

Node I 16 46.00 31.00 32.73 22.94

38.3%

Node L* 18 117.60 121.42

0.1% 117.75

*The voltage measurements were taken at the customer's meter. Therefore, an average voltage drop of 3.67 V was included to reduce the primary voltage to the comparison shown in the table. 

Simulation

PF

Field Data - 7/17/06 - Circuit Peak 17:43

Current Voltage Current Voltage

 68
 69

Location Node

A B C

A B C

A B C

A B C

A B 

Start of Circuit 1 562.00 635.00 615.00 564.59 636.69 616.03 126.46 126.34 

0.3%

Generator 10 8.07 7.08 6.84 122.25 122.25 120.75 0.98 0.98 0.98 7.30 7.19 7.22 121.76 123.67 

5.7% 1.2% 0.0%

VR 1 9 312.30 296.60 279.70 124.20 123.75 123.56 0.93 0.94 0.93 316.35 299.29 284.46 123.92 124.46 

1.3% 0.5% 3.4%

Cap 1 6 50.45 103.25 10.34 49.80 100.44 11.10

2.6%

Cap 2 12 54.29 13.02 24.72 54.48 13.91 25.46

2.0%

Cap 3 13 50.05 18.57 47.97 49.11 19.56 45.29

4.0%

Simulation

PF

Field Data - 7/29/06 - Circuit Peak 17:11

Current Voltage Current Voltage

Location Node

A B C

A B C

A B C

A B C

A B 

Start of Circuit 1 604.00 696.00 678.00 608.76 699.59 678.97 126.26 126.14 12

0.5%

Generator 10 N/A N/A N/A N/A N/A N/A N/A N/A N/A

VR 1 9 369.00 355.00 314.00 123.00 124.78 124.35 0.94 0.94 0.93 376.11 362.02 324.22 122.93 123.63 12

2.3% 0.5% 3.4%

Cap 1 6 51.53 108.84 11.78 50.82 105.52 12.49

2.8%

Cap 2 12 49.30 14.93 23.96 48.74 15.69 24.55

2.2%

Cap 3 13 62.18 27.37 54.05 60.45 28.03 49.14

5.1%

Simulation

PF

Field Data - 7/31/06 - Circuit Peak 17:56

Current Voltage Current Voltage

Table 18. Field Verification Data – 7/29/06 – DR Generation On 

Table 19. Field Verification Data – 7/31/-6 – DR Generation Off 
6.5 Summary of Variance Results 

From Table 20, we see that the percent variance between actual field measured data and 

the simulated data is less than 6% (5.7% actual) for the phase currents throughout the 

circuit, except for node I, where the percent unbalanced current is very high. The highest 

percent variance for the phase voltages throughout the circuit is 1.5%, whereas the highest 

percent variance for the power factor is 5.7% at the start of the circuit. Note that the lowest 

percent variances occur on the load side of the step regulators, because these regulators are 

individually phase-controlled and not gang-operated, as the LTC at the substation. 

As shown in the percent variance data in Table 20, the simulated data closely match actual 

field-measured data in most cases; the models developed in Section 3.0 can be used to 

estimate phase currents, phase voltages, and power factors without the use of extensive 

circuit metering. However, knowing the phasing of the loads is paramount in accurate data 

simulation. No significant difference in the percent variance occurred when the variance 

data were compared for the situation in which the DG is turned on and that in which it is 

turned off. 

 70
Table 20. Percent Variance Between Actual and Simulated 

Currents, Voltages, and Power Factors 

Location Node Current Voltage P.F. 

Test Day 7/17/06 DR Generation Running 

Start of Circuit 

Generator 

Voltage Regulator 

Capacitor 1 

Capacitor 2 

Capacitor 3 

Node I 

Node L 

1 

10 

9 

6 

12 

13 

16 

18 

2.2 

2.3 

3.7 

3.0 

2.0 

3.9 

38.3 

-- 

0.3 

1.1 

0.2 

0.7 

1.5 

1.2 

-- 

0.1 

5.7 

0.0 

2.4 

-- 

-- 

-- 

-- 

-- 

Test Day 7/29/06 DR Generation Running

Start of Circuit 

Generator 

Voltage Regulator 

Capacitor 1 

Capacitor 2 

Capacitor 3 

1 

10 

9 

6 

12 

13 

0.3 

5.7 

1.3 

2.6 

2.0 

4.0 

-- 

1.2 

0.5 

-- 

-- 

-- 

-- 

0.0 

3.4 

-- 

-- 

-- 

Test Day 7/31/06 DR Generation Off 

Start of Circuit 

Generator 

Voltage Regulator 

Capacitor 1 

Capacitor 2 

Capacitor 3 

1 

N/A 

9 

6 

12 

13 

0.5 

N/A 

2.3 

2.8 

2.2 

5.1 

-- 

N/A 

0.5 

-- 

-- 

-- 

-- 

N/A 

3.4 

-- 

-- 

-- 

 71
7.0 Conclusions and Recommendations 

Conclusions 

The distribution circuit selected was a 13.2 kV three-phase wye-multigrounded system 

that serves approximately 76.2% residential, 4.0% commercial, and 19.8% light industrial 

customers. The load on the summer peak day was 15.3 MVA, and the summer peak day 

minimum load was 5.91 MVA. A 1000 kW synchronous generator, installed at the 

midpoint of the circuit, offset the peak load on July 17, 2006, by 449.5 kVA. There were 

no voltage violations on the circuit. The lowest single-phase measured voltage was 117.60 

V on phase C versus a simulated value of 117.75 V. 

Models were developed for the 10 MVA LTC delta wye-grounded substation transformer, 

the 167 kVA bidirectional voltage regulators, the three wye-grounded capacitors, line 

impedances, and all distribution circuit transformer connections (9 cases). 

The line loss model was validated using three line configurations and balanced and 

unbalanced load conditions. The three line configurations studied were the balanced 

impedance triangular spacing configuration, or equilateral configuration; the flat 

configuration nontransposed; and the flat configuration transposed. The purpose of the 

validation was to show that with a balanced line impedance and balanced load, kW losses 

are the same in each phase, and the total kW losses are the lowest. In addition, an example 

was provided to show that even though the kW losses per phase are not correct (owing to 

the Kron reduction process from a 4x4 matrix to a 3x3 matrix), the total losses for the three 

phases are in fact correct. 

Models were developed for the 1000 kW synchronous generator, the self-excited 400 kW 

induction generator, and the 400 kW high-speed generator and inverter generator system. 

The line, equipment, and generation models were verified with the maximum phase current 

variance of 3.9% on July 17, 2006, and 5.7% on July 29, 2006, except for node I, where 

the percent unbalance current was very high. The maximum phase voltage variance was 

1.5% on July 17 and 1.2% on July 29. The maximum power factor variance did not exceed 

5.7% on July 17 and 3.4% on July 29. 

Three voltage control strategies were tested for the 400 kW induction generator and were 

tested at low and high substation primary voltage for a total of 6 simulations. Three voltage 

control strategies for the 400 kW inverter-based generation were tested at low and high 

substation primary voltage, for a total of 6 simulations. And 13 voltage control strategies 

were simulated for high voltage and 13 for low voltage for the synchronous generator 

located at the beginning location of the circuit, at the mid-location, and at the end location, 

for a total of 78 simulations. 

The maximum released capacity of 10.44% was achieved with the 1000 kW synchronous 

generator, with P = 100% and Q = 100%. 

 72
The optimum location for the DG with the highest released capacity of 10.44% is at the 

source of the circuit, because it directly offsets the load current and load losses of the 

circuit. The optimum DG location to achieve the greatest loss reduction is at the midpoint 

location of the circuit, because adding generation here reduces, on a pro rata basis, the load 

and the length of the circuit, and thus there are reduced losses. There is little difference 

between locating the DG at the midpoint or end of the circuit to improve the voltage 

regulation; there is a slightly better improvement at the midpoint for circuits in which the 

conductor size of the entire three-phase backbone is the same. 

Recommendations 

Distributed generation can be used to reduce real peak losses and to improve voltage 

regulation and release capacity. This project used validated circuit models and a validated 

VDC load model to test 78 synchronous generator voltage control strategies. The voltagedependent current model best represented the load characteristics of the circuit. 

Future research should fund the development of a real-time optimal control of one or more 

DG units to accomplish the optimum generator voltage control condition, including the 

control of the substation transformer LTC, the bidirectional step regulators, and the 

switched capacitors. The cost to perform this development work on the Milford Circuit DC 

8103 would be substantially lower than installing DG units and metering equipment on a 

new circuit, because all the equipment is already installed and tested. Furthermore, the 

newly developed control strategies can be tested in summer 2008. The estimated cost of 

this project would be less than $500,000, and the development, testing, and analysis 

schedule is estimated to be approximately 1.5 years. 

Benefits to California 

Models have been developed and validated with measured test data. This project shows 

that adding DG units to a distribution circuit can reduce peak losses and improve 

generation capacity (released capacity), and these results can be achieved in 1 year rather 

than the 7 to 10 years associated with traditional central station generation. Also, 

improved voltage regulation results in fewer service voltage criteria violations and fewer 

distribution energy losses (approximately 10%). 

 73
8.0 References 

Davis, M.W. Determine the Market Penetration Limits of Distributed Energy Resources 

(by Distributed Generation Type) by the Year 2015. Work performed under contract no. 

ADD-3-33906-01. Golden, CO: National Renewable Energy Laboratory; December 2003. 

Davis, M.W.; Krupa, T.J.; Diedzic, M.J. “The Economics of Load Management on the 

Design and Operation of the Distribution System, Part II, Load Characteristics.” IEEE 

Transactions on Power Apparatus and Systems, Vol. PA 8-102, No. 3, March 1983; pp. 

654-674. 

Kersting, W.H. “The Computation of Neutral and Dirt Currents and Power Losses.” IEEE 

Power Engineering Society Power Systems Conference and Exposition 2004, Vol. 1, 10-

13, October 2004; pp. 213-218. 

 74
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PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ORGANIZATION.

1. REPORT DATE (DD-MM-YYYY)

July 2007 

2. REPORT TYPE

Annual Subcontract Report 

3. DATES COVERED (From - To)

June 2006-June 2007 

5a. CONTRACT NUMBER

DE-AC36-99-GO10337 

5b. GRANT NUMBER

4. TITLE AND SUBTITLE

Modeling and Verification of Distributed Generation and Voltage 

Regulation Equipment for Unbalanced Distribution Power Systems 

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

NREL/SR-581-41885 

5e. TASK NUMBER

WW88.2003 

6. AUTHOR(S)

M.W. Davis, R. Broadwater, and J. Hambrick 

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

M. W. Davis, DTE Energy, Detroit, MI; R. Broadwater, Electrical Distribution 

Design, Inc., Blacksburg, VA; J. Hambrick, Virginia Polytechnic Institute and 

State University, Blacksburg, VA 

8. PERFORMING ORGANIZATION 

REPORT NUMBER

ZAT-5-32616-06 

10. SPONSOR/MONITOR'S ACRONYM(S)

NREL 

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National Renewable Energy Laboratory 

1617 Cole Blvd. 

Golden, CO 80401-3393 11. SPONSORING/MONITORING 

AGENCY REPORT NUMBER

NREL/SR-581-41885 

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13. SUPPLEMENTARY NOTES

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14. ABSTRACT (Maximum 200 Words)

This report summarizes the development of models for distributed generation and distribution circuit voltage 

regulation equipment for unbalanced power systems. The project also included verification, through actual field 

measurements, that the models accurately represent the power distribution system. The variance between actual 

field measurements and model simulation data did not exceed 5.7% for phase currents and 1.5% for phase voltages. 

15. SUBJECT TERMS

modeling; distributed generation; unbalanced distribution power systems; voltage regulation equipment; validating 

distribution circuit models; National Renewable Energy Laboratory; NREL; DTE Energy 

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