IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007 1019

Development of Three-Phase Unbalanced PowerFlow Using PV and PQ Models for Distributed

Generation and Study of the Impact of DG ModelsSarika Khushalani, Student Member, IEEE, Jignesh M. Solanki, Student Member, IEEE, and

Noel N. Schulz, Senior Member, IEEE

Abstract—With the increased installations of distributed genera-tors (DGs) within power systems, load flow analysis of distributionsystems needs special models and algorithms to handle multiplesources. In this paper, the development of an unbalanced three-phase load flow algorithm that can handle multiple sources is de-scribed. This software is capable of switching the DG mode of op-eration from constant voltage to constant power factor. The algo-rithm to achieve this in the presence of multiple DGs is proposed.Shipboard power systems (SPS) have other special characteristicsapart from multiple sources, which make the load flow difficultto converge. The developed software is verified for a distributionsystem without DG using the Radial Distribution Analysis Package(RDAP). The developed software analyzes an IEEE test case andan icebreaker ship system. System studies for the IEEE 37-nodefeeder without the regulator show the effect of different models andvarying DG penetration related to the increase in loading. Systemlosses and voltage deviations are compared.

Index Terms—Distributed generation, IEEE 37-node feeder, ra-dial distribution analysis package, shipboard power systems.

I. INTRODUCTION

ONE of the key calculations for any system is the deter-mination of the steady-state behavior, which is termed

as distribution power flow for a distribution system. Distribu-tion automation needs fast and efficient power flow solutions.The loading of a distribution feeder is inherently unbalanceddue to a large number of unequal single-phase loads and thenonsymmetrical conductor spacing of three-phase undergroundand overhead line segments. Due to these factors, conventionalpower flow programs used for transmission system studies donot show good convergence properties for distribution systems.These programs also assume a perfectly balanced system so thata single-phase equivalent can be used. The rise in power de-mand has led to installation of small power units called dis-tributed generators (DGs), which give high fuel flexibility. ADG, if properly planned and controlled, can be beneficial to the

Manuscript received March 24, 2006; revised February 20, 2007. Thiswork was supported in part by NSF Career under Grant ECS 0196559 andin part by the Office of Naval Research under Grants N00014-02-1-0623 andN00014-03-1-0744. Paper no. TPWRS-00158-2006.

S. Khushalani and J. M. Solanki are with Mississippi State University,Mississippi State, MS 39762 USA (e-mail: sakru1@yahoo.com; jigneshm-solanki@yahoo.com).

N. N. Schulz is with the Department of Electrical and Computer Engi-neering, Mississippi State University, Mississippi State, MS 39762 USA(e-mail: schulz@ece.msstate.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2007.901476

power industry, help defer the costs of expansion, and have pos-itive environmental impacts. A DG alters the power flow of thesystem, thus impacting the overall system losses and voltageprofile of the system. A lot of work has been done for radialpower flow solution [1]–[10]. Cheng and Shirmohammadi [11]address the load flow solution, incorporating DG for terrestrialdistribution systems. Initial development of such a power flowfor radial terrestrial distribution systems with distributed gen-erator was addressed in [12]. Butler et al. [13] developed athree-phase load flow algorithm for shipboard power systems(SPS) and handle multiple sources by collapsing them into asingle source. A comparison of distribution power flows fora balanced SPS is addressed by Lewis and Baldwin [14]. AnSPS is an ungrounded delta-connected system where genera-tion, transmission, and distribution are tightly coupled. Anal-ysis of SPS, due to its distinctive characteristics, leads to a fur-ther complication in distribution power flow because of almostthe same nominal voltages of all generator nodes. This paperdetails the power flow development for three-phase unbalancedterrestrial distribution systems and SPS with distributed gener-ator nodes modeled as PQ and PV nodes. Unlike the develop-ment in [12], this development can handle multiple DGs andallows for switching the DG mode from constant voltage to con-stant power factor. Comparisons of the power flow results withstandard distribution power flow software are made, and lim-itations of the standard software are addressed. An SPS loadflow solution is obtained. System studies showing the impactof considerable DG penetration on steady-state behavior of theCalifornia distribution feeder are shown. PQ and PV represen-tations along with different penetration levels are incorporatedin the case studies.

II. COMPONENT MODELING

Because of the limited use of matrix operations, the ladderiterative method is selected for the load flow. Reference [12]reviews previous work related to the ladder iterative method.This method involves two sweeps of calculations. In the forwardsweep, the end voltages are initialized for the first iteration, andcurrents are calculated starting at the buses at the load end of theradial branch and solved up to the source bus by applying thecurrent summation method. The backward sweep starts at thesource bus and calculates voltages using the current calculatedfrom the forward sweep until the load end of the radial branches.The voltages from the backward sweep are used for the next it-eration in the forward sweep calculations. Convergence occurswhen the calculated source voltage in the backward sweep cor-responds to specified source voltage.

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1020 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007

TABLE ICOMPONENT MODELS

Modeling equations for various components of the distribu-tion system are tabulated in Table I. Details of the models canbe found in [12] and [15]. Reference [16, Tables II and III] helpsto calculate the impedance of overhead lines from the givenphasing, space ID, material, and stranding. Reference [16, Ta-bles IV–VI] used the tables to calculate the impedance of under-ground concentric or tape shield cables. Transformers are mod-eled as in [15], and only the updated equations are given here.Four different types of transformer connections have been mod-eled: delta-wye, wye-delta, wye-wye, and delta-delta.

The DG connections can be wye or delta. Depending on thecontrol, the DG may be set to output power at either constantpower factor for small DG or constant voltage for large DG.Thus, two types of DG models need to be developed: constantPQ, modeled as negative load with currents injecting into thenode, and PV nodes for which the calculations are as below.

1) Initially, the generator real power and positive sequencevoltage are specified. The reactive power is initialized tozero. After a load flow has converged, the positive sequencevoltage magnitude mismatch at the PV node is checked

(1)

where of PV nodes.

2) If the voltage mismatch is within the specified tolerance,the PV node voltage has converged to the specified value.If a voltage mismatch at the PV node is not less than thespecified tolerance, then reactive power compensation Qgenerated by that PV node in order to maintain the voltageat specified value needs to be calculated as follows:

(2)

where is the positive sequence sensitivity impedancematrix whose size is . The diagonal elements of thismatrix are the absolute value of the positive sequence of thesum of series line impedances between each PV node andthe source node. The off-diagonal elements are the sums ofthe common series line impedances between two PV nodesand source node. is . Thus, is andis the magnitude of reactive current injection. The DG canoperate in lagging as well as leading power factor mode.Thus, the injection of current will depend on the error dif-ference . If is positive, then reactive power issupplied by DG, and is negative then reactive poweris absorbed by DG.The reactive current injection is thus

(3)

where , and are the angles of the convergedvoltage at the th node.

3) There is a limit to which the DG can produce reactivepower. This limit is decided in this program by setting thepower factor limits between 0.8 and 1, lagging/leading

(4)

If during computation the reactive power of any of the DGsgoes outside its limits, it is fixed at the limiting value, andthis node is now treated as a PQ node. The limiting valueis calculated as the three-phase reactive power limit; thus,the total per-phase reactive current that the DG can injectbefore its limit is hit is given by

(5)

4) These currents are then added, to the load currents andcurrents calculated due to DG real power injection ,at the th node

(6)

Load flow runs again to check the voltage magnitudes andnew . If after load flow the of the PV node con-

KHUSHALANI et al.: DEVELOPMENT OF THREE-PHASE UNBALANCED POWER FLOW 1021

Fig. 1. Flow chart for load flow with multiple DGs.

verted to PQ node is within the limits, the node is switchedback to PV node. Fig. 1 summarizes the solution algorithm.

III. SYSTEM DESCRIPTIONS

The IEEE 37-node feeder is an actual feeder in California.The data for the feeder were obtained from IEEE test casearchive for distribution feeders [17]. The diagram of the feederis shown in Fig. 2. The regulator was removed in order toclearly see the effect of the DG on the system. The data arecharacterized by

Loads—Spot loads, single-phase and three-phase bal-anced and unbalanced loads, delta connected, constantkW, kVAR, constant Z and constant I type;Overhead and Underground Lines—Three-phase lineswith different spacing of phases;Transformer—Substation and inline transformers aredelta-delta.

The 18-node SPS of an icebreaker ship was also analyzed forwhich the data were obtained from [14]. The diagram of thesystem is shown in Fig. 3.

The system was represented and renumbered as in Fig. 4 sothat the analysis could be done using the developed unbalancedpower flow software. The data are characterized by

Loads—Spot loads, three-phase balanced loads, delta con-nected, constant kW, kVAR;

Fig. 2. IEEE 37-node feeder.

Fig. 3. Shipboard Power System.

Fig. 4. Renumbered Shipboard Power System.

Overhead and Underground Lines—Three-phasecables;Transformer—Inline transformer is delta-delta with tapratio of one.

1022 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007

TABLE IIMODIFIED IEEE 37–NODE VOLTAGES

IV. RESULTS

A. Load Flow

The unbalanced load flow software was developed inMATLAB. Using Radial Distribution Analysis Package(RDAP) [18]), a commercially available distribution powerflow package, the authors compared results on a commonIEEE distribution test system [17] to validate the results of thedeveloped software.

1) Results of the original feeder (without DG) obtained bythe developed software and those obtained by RDAP arecompared. Voltages are as shown in Table II, and the cur-rents are shown in Table VI in the Appendix. Results ob-tained from the developed program closely match the re-sults obtained from RDAP for the original feeder. The loadflow took one iteration to converge. RDAP can only handlemodeling a DG as a negative load, so results are not avail-able for the PV model.

2) The voltages in per-unit and currents in amperes of 18 nodeSPS are shown in Table III. Node 1 is slack node, and nodes2–4 are modeled as PV nodes. The generation values areshown in Table IV. The results were compared to results in[14]. These results are obtained with 0.001 p.u. tolerancefor load flow as well as PV node convergence. The cablesare short, and even a very small voltage difference leads to

TABLE IIISPS VOLTAGES AND CURRENTS

TABLE IVSPS GENERATION

a large reactive current injection. The load flow took fouriterations to converge.

B. System Studies

The IEEE 37-node feeder [17] without the regulator wasstudied under different DG modeling and varying penetration.This study helps to demonstrate how the percentage of thedistributed generation as well as the loading on the systemcoupled with the DG model affect the final voltage results.

This highlights the variation in results depending on the DGmodel.

1) DG is connected to node 734.2) DG penetration is defined as

% where

3) Anticipating the future load growth, DG penetration is in-creased by increasing the real and reactive power of loadsin all the phases of nodes 727, 728, 729, 730, 731, 732, 733,735, 736, 737, 738, 740, 741, 742, and 744. The penetra-tion of DG is increased in steps of 3.5% up to 35%, whichcorresponds to an increase of loading in steps of 5% from5% to 50%, respectively

4) Fig. 5(a)–(c) shows a comparison of the voltage deviationfrom 1 p.u. for different DG models and varying DG pen-etration. The x-axis of Fig. 5(a)–(c) represents the nodenumber, the y-axis of Fig. 5(a) and (b) represents the per-centage DG penetration, the y-axis of Fig. 5(c) representsthe percentage load increase, and the z-axis of Fig. 5(a)–(c)represents the voltage deviation from 1 p.u.Fig. 5(a) is obtained with a DG modeled as a constantPQ node. The surface plot of Fig. 5(a) clearly indicatesthe voltage deviation is low for the downstream nodes ofthe feeder. Voltage deviation is high for nodes 744 and728 because load at 728 is a three-phase load, and hence,

KHUSHALANI et al.: DEVELOPMENT OF THREE-PHASE UNBALANCED POWER FLOW 1023

Fig. 5. IEEE 37-node feeder study with PQ and PV models and varying DG penetration level (a) PQ model. (b) PV model. (c) PV model with 50% penetration.(d) Loss comparison.

the loading is increased in all three phases and 744 is di-rectly connected to this node. The minimum voltage de-viation is 0.01933 p.u. Fig. 5(b) is obtained with a DGmodeled as a PV node that switches to PQ node in caseof a reactive power limit hit. The same observations can bemade for surface plot of Fig. 5(b). The minimum voltagedeviation is 0.0169 p.u. Comparing surfaces of Fig. 5(a)and (b) demonstrates that the difference in voltage devia-tion for downstream nodes during low DG penetration isnot much. However, the difference in voltage deviation fordownstream nodes during high DG penetration is consid-erable. Fig. 5(c) is obtained with the DG modeled as PVnode and constant penetration, i.e., 50%, which switchesto PQ node in case of a reactive power limit hit. However,there was no reactive power limit hit for the load increasesshown. The surface plot of Fig. 5(c) clearly indicates thatthe voltage deviation is low for the downstream nodes ofthe feeder, but it increases with an increase in loading. Theminimum voltage deviation is 0.0016 p.u. Based on theseobservations, for an increase in load, the voltage deviation

is the least when DG is modeled as PV node with 50% DGpenetration.

5) Twelve cases as shown in Table V are defined as follows.a) Cases 0–3 have 0% DG penetration.b) Cases 4–6 have varying DG penetration but are mod-

eled as PQ node.c) Cases 7–9 have varying DG penetration but are mod-

eled as PV node, which switches to PQ node in caseof a limit hit.

d) Cases 10–12 have 50% DG penetration but are mod-eled as PV node, which switches to PQ node in caseof a limit hit. However, for all three cases, the limitdoes not hit with this penetration level.

6) Fig. 5(d) shows the system loss comparison for all thesecases. Losses for cases 1–3, which correspond to no DGpenetration, are quite high. Losses for cases 4–6, whichcorrespond to the PQ model, are slightly higher than cases7–9, which correspond to the PV model. For 50% DG pen-etration, losses are much less. Losses for case 12 are morethan that of cases 10 and 11 due to an increase in loading.

1024 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007

TABLE VCASE SCENARIOS

Fig. 6. Real power contributions from DG and substation along with total realpower loading.

For a 50% increase in load, losses decrease by 3.4% due to50% DG penetration.

7) Fig. 6 shows the real power generated by the DG relativeto that from the substation in all the 12 cases. This graphis also a representation of the DG injection. For cases 0–3,there is no DG penetration, and hence, the entire real powercontribution is from the substation. For cases 4–6 and 7–9,the DG penetration increases, and hence, the substationcontribution decreases. For cases 10–12, DG penetrationis 50%, and thus, the substation contributes 50%.

8) The power flow took one iteration to converge for all caseswhere the DG was modeled as a PQ node, whereas a max-imum of three iterations were required for all cases whereDG was modeled as a PV node.

V. CONCLUSION

This paper describes a revised general three-phase unbal-anced power flow algorithm that allows for the incorporationof DGs modeled as either PV or PQ nodes. The algorithm wastested on both an IEEE 37-node test feeder and an 18-nodeicebreaker shipboard power system. Comparing the resultsof the unbalanced power flow without DG with RDAP, anestablished software package, demonstrates the accuracy of thealgorithm for an established test case. Advanced studies on theIEEE 37-node test case with DG demonstrate the impact of DGmodel type, size, and load variations on the results.

Further studies, including using the recently published paperon induction machine models and the IEEE 34-node test case

TABLE VIMODIFIED IEEE 37-NODE CURRENTS

[19], will provide additional opportunities to study the impactof various DG models on unbalanced power flow analysis.

APPENDIX

Table VI shows the modified IEEE 37-node currents.

ACKNOWLEDGMENT

The authors would like to thank Dr. T. Baldwin from FloridaState University for the icebreaker shipboard power system data.

REFERENCES

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[7] W. Xu et al., “A generalized three-phase power flow method for theinitialization of EMTP simulations,” in Proc. Int. Conf. Power SystemTechnology, 1998, vol. 2, pp. 875–879.

[8] H. M. Mok et al., “Power flow analysis for balanced and unbalancedradial distribution systems,” [Online]. Available: http://www.itee.uq.edu.au/~aupec/aupec99/mok99.pdf.

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[9] M. A. Laughton, “Analysis of unbalanced polyphase networks bymethod of phase co-ordinates,” Proc. Inst. Elect. Eng., vol. 15, no. 8,pp. 1163–1172, Aug. 1968.

[10] W. H. Kersting, Distribution System Modeling and Analysis. BocaRaton, FL: CRC, 2002.

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Sarika Khushalani (S’06) received the B.E. degreefrom Nagpur University, Nagpur, India, in 1998,the M.E. degree from Mumbai University, Mumbai,India, in 2000, and the Ph.D. degree from theElectrical and Computer Engineering Departmentof Mississippi State University (MSU), MississippiState, MS, in 2006.

She is now an engineer working for Open SystemsInternational, Minneapolis, MN. She was involvedin research activities at IIT Bombay, Bombay, India.Her research interests are computer applications in

power system analysis and power system control.Ms. Khushalani received a Honda Fellowship Award at MSU.

Jignesh M. Solanki (S’06) received the B.E. degreefrom V.N.I.T., Nagpur, India, in 1998, the M.E.degree from Mumbai University, Mumbai, India, in2000, and the Ph.D. degree from the Electrical andComputer Engineering Department of MississippiState University, Mississippi State, MS, in 2006.

He is now an engineer working for Open SystemsInternational, Minneapolis, MN. He was involved inresearch activities at IIT Bombay, Bombay, India. Hisresearch interests are power system analysis and itscontrol.

Noel N. Schulz (SM’00) received the B.S.E.E. andM.S.E.E. degrees from Virginia Polytechnic Instituteand State University, Blacksburg, in 1988 and 1990,respectively, and the Ph.D. degree in electricalengineering from the University of Minnesota,Minneapolis, in 1995.

She has been an Associate Professor in theElectrical and Computer Engineering Departmentat Mississippi State University, Mississippi State,MS, since July 2001. Her research interests are incomputer applications in power system operations,

including artificial intelligence techniques.Dr. Schulz holds the TVA Endowed Professorship in Power Systems Engi-

neering. She is an NSF CAREER award recipient. She has been active in theIEEE Power Engineering Society and is serving as Secretary for 2004–2007.She was the 2002 recipient of the IEEE/PES Walter Fee Outstanding YoungPower Engineer Award.