OpenDSS Project

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

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

Generation and Study of the Impact of DG Models Sarika 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 distribution systems needs special models and algorithms to handle multiple sources. 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 characteristics apart from multiple sources, which make the load flow difficult to converge. The developed software is verified for a distribution system without DG using the Radial Distribution Analysis Package (RDAP). The developed software analyzes an IEEE test case and an icebreaker ship system. System studies for the IEEE 37-node feeder without the regulator show the effect of different models and varying DG penetration related to the increase in loading. System losses 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 unbalanced due to a large number of unequal single-phase loads and the nonsymmetrical conductor spacing of three-phase underground and overhead line segments. Due to these factors, conventional power flow programs used for transmission system studies do not show good convergence properties for distribution systems. These programs also assume a perfectly balanced system so that a 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. A DG, if properly planned and controlled, can be beneficial to the

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

S. Khushalani and J. M. Solanki are with Mississippi State University, Mississippi State, MS 39762 USA (e-mail: [email protected]; jigneshm- [email protected]).

N. N. Schulz is with the Department of Electrical and Computer Engi- neering, Mississippi State University, Mississippi State, MS 39762 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available online at 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 the system, thus impacting the overall system losses and voltage profile of the system. A lot of work has been done for radial power flow solution [1]–[10]. Cheng and Shirmohammadi [11] address the load flow solution, incorporating DG for terrestrial distribution systems. Initial development of such a power flow for radial terrestrial distribution systems with distributed gen- erator was addressed in [12]. Butler et al. [13] developed a three-phase load flow algorithm for shipboard power systems (SPS) and handle multiple sources by collapsing them into a single source. A comparison of distribution power flows for a balanced SPS is addressed by Lewis and Baldwin [14]. An SPS 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 almost the same nominal voltages of all generator nodes. This paper details the power flow development for three-phase unbalanced terrestrial 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 and allows for switching the DG mode from constant voltage to con- stant power factor. Comparisons of the power flow results with standard distribution power flow software are made, and lim- itations of the standard software are addressed. An SPS load flow solution is obtained. System studies showing the impact of considerable DG penetration on steady-state behavior of the California distribution feeder are shown. PQ and PV represen- tations along with different penetration levels are incorporated in the case studies.

II. COMPONENT MODELING

Because of the limited use of matrix operations, the ladder iterative 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 forward sweep, the end voltages are initialized for the first iteration, and currents are calculated starting at the buses at the load end of the radial branch and solved up to the source bus by applying the current summation method. The backward sweep starts at the source bus and calculates voltages using the current calculated from 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 occurs when the calculated source voltage in the backward sweep cor- responds to specified source voltage.

0885-8950/$25.00 © 2007 IEEE

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

TABLE I COMPONENT MODELS

Modeling equations for various components of the distribu- tion system are tabulated in Table I. Details of the models can be found in [12] and [15]. Reference [16, Tables II and III] helps to calculate the impedance of overhead lines from the given phasing, 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 the control, the DG may be set to output power at either constant power factor for small DG or constant voltage for large DG. Thus, two types of DG models need to be developed: constant PQ, modeled as negative load with currents injecting into the node, and PV nodes for which the calculations are as below.

1) Initially, the generator real power and positive sequence voltage are specified. The reactive power is initialized to zero. After a load flow has converged, the positive sequence voltage 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 the specified tolerance, then reactive power compensation Q generated by that PV node in order to maintain the voltage at specified value needs to be calculated as follows:

(2)

where is the positive sequence sensitivity impedance matrix whose size is . The diagonal elements of this matrix are the absolute value of the positive sequence of the sum of series line impedances between each PV node and the source node. The off-diagonal elements are the sums of the common series line impedances between two PV nodes and source node. is . Thus, is and is the magnitude of reactive current injection. The DG can operate 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 is supplied by DG, and is negative then reactive power is absorbed by DG. The reactive current injection is thus

(3)

where , and are the angles of the converged voltage at the th node.

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

(4)

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

(5)

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

(6)

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

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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 switched back 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 case archive for distribution feeders [17]. The diagram of the feeder is shown in Fig. 2. The regulator was removed in order to clearly see the effect of the DG on the system. The data are characterized by

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

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

The system was represented and renumbered as in Fig. 4 so that the analysis could be done using the developed unbalanced power 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-phase cables; Transformer—Inline transformer is delta-delta with tap ratio of one.

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

TABLE II MODIFIED IEEE 37–NODE VOLTAGES

IV. RESULTS

A. Load Flow

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

1) Results of the original feeder (without DG) obtained by the developed software and those obtained by RDAP are compared. 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 load flow took one iteration to converge. RDAP can only handle modeling 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 node SPS are shown in Table III. Node 1 is slack node, and nodes 2–4 are modeled as PV nodes. The generation values are shown in Table IV. The results were compared to results in [14]. These results are obtained with 0.001 p.u. tolerance for load flow as well as PV node convergence. The cables are short, and even a very small voltage difference leads to

TABLE III SPS VOLTAGES AND CURRENTS

TABLE IV SPS GENERATION

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

B. System Studies

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

This highlights the variation in results depending on the DG model.

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 loads in 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%, which corresponds to an increase of loading in steps of 5% from 5% to 50%, respectively

4) Fig. 5(a)–(c) shows a comparison of the voltage deviation from 1 p.u. for different DG models and varying DG pen- etration. The x-axis of Fig. 5(a)–(c) represents the node number, the y-axis of Fig. 5(a) and (b) represents the per- centage DG penetration, the y-axis of Fig. 5(c) represents the 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 constant PQ node. The surface plot of Fig. 5(a) clearly indicates the voltage deviation is low for the downstream nodes of the feeder. Voltage deviation is high for nodes 744 and 728 because load at 728 is a three-phase load, and hence,

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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 DG modeled as a PV node that switches to PQ node in case of a reactive power limit hit. The same observations can be made for surface plot of Fig. 5(b). The minimum voltage deviation 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 is not much. However, the difference in voltage deviation for downstream nodes during high DG penetration is consid- erable. Fig. 5(c) is obtained with the DG modeled as PV node and constant penetration, i.e., 50%, which switches to PQ node in case of a reactive power limit hit. However, there was no reactive power limit hit for the load increases shown. The surface plot of Fig. 5(c) clearly indicates that the voltage deviation is low for the downstream nodes of the feeder, but it increases with an increase in loading. The minimum voltage deviation is 0.0016 p.u. Based on these observations, for an increase in load, the voltage deviation

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

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 case of a limit hit.

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

6) Fig. 5(d) shows the system loss comparison for all these cases. Losses for cases 1–3, which correspond to no DG penetration, are quite high. Losses for cases 4–6, which correspond to the PQ model, are slightly higher than cases 7–9, which correspond to the PV model. For 50% DG pen- etration, losses are much less. Losses for case 12 are more than that of cases 10 and 11 due to an increase in loading.

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

TABLE V CASE SCENARIOS

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

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

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

8) The power flow took one iteration to converge for all cases where the DG was modeled as a PQ node, whereas a max- imum of three iterations were required for all cases where DG 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 incorporation of DGs modeled as either PV or PQ nodes. The algorithm was tested on both an IEEE 37-node test feeder and an 18-node icebreaker shipboard power system. Comparing the results of the unbalanced power flow without DG with RDAP, an established software package, demonstrates the accuracy of the algorithm for an established test case. Advanced studies on the IEEE 37-node test case with DG demonstrate the impact of DG model type, size, and load variations on the results.

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

TABLE VI MODIFIED IEEE 37-NODE CURRENTS

[19], will provide additional opportunities to study the impact of 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 Florida State University for the icebreaker shipboard power system data.

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approach,” IEEE Trans. Power Del., vol. 6, no. 3, pp. 1146–1152, Jul. 1991.

[2] R. D. Zimmerman and H. D. Chiang, “Fast decoupled power flow for unbalanced radial distribution systems,” IEEE Trans. Power Syst., vol. 10, no. 4, pp. 2045–2052, Nov. 1995.

[3] J.-H. Teng, “A direct approach for distribution system load flow solu- tions,” IEEE Trans. Power Del., vol. 18, no. 3, pp. 882–887, July 2003.

[4] W. M. Lin et al., “Three-phase unbalanced distribution power flow so- lutions with minimum data preparation,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1178–1183, Aug. 1999.

[5] Y. J. Jang and J. K. Park, “Three-phase power flow method based on fast-decoupled method for unbalanced radial distribution system,” [On- line]. Available: http://eeserver.korea.ac.kr/~bk21/arch/bk21conf/54. pdf.

[6] J.-H. Teng, “A network-topology-based three-phase load flow for dis- tribution systems,” Proc. Nat. Sci. Council, vol. 24, no. 4, pp. 259–264, 2000.

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

[8] H. M. Mok et al., “Power flow analysis for balanced and unbalanced radial 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 by method 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. Boca Raton, FL: CRC, 2002.

[11] C. S. Cheng and D. Shirmohammadi, “A three-phase power flow method for real-time distribution system analysis,” IEEE Trans. Power Syst., vol. 10, no. 2, pp. 671–679, May 1995.

[12] S. Khushalani and N. N. Schulz, “Unbalanced distribution power flow with distributed generation,” in Proc. IEEE Transmission and Distri- bution Conf., Dallas, TX, May 2006.

[13] M. M. Medina, L. Qi, and K. L. Butler-Purry, “A three-phase load flow algorithm for shipboard power systems (SPS),” in Proc. IEEE Power Eng. Soc. Transmission and Distribution Conf. Expo., Sep. 2003, vol. 1, pp. 227–233.

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[15] W. H. Kersting, Distribution System Modeling and Analysis. Boca Raton, FL: CRC, 2002.

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[17] Radial Distribution Test Feeders. [Online]. Available: http://www.ewh. ieee.org/soc/pes/dsacom/testfeeders.html.

[18] WH Power Consultants, RDAP User Manual, ver. 3.0. Las Cruces, NM, Sep. 1999. [Online]. Available: http://www.zianet.com/whpower.

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

She is now an engineer working for Open Systems International, Minneapolis, MN. She was involved in 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. degree from V.N.I.T., Nagpur, India, in 1998, the M.E. degree from Mumbai University, Mumbai, India, in 2000, and the Ph.D. degree from the Electrical and Computer Engineering Department of Mississippi State University, Mississippi State, MS, in 2006.

He is now an engineer working for Open Systems International, Minneapolis, MN. He was involved in research activities at IIT Bombay, Bombay, India. His research interests are power system analysis and its control.

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

She has been an Associate Professor in the Electrical and Computer Engineering Department at Mississippi State University, Mississippi State, MS, since July 2001. Her research interests are in computer 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 the IEEE Power Engineering Society and is serving as Secretary for 2004–2007. She was the 2002 recipient of the IEEE/PES Walter Fee Outstanding Young Power Engineer Award.

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Project 1/New_Load_files_for Project1.csv

Load for Node 634abc (kW) pf=0.928477,"Multiplier for load at node 671abc Base Load= 1500kW, 1600 kVar","Multiplier for load at node 645b Base Load= 370kW,125 kVar","Multiplier for load at node 646bc Base Load= 230kW,132 kVar","Multiplier for load at node 692ac Base Load=170kW,1151 kVar","Multiplier for load at node 675abc Base Load_a=485kW,190 kVar, Base Load_b=68kW,660 kVar, Base Load_c=290kW,1212kVar","Multiplier for load at node 611c Base Load=170kW,80 kVar","Multiplier for load at node 652a kW=128, kVar=86","Multiplier for load at node 670abc Base Load_a=170kW,10 kVar, Base Load_b=150kW,738 kVar, Base Load_c=117kW,468kVar" 518.7029445,0.345,0.345,0.345,0.345,0.345,1,0.345,0.345 529.746312,0.353,0.353,0.353,0.353,0.353,1,0.353,0.353 537.7013205,0.358,0.358,0.358,0.358,0.358,1,0.358,0.358 553.478631,0.368,0.368,0.368,0.368,0.368,1,0.368,0.368 554.2211805,0.369,0.369,0.369,0.369,0.369,1,0.369,0.369 572.8086,0.381,0.381,0.381,0.381,0.381,1,0.381,0.381 621.3382245,0.414,0.414,0.414,0.414,0.414,1,0.414,0.414 689.3123295,0.459,0.459,0.459,0.459,0.459,1,0.459,0.459 638.77359,0.425,0.425,0.425,0.425,0.425,1,0.425,0.425 611.3455845,0.407,0.407,0.407,0.407,0.407,1,0.407,0.407 703.1069445,0.468,0.468,0.468,0.468,0.468,1,0.468,0.468 582.5666895,0.388,0.388,0.388,0.388,0.388,1,0.388,0.388 567.7961535,0.378,0.378,0.378,0.378,0.378,1,0.378,0.378 530.762031,0.353,0.353,0.353,0.353,0.353,1,0.353,0.353 514.4988795,0.342,0.342,0.342,0.342,0.342,1,0.342,0.342 517.668039,0.345,0.345,0.345,0.345,0.345,1,0.345,0.345 532.3037025,0.354,0.354,0.354,0.354,0.354,1,0.354,0.354 671.613033,0.447,0.447,0.447,0.447,0.447,1,0.447,0.447 594.235743,0.396,0.396,0.396,0.396,0.396,1,0.396,0.396 571.6887195,0.381,0.381,0.381,0.381,0.381,1,0.381,0.381 559.121889,0.372,0.372,0.372,0.372,0.372,1,0.372,0.372 552.6093885,0.368,0.368,0.368,0.368,0.368,1,0.368,0.368 541.1278125,0.36,0.36,0.36,0.36,0.36,1,0.36,0.36 533.0840775,0.355,0.355,0.355,0.355,0.355,1,0.355,0.355