04162603.pdf

624 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 2, MAY 2007

Optimized Restoration of UnbalancedDistribution Systems

Sarika Khushalani, Student Member, IEEE, Jignesh M. Solanki, Student Member, IEEE, andNoel N. Schulz, Senior Member, IEEE

Abstract—A novel formulation for service restoration algorithmfor unbalanced three phase distribution systems is described. Thisproblem is a constrained multiobjective optimization formulatedas a mixed integer non-linear programming problem. A compar-ison of the solutions with and without switch pairs has been made.The formulation was first validated using already developed three-phase unbalanced power flow software. The three-phase unbal-anced power flow equations were embedded in the formulation,and hence separate calculations were not needed. Simulation re-sults are presented for modified IEEE 13-node and IEEE 37-nodetest cases.

Index Terms—Load flow analysis, optimization methods, powerdistribution, power distribution control, power system restoration,shipboard power system.

I. INTRODUCTION

WITH the advent of fast computers and changing tech-nology, there is a surge of interest in the field of dis-

tribution automation. Service reliability and increased customersatisfaction are the major areas where most of the efforts arefocusing at present. This paper enhances these efforts by thedevelopment of restoration schemes for an unbalanced distri-bution system. Reconfiguration is the process of changing theopen/closed status of switches and is done for volt/var support,loss reduction, load balancing and restoration. Reconfigurationfor restoration is a combinatorial problem involving searchingan enormous space of solutions. The problems with integer vari-ables are NP hard, meaning no known algorithm exists to solvethese problems in polynomial time. However, reconfigurationfor restoration problem is both NP hard and NP and hence be-longs to the class of NP complete problems. For such kind ofproblems, the solution time increases with an increase in thenumber of integer variables. However, the solution time gener-ally depends on the formulation. The complexity of the problemcan be reduced by reducing the number of integer variables. Thisreduction can be achieved by formulating problem without in-

Manuscript received November 12, 2006; revised February 13, 2007.This work was supported by the Office of Naval Research under GrantsN00014-02-1-0623 and N00014-03-1-0744. Paper no. TPWRS-00464–2006.

S. Khushalani and J. M. Solanki are with Mississippi State Univer-sity, Mississippi State, MS 39762 USA (e-mail: [email protected];[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 onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2007.894866

teger restrictions and later rounding it, but this rounding mayresult in a suboptimal or infeasible solution.

This problem has been approached using heuristics [1]–[3],mathematical programming [4], [5], meta-heuristics (genetic al-gorithms, tabu search, simulated annealing) [6]–[8] and expertsystems [9], [10]. A combination of approaches [11], [12] hasalso been used. However, most of the approaches use resistivemodels of loads and lines and simplify the distribution system.In practice, a distribution system is normally unbalanced dueto varied customer demands, un-transposed and mutually cou-pled cables and lines. Reference [13] attempts service restora-tion for unbalanced distribution system using hybrid flow pat-tern, a heuristic strategy based on series of switch operations andsolves power flow only once. Other [14]–[17] do reconfigurationfor loss minimization or load balancing or both, for unbalanceddistribution systems using one or several power flow solutions.

This paper formulates the reconfiguration for restorationproblem as a multi-objective mixed integer non-linear pro-gramming problem. Distribution systems can be in normal,faulted or restored operating states. After fault isolation, someof the un-faulted sections of the system may be left unsupplied.Restoration will attempt to restore most, or if possible, all ofthese loads. Thus, in a restored state the equality constraintsmay be satisfied, whereas some of the inequality constraintsmay not be satisfied. The equality constraints are the threephase unbalanced power flow equations; the inequality con-straints are the operation restrictions imposed by the equipment.The important aspects of this formulation are that a separatethree-phase unbalanced power flow calculation is not required,equations are not linearized and mutually coupled cables areconsidered. Distribution systems have a meshed structure withnormally closed switches called sectionalizing switches andnormally open switches called tie switches. The operation ofdistribution systems is radial due to complications in protectioncoordination, and therefore, it is appropriate to keep the systemradial even after restoration. This radiality constraint is ex-plained in detail in Section II where the problem is formulated.Solution method is detailed in Section III. Section IV describesthe test systems and results obtained.

II. PROBLEM FORMULATION

This section presents two different formulations for theproblem. The first formulation uses real power and reactivepower and is highly nonlinear as compared to the secondformulation that utilizes voltage and current.

A. Formulation I

The outline of the first formulation is given below.

0885-8950/$25.00 © 2007 IEEE

KHUSHALANI et al.: OPTIMIZED RESTORATION OF UNBALANCED DISTRIBUTION SYSTEMS 625

Fig. 1. Test case.

TABLE IRESULTS FOR TEST CASE

Objective

(i)

subject toEquality Constraints1) .2)

.3)

.4) for fixed load.

Inequality Constraints5) , generator real power limit.6) , generator reactive power limit.7) , line limits.8) , voltage magnitude limits.9) , voltage angle limits.

where set of phases a, b, and c, and set of load nodes.and are the net active and reactive power injections in

phase of node . is the three-phase apparent power. isthe voltage of phase of node . and are the real andreactive parts of the 3 3 admittance matrix of branch betweennode and node , whereas is the difference in voltageangle between phases and of nodes and . is a bi-nary variable. is a weighting factor for vital load that isgreater that weighting factor for semi-vital load . Themodel was first tested on a small unbalanced system shown inFig. 1. The results are shown in Table I. When this formula-tion was used for restoration of systems with larger number ofnodes the solution failed to converge. This failure occurred be-cause the distribution systems are ill-conditioned systems, andthus the Jacobian matrix tends to become singular. A decouplingprocedure was also tried; it split the problem into two subprob-lems; and but due to high R/X ratio of the linesit still failed to converge. However, other available decouplingmethods [19], [20] show convergence even for R/X ratio of 2.

TABLE IILOAD MODEL

TABLE IIIRESULTS VERIFICATION FOR MODIFIED IEEE 13- NODE

TABLE IVRESTORATION OF MODIFIED IEEE 13-NODE SYSTEM

B. Formulation II

To counter the shortcomings of the previous algorithm, a newalgorithm was formulated. Component models for the distribu-tion system were needed.

1) Distribution System Line Model: The impedance of over-head lines and underground cables was calculated. Tables II andIII in [18] were used to calculate the impedance of overheadlines from the given phasing, space ID, material and stranding.Tables IV–VI in the same reference were used to calculate theimpedance of underground concentric or tape shield cables. Thephase admittance matrix does not have a significant contribu-tion, and hence it is neglected.

2) Load Model: One-, two-, or three-phase loads with wyeor delta connections can exist. Only constant current loads areconsidered where the magnitude of the load current is calculatedby the specified real and reactive power at nominal voltage andis kept constant. Table II shows the model equations.

626 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 2, MAY 2007

TABLE VRESULTS VERIFICATION FOR MODIFIED IEEE 37-NODE

TABLE VIRESTORATION OF MODIFIED IEEE 37-NODE SYSTEM

3) Distributed Load Model: Sometimes, the primary feedersupplies loads through distribution transformers tapped at var-ious locations along line section. If every load point is modeledas a node, then the system will have a large number of nodes.So, these loads are represented as lumped loads.

1) A dummy node is created at one-fourth length of line fromsending node where two-thirds of the load is connected.

2) One-third load is connected at the receiving node.Objective

(II)

subject toEquality Constraints

10)

.11) .

12) .

13) , for fixed load.

Inequality Constraints

14) , voltage magnitude limits.

15) , for variable load.

16) , line limits.

where is the closed switch, and is the open switch.is the load current flowing in phase of node and is theweighting factor, which is less than 1 so that maximization oftotal load is the priority. This part of the expression is neededto minimize switching for cases that have the same total load.

is the switch between nodes and . is the mutuallycoupled 3 3 impedance matrix of the branch between nodes

and . is a binary variable. is the set of branches withcurrents going into the node , and is the set of branches withcurrents going out of the node .

Equations (III) and (IV) are the governing equations ofvoltage at node in Fig. 2 that can be fed by two branchesdepending on which switch is closed. The second term on theRHS of (10) is added to (III) and (IV) in order to validate bothequations simultaneously

(III)

(IV)

The fault is detected and isolated before performing restoration.This is simulated by setting the corresponding andthereby . The variable slackcurr ensures that constraints(11) and (12) are valid when .

C. Switch Pairs

Two schemes, with the switch pair scheme (WSPS) andwithout the switch pair scheme (WOSPS), are considered. Forthe WSPS, an additional constraint (V) is introduced in theformulation (II)

(V)

where is the set of switches that make switch pairs. Noradiality constraint must be enforced, as the system is alwaysradial. For the WOSPS radiality constraint, (VI) is added toformulation (II).

(VI)

is the set of switches that result in a loop. WSPS involvesa search space of solutions smaller than WOSPS. However,WSPS fails to provide a restoration scheme for several faults as

KHUSHALANI et al.: OPTIMIZED RESTORATION OF UNBALANCED DISTRIBUTION SYSTEMS 627

Fig. 2. Illustration of switch pairs.

shown in Fig. 2. For a fault, F1, and SW15 and SW45 as pairs,the restoration path to L3 is through SW12-SW23-SW34-SW45and SW56. For the same switch pairs and fault at F2 there isno restoration path to L1 and L2. The WSPS scheme needs noadditional radiality constraint to be reinforced as the searchspace of solutions is smaller, but it fails to provide a restorationpath even if one exists.

The WOSPS scheme needs the reinforcement of radialityconstraint, so the search space of solutions is large but it pro-vides a restoration path for all faults if one exists. This shows aclear trade off between both schemes.

III. SOLUTION PROCESS

Formulation II requires the solution of a mixed integernon-linear optimization problem for which LINGO com-mercial optimization package from LINDO Systems Inc[19] is used. Branch-and-bound type techniques cannot bedirectly applied unless the problems are convex. LINGOhas a direct solver, a linear solver, a non-linear solver and abranch-and-bound manager. Integer restrictions in the probleminvoke a branch-and-bound manager, which in turn invokes alinear or nonlinear solver depending on the nature of the formu-lation. LINGO uses revised simplex method for its linear solverand successive linear programming, as well as a generalizedreduced gradient for its non-linear solver. LINGO can solveproblems with unlimited constraints and variables but cannothandle complex numbers so the problem is reformulated

subject toEquality Constraints

17)

.18)

.

19) .

20) .

21) .

22) .

23) .24) .25) .26) .

27) .

Inequality Constraints

28) , voltage magnitude limits;

29)and 0, for

variable load;30)

and 0, for

variable load;31) , for the line limits;

32) , for the WOSPS scheme;

33) , for the WSPS scheme;

where are the set of loads with positive real and imagi-nary parts and are those with negative parts. The resultsin Section IV validate this formulation.

IV. SIMULATION RESULTS

IEEE 13-node distribution system is considered and some as-sumptions are made.

1) The regulator is removed from the system.2) Capacitors are not modeled, and hence they are removed

from the system.3) A dummy node “7” is introduced, due to distributed load

modeling.4) Steps 1 and 2 lead to lower and unacceptable voltages in

the system. Load is reduced in order to bring the voltageswithin limits of 0.95 and 1.05.

5) All loads are considered as constant current loads.6) Normally-open tie-switches are introduced in the system

for creating restoration scenarios.Fig. 3 shows the one line diagram of modified IEEE 13-nodesystem, where the phases in each branch are indicated. As shownin Fig. 3, a switch is introduced in each branch. To minimizeclutter on the diagram, the switches are not directly labeledin Fig. 3. However in the text they are referred to using theirto/from nodes. For instance the normally open switch betweennodes 10 and 14 is called SW1014. There are three normallyopen switches and thirteen normally closed switches. Initiallythe optimization was run under unfaulted condition and for theconfiguration of switches as shown in Fig. 3. The solution ob-tained was compared with a developed unbalanced power flowprogram in MATLAB [20]. A comparison of voltage magni-tudes obtained in per unit and angles obtained in degrees fromdeveloped program, and the optimization solution is in Table III.These results demonstrate the validity of optimization formula-tion. Fault scenarios can now be simulated. If a fault exists on the

628 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 2, MAY 2007

Fig. 3. Modified IEEE 13-node system.

Fig. 4. Voltages after restoration for modified IEEE 13-Node system.

branch 8–9 and SW89 is opened, then load at node 9, which isa single-phase load, and load at node 10, which is a three-phaseload, are left without supply. The solution obtained after opti-mization is closing SW1014 in order to supply both loads. Afterrestoration for a fault at 2–3, the magnitude of current flowingin branch 3–4 is 0.5825, 0.2250, 0.2250 pu in phases a, b, and c.In order to show that the developed formulation can shed loadsif all of the loads cannot be supplied the limit of cable 3–4 was

Fig. 5. Modified IEEE 37-Node System.

reduced to 0.4 and partial shedding of the variable three- phaseload 4 occurs under the same fault. Table IV provides a resultcomparison of the two techniques for several fault simulationsincluding those switches whose status changes after restoration.NSF indicates that no solution was found.

Table IV shows that the time required to find a global optimalsolution for the WSPS scheme is much less than the time re-quired for the WOSPS scheme. However, there are several faultscenarios for which no feasible solution exists for the WSPSscheme. A plot of voltages for some critical nodes, after therestoration for the WSPS and WOSPS schemes are as shownin Fig. 4(a) and (b). The voltages are within the prescribed tol-erances of 0.95 and 1.05.

The IEEE 37-node distribution system, which is an actualdistribution system in California was also considered and someassumptions were made.

1) Regulator was removed from the system.2) All loads were considered as constant current loads.3) Normally-open tie-switches as shown in Fig. 5 were intro-

duced in the system for creating restoration scenarios.There are six normally open switches and thirty-six normally

closed switches. Initially the optimization is run under unfaultedcondition and for the configuration of switches as shown inFig. 5. As indicated for Fig. 3, the switches are not directly la-beled on the diagram but are referred to by their to/from nodes.The comparison of solution with developed unbalanced powerflow program in MATLAB is shown in Table V.

Fault simulations in several locations and the time to find aglobal optimal along with switches whose status change afterrestoration are shown in Table VI. A plot of voltages for somecritical nodes after the restoration are shown in Fig. 6. Since

KHUSHALANI et al.: OPTIMIZED RESTORATION OF UNBALANCED DISTRIBUTION SYSTEMS 629

Fig. 6. Voltages after restoration for IEEE 37 Node Case.

TABLE VIISIZE OF OPTIMIZATION PROBLEM

all nodes in this test system are three phase nodes, the positivesequence voltage is plotted.

For this test case, only the switches downstream of the faultedbranch are considered as variables along with normally openswitches. Also only the WOSPS scheme was used. Thus, thesearch space varies from a minimum of six to a maximum of 41integer variables as shown in Table VII. Table VII also showsthe number of linear, non-linear and integer variables and con-straints for both the test systems. The simulation results pre-sented were obtained by 2 GHz Pentium® 4 PC. Accuracy of theresults illustrates that this formulation is valuable for restorationof unbalanced distribution systems.

V. CONCLUSIONS

The service restoration problem is formulated for three phaseunbalanced distribution systems. The equations are non- linearwith integer variables, making it a hard problem to solve. Globaloptimal solution was obtained under several fault cases for theIEEE-13 and 37 node distribution systems which demonstratethe accuracy of the formulation. A comparison was made be-tween WSPS and WOSPS schemes and a trade off was indi-cated. The formulation does not require separate load flow cal-culations and gives a complete solution of three phase voltagesand currents with best switching configuration. Efforts are underway to optimize the formulation for further reduced timings.

REFERENCES

[1] A. L. Morelato and A. J. Monticelli, “Heuristic search approach to dis-tribution system restoration,” IEEE Trans. Power Delivery, vol. 4, pp.2235–2241, Oct. 1989.

[2] J. S. Wu, K. L. Tomsovic, and C. S. Chen, “A heuristic search approachto feeder switching operations for overload, faults, unbalanced flowand maintenance,” IEEE Trans. Power Delivery, vol. 6, pp. 1579–1586,Oct. 1991.

[3] D. Shirmohammadi, “Service restoration in distribution networks vianetwork reconfiguration,” IEEE Trans. Power Delivery, vol. 7, pp.952–958, Apr. 1992.

[4] K. L. Butler, N. D. R. Sarma, and R. Prasad, “Network reconfigurationfor service restoration in shipboard power distribution systems,” IEEETrans. Power Systems, vol. 16, pp. 653–661, Nov. 2001.

[5] T. Nagata and H. Sasaki, “An efficient algorithm for distribution net-work restoration,” in Proc. IEEE Power Eng. Soc. Summer Meeting,Jul. 2001, vol. 1, pp. 54–59.

[6] S. Toune, H. Fudo, T. Genji, Y. Fukuyama, and Y. Nakanishi, “A reac-tive tabu search for service restoration in electric power distributionsystems,” in Proc. IEEE Int. Conf. Evolutionary Computation, May4–9, 1998, pp. 763–768.

[7] Y. Hsiao and C. Chien, “Enhancement of restoration service in distri-bution systems using a combination fuzzy-GA method,” IEEE Trans.Power Syst., vol. 15, pp. 1394–1400, Nov. 2000.

[8] S. Toune, H. Fudo, T. Genji, Y. Fukuyama, and Y. Nakanishi, “Com-parative study of modern heuristic algorithms to service restorationin distribution systems,” IEEE Trans. Power Delivery, vol. 17, pp.173–181, Jan. 2002.

[9] C. Chao-Shun, C.-H. Lin, and T. Hung-Ying, “A rule-based expertsystem with colored petri net models for distribution system servicerestoration,” IEEE Trans. Power Systems, vol. 17, pp. 1073–1080, Nov.2002.

[10] K. L. Butler, J. A. Momoh, and L. G. Dias, “Expert system assistedidentification of line faults on delta-delta distribution systems,” in Proc.35th Midwest Symp. Circuits and Systems, Aug. 9–12, 1992, vol. 2, pp.1208–1213.

[11] Q. Zhou, D. Shirmohammadi, and W. Liu, “Distribution feeder re-configuration for service restoration and load balancing,” IEEE Trans.Power Syst., vol. 12, pp. 724–729, May 1997.

[12] C. S. Chen, C. H. Lin, C. J. Wu, and M. S. Kang, “Feeder reconfigura-tion for distribution system contingencies by object oriented program-ming,” in Proc. IEEE Power Eng. Soc. Summer Meeting, Jul. 16–20,2000, vol. 1, pp. 431–436.

[13] L. Jiansheng, D. Youman, H. Ying, and Z. Boming, “Network reconfig-uration in unbalanced distribution systems for service restoration andloss reduction,” in Proc. IEEE Power Eng. Soc. Winter Meeting, Jan.23–27, 2000, vol. 4, pp. 2345–2350.

[14] J.-C. Wang, H.-D. Chiang, and G. R. Darling, “An efficient algorithmfor real-time network reconfiguration in large scale unbalanced distri-bution systems,” IEEE Trans. Power Syst., vol. 11, pp. 511–517, Feb.1996.

[15] V. Borozan, D. Rajicic, and R. Ackovski, “Minimum loss reconfigu-ration of unbalanced distribution networks,” IEEE Trans. Power De-livery, vol. 12, pp. 435–442, Jan. 1997.

[16] J. Zhu, M.-Y. Chow, and F. Zhang, “Phase balancing using mixed-in-teger programming,” IEEE Trans. Power Syst., vol. 13, pp. 1487–1492,Nov. 1998.

[17] M. E. Baran and F. F. Wu, “Network reconfiguration in distributionsystems for loss reduction and load balancing,” IEEE Trans. Power De-livery, vol. 4, pp. 1401–1407, Apr. 1989.

[18] W. H. Kersting, “Radial distribution test feeders,” in Proc. IEEE PESWinter Meeting, 2001, vol. 2, pp. 908–912.

[19] Release-9 LINDO Syst., Inc., Chicago, IL, 2003.[20] 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.

Sarika Khushalani (S’06) received the B.E. de-gree from Nagpur University and the M.E. degreefrom Mumbai University, India, in 1998 and 2000,respectively. She is currently pursuing the Ph.D.degree in the Electrical and Computer EngineeringDepartment, Mississippi State University.

She was involved in research activities at the In-dian Institute of Technology, Bombay. Her researchinterests are computer applications in power systemanalysis and power system control.

Ms. Khushalani is a Honda Fellowship Award re-cipient from Mississippi State University.

630 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 2, MAY 2007

Jignesh M. Solanki (S’06) received the B.E. degreefrom V.N.I.T., Nagpur, India, and the M.E. degreefrom Mumbai University, India, in 1998 and 2000, re-spectively. He is currently pursuing the Ph.D. degreein the Electrical and Computer Engineering Depart-ment, Mississippi State University.

He was involved in research activities at the IndianInstitute of Technology, Bombay. His research inter-ests are power system analysis and its control.

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. She received the Ph.D. in electricalengineering from the University of Minnesota,Minneapolis, in 1995.

She has been an Associate Professor in theElectrical and Computer Engineering Department,Mississippi State University, since July 2001 andholds the TVA Endowed Professorship in PowerSystems Engineering. Prior to that, she spent six

years on the faculty of Michigan Tech. Her research interests are in computerapplications in power system operations including artificial intelligencetechniques.

Dr. Schulz is a 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. She is a member of Eta Kappa Nu and Tau Beta Pi.