Power Flow Analysis for Balanced andUnbalanced Radial Distribution Systems

H.M. Mok and S.Elangovan M.M.A. Salama Dr Cao LongjianNational University of Singapore University of Waterloo Power Automation Pte Ltd

Abstract

This paper reports on an efficient method of power flow analysis for solvingbalanced and unbalanced radial distribution systems. The radial distributionsystem is modelled as a series of interconnected single feeders. Using Kirchoff’slaws, a set of iterative power flow equations was developed to conduct the powerflow studies. Due to the voltage dependency of loads in distribution systems,various static load models are incorporated in the power flow algorithm to obtainbetter and more accurate results. The computer software implemented in thispower flow method was then developed using C++ and successfully applied toseveral practical radial distribution networks. The convergence patterns of thispower flow method for the various load models were also analysed. In addition,the effects of system characteristics on the convergence of the method arehighlighted.

1.0 INTRODUCTION

Power flow analysis is a very important and basic toolfor the analysis of any power system as it is used inthe planning and design stages as well as during theoperational stages. Some applications, especially inthe fields of optimization of a power system anddistribution automation, need repeated fast power flowsolutions. In these applications, it is imperative thatthe power flow analysis is solved as efficiently aspossible. With the invention and widespread use ofdigital computers in 1950s, many methods for solvingthe power flow problem have been developed such asindirect Gauss-Seidel (bus admittance matrix), directGauss-Seidel (bus impedance matrix), Newton-Raphson and its decoupled versions[1]. However,these algorithms have been designed for transmissionsystems, and therefore their application to thedistribution systems usually does not provide goodresults and very often, the solution diverges[1]. One ofthe reasons why these methods are unsuitable fordistribution systems is that they are mostly based onthe general meshed topology of a typical transmissionsystem whereas most distribution systems have aradial or tree structure. Another reason is due to thehigh R/X ratio of distribution systems. This is a factorwhich causes the distribution systems to be ill-conditioned for conventional power flow methods,especially the fast-decoupled Newton method, whichdiverges in most cases. Lastly, the active and reactiveloads of a distribution system are dependent on the busvoltage. However, most of the conventional powerflow methods for both transmission and distributionsystems, consider power demands as specifiedconstant values. The constant power load model ishighly questionable and it is so especially for adistribution system because the bus voltages are not

controlled. Therefore, there is a need for a power flowmethod that takes this aspect into consideration toobtain better and more accurate results.Though considerable efforts has been directed to thedevelopment of solution algorithm for power flowanalysis of transmission systems with great success[1], in contrast, comparatively fewer solutionalgorithms have been developed for power flowanalysis of distribution systems. Therefore, thegrowing need of distribution companies for morecomplete studies and the increase in systemautomation have motivated the development of aspecialized algorithm for distribution systems thatconsider all their particular characteristics.

2.0 DISTRIBUTION SYSTEM MODELLING

For the purpose of power flow study, the radialdistribution system is modelled as a network of busesconnected by distribution lines or switches connectedto a voltage specified source bus. Each bus may alsohave a corresponding bus load, compensating load(shunt capacitor or inductor), lateral load and/orcogenerator connected to it.

2.1 Line ModelGenerally, the length of most distribution line sectionsis short (<80km). In addition, the voltage level ofdistribution systems is lower than that of transmissionsystems and hence the π model is not used becauseline shunt capacitance is found to be negligible.Instead, the short line model is used to represent theline sections for distribution systems whereby thedistribution line sections are modelled as seriesimpedances (Z=rl+jxl).

2.2 Bus LoadThis is the load drawn at a bus. The load model usedis a general model that allows each load to bemodelled as constant power, constant current, constantimpedance or exponential model. The variation of thereal and reactive power of a load with respect to thevoltage magnitude is given in Equations (1) and (2).

1koVPP = (1)

2koVQQ = (2)

(i) k1=k2=0 for constant power loads(ii) k1=k2=1 for constant current loads(iii)k1=k2=2 for constant impedance loads(iv)k1=1.38, k=3.22 for exponential loads

where Po and Qo represent the specified active andreactive powers at nominal voltage and V is the actualvoltage magnitude in per unit. In addition, k1 and k2need not be equal to each other because the real andreactive parts of the load can be modelled withdifferent voltage sensitivities. A more detaileddescription of static load models can be found inReference [2].

( )38.13

221 VaVaVaaPP oo +++= (3)

( )22.33

221 VbVbVbbQQ oo +++= (4)

where13210 =+++ aaaa (5)

13210 =+++ bbbb (6)

3.0 SOLUTION TECHNIQUE

3.1 Balanced Distribution SystemUsing the per phase equivalent circuit in Fig. 1, a setof power flow equations was derived to calculate thevoltages at bus ‘k’ on a lateral for the balanced radialdistribution network and it is given in Equation (7).

Fig. 1: Per phase equivalent circuit of a lateral

VS: Voltage of the source bus of the lateralVk: Voltage at the k th busIt : Total current drawn by the lateral

∑ ∑ ∑=

−

= +=

+−=

k

j

k

j

k

jiijjtsk ZIZIVV

1

1

1 1 (7)

3.2 Unbalanced Distribution SystemFor the three-phase power flow analysis of anunbalanced distribution system, the voltages at bus ‘n’for each phase on a lateral of an unbalanceddistribution system was derived similarly and is givenin (8-10). This is done by modelling the mutualimpedances as current controlled voltage sourcesalong the sections of the lateral as shown in Fig 2.

∑ ∑

∑+−

∑ ∑

∑+−

∑ ∑

∑+−=

=

−

= +=

=

−

= +=

=

−

= +=

n

k

n

k

n

kiacickacktc

n

k

n

k

n

kiabibkabktb

n

k

n

k

n

kiaaiakaaktaasan

ZIZI

ZIZI

ZIZIVV

1

1

1 1

1

1

1 1

1

1

1 1

(8)

∑ ∑

∑+−

∑ ∑

∑+−

∑ ∑

∑+−=

=

−

= +=

=

−

= +=

=

−

= +=

n

k

n

k

n

kibcickbcktc

n

k

n

k

n

kiabiakabkta

n

k

n

k

n

kibbibkbbktbbsbn

ZIZI

ZIZI

ZIZIVV

1

1

1 1

1

1

1 1

1

1

1 1

(9)

∑ ∑

∑+−

∑ ∑

∑+−

∑ ∑

∑+−=

=

−

= +=

=

−

= +=

=

−

= +=

n

k

n

k

n

kibcibkbcktb

n

k

n

k

n

kiaciakackta

n

k

n

k

n

kiccickccktccscn

ZIZI

ZIZI

ZIZIVV

1

1

1 1

1

1

1 1

1

1

1 1

(10)

Fig 2:Three phase equivalent circuit of a lateral

3.3 Iterations on a single feederThe approach used is iterative and is based on the factthat, given the voltage at one end of a distributionfeeder and the complex load demands at each bus onthe feeder, it is possible to calculate the voltages andcurrents at each of the buses. To illustrate the aboveformulation, a single feeder in Fig.3 is considered.Firstly, the voltage of each bus is initialized to Vs, the

voltage specified at the source bus which is usually1.0∠0.0° p.u. Since the loads at each of the buses areknown, using equation (11), the currents due to busload, compensating loads, cogeneration and/or lateralloads at every bus of the lateral are computed andsummed up so that the currents drawn at each bus canbe calculated. Then by summing up the individualbus currents, It can be obtained.

Using the power flow equation from (7), the voltage ateach bus of the feeder can be calculated. With thenew voltages, the various types of loads at each busare updated using Eq. (3) and (4) depending on theload modelling constants of each load. Using the newupdated loads and voltages, the currents arerecalculated so as to obtain the next set of voltages.The new set of voltages are then compared with theprevious set and if the absolute differences are allwithin a certain tolerance as in (12), the power flowsolution of the single feeder is obtained, otherwise theprocess above is repeated until (12) is satisfied.

Fig.3: A single feeder

∗

=

n

nn V

SI (11)

ε<− −1ii VV (12)

For the three-phase power flow analysis of unbalanceddistribution systems, the iterations on a single feeder issimilar to that of the balanced load flow study; thedifference being only that it is conducted on phases‘a’, ‘b’ and ‘c’ respectively for each feeder and thepower flow equations (8-10) are used instead.

3.4 Power flow AlgorithmA distribution system is made up of more than just onesingle feeder. Most distribution systems have severalfeeders or laterals connected in a radial configurationwith the main feeder connected to the source of thenetwork. There will be laterals branching off the mainfeeder and the sub-laterals will tap off from them.Firstly, all the buses are initialized to the voltagespecified at the source bus of the network which isusually 1.0∠0° p.u. A backward sweep is done fromthe highest level of laterals to the main lateral tocalculate the total lateral load drawn by each lateral.This is done by summing the individual loads drawn ateach bus of the lateral and the losses along thesections from the end bus of the lateral up to thesource bus of the lateral as shown in Fig. 4 by usingequations (13-15).

losskrksk SSS += (13)

loadkskrk SSS += +1 (14)

k

r

rklossk Z

V

SS

2

2

= (15)

Fig.4: The short line model of a branch on a lateral

Initially, the losses along the sections were calculatedbased on the bus voltage of 1.0∠0° pu. However, thelosses usually account for less than 5% of the systemdemand, therefore the initial error incurred isinsignificant.The aggregated lateral load modelling constants foreach total lateral load drawn by each lateral are thencalculated based on the load modelling constants andload weightage of each individual bus on thecorresponding lateral. The lateral currents on the mainfeeder (level n lateral, where n=1 for the main feeder)are calculated and the iterative technique in theprevious section is performed on the main feeder. Forthe next higher level of laterals, the bus voltages oneach of these laterals are initialized to the bus voltageof the source node of the lateral. The bus currents arerecalculated using equation (11) and iterations are thenperformed on these laterals. This is repeated untiliterations are performed on the highest level oflaterals.With the new set of bus voltages, the backward sweepis done again to recalculate the lateral loads.Subsequently, a loop is performed until convergenceis achieved. The bus voltages are first stored in adummy array and iterations are performed from themain feeder to the highest level of laterals. Thebackward sweep is then executed and the lateralcurrents recalculated. The new set of bus voltages isthen compared to the previous one stored in thedummy array and if the magnitude of difference forevery bus in the system is within the specifiedtolerance, convergence is achieved and the power flowsolution is obtained.

4.0 VALIDATION OF RESULTS

The power flow algorithm described above wasimplemented using Borland C++ builder as aWindows 98 software named ‘RadDisFlow’. Toensure the validity of the results obtained, test resultson various radial distribution systems was validatedagainst results obtained using an existing power flow

…

Vs

Ssk …

Vr

Slossk

Srk Ssk+1

Sloadk

Srk-1

Sloadk-1

SourceBus

ItVs Vn

EndBus

software, ERACS, which includes the NewtonRaphson, Gauss Seidel and Fast- Decoupled methods,was used for the validation process. For the balancedpower flow analysis, the networks used were themodified IEEE 13-bus system from [3], the 34-bussystem from [4] and a self constructed 98-bus system.A comparison was done and the results obtained werein agreement with the Newton Raphson and GaussSeidel method. The Fast-Decoupled method divergeddue to the high R/X ratio of the network. For thethree-phase power flow analysis, the results of apower flow study on the IEEE 13-bus Radial TestFeeder from [3] was compared those obtained usingthe Newton Raphson method. It was found that thedifferences between the two sets of results wereinsignificant. Hence, the accuracy and validity of thenew method of power flow analysis for balanced andunbalanced radial distribution system is confirmed

4.1 Comparison of PerformanceTo gauge the performance of RaDisFlow, it iscompared with the conventional methods (GaussSeidel, Newton Raphson and Fast Decoupled) usingERACS. First, a 11kV,29-bus radial distributionsystem is used. The power flow solution using theGauss Seidel and Fast Decoupled method diverged.As for the Newton Raphson method, the solution wasobtained in 1.0 sec after 3 iterations. WhenRaDisFlow was used, the power flow solution wasobtained in 0.032 sec after 3 iterations. ThoughRaDisFlow took the same number of iterations as theNewton Raphson method, it is 28.57 times faster. Inaddition, the memory requirement of proposed methodis only 35.3% of the Newton Raphson method.Hence, it can be seen that the performance ofRaDisFlow is more superior to the Newton Raphsonmethod when solving the radial distribution system. Itcan also be noticed that the convergencecharacteristics of RaDisFlow are better than the GaussSeidel and Fast Decoupled methods. The performanceof RaDisFlow was also measured against more recentmethods of power flow study for distribution systemsin [5] and [6] using a 11kV, 98-bus radial distributionsystem. It was found that it is 2.39 times faster thanthe method in [6] and took 2 iterations while themethod in [6] took 6 iterations. When the comparedto the method in [5], it is 2.17 times faster and took 3iterations less.For the three-phase power flow analysis, the 37-bussystem from [3] was used. Table 1 shows the resultsof the comparison.

Table 1. Results of the ComparisonMethods of Power

Flow AnalysisNumber ofiterations

Computation Time/seconds

Newton Raphson 3 1.84Fast Decoupled Diverged DivergedImplicit Zbus Gauss 5 4.35New Method 3 0.31

4.2 Effects of Load ModellingThis paper also compares the power flow results of adistribution system for the different voltage dependentload models. The 34-bus radial distribution systemfrom [4] was used to examine the effects of variousload. The load models used are the constant-power,constant-current, constant impedance and theexponential model. The voltage profiles obtained forthe 4 different load models are given in Fig. 1. It canbe observed that the constant-power load model hasthe lowest voltage profile while the constant-impedance load model gives the highest voltageprofile. The total active and reactive power suppliedby source bus of the system and the system losses forthe various load models are shown in Table 1. Thereis a substantial reduction in the total load demand aswell as losses of the system for the other three loadmodels. The reduction of the system load and lossesis due to the lower voltage profile of the other loadmodels. It can be seen that the use of the variousstatic load models have a significant effect on thepower flow results.

4.3 Convergence AnalysisThe convergence characteristic of the new method ofpower flow analysis was examined using the 34-busradial distribution system distribution system in [4] forthe 4 different load models. The convergence patternof the power flow method for the various static loadmodels is shown in Fig. 2. The solution converges inthe shortest number of iterations for the constant-power load model. For the other 3 load models, themethod required a slightly higher number of iterationsto converge because the load values were adjustedduring each iteration when the voltage magnitudeschange. The power flow analysis of the system wasalso performed by modelling the active and reactivepowers of the load demand separately as shown inEquations (1) and (2). The number of iterationsneeded to perform the power flow study fordifferent values of k1 and k2 are given in Table 2. Itcan be observed that the convergence characteristic ofthe power flow method is very much dependent on thevoltage sensitivity of the real and reactive loads of thenetwork. The more sensitive the loads are to the busvoltages, the longer it takes for the solution toconverge.The convergence property of the power flow methodwas also examined in terms of the following systemparameters:-

(a) Number of buses(b) Percentage loading of the system(c) Number of levels of laterals(d) The R/X ratio of the conductors

Table 2: System load and losses

Const P&Q Const Current Const Z Exp Model

(Base Case) % diff % diff % diff

Total P Supplied 1.005428 0.931942 -7.308 0.873564 -13.115 0.906735 -9.816

Total Q Supplied 0.597567 0.555588 -7.024 0.52204 -12.639 0.481882 -19.359

Total P Loss 0.098724 0.083383 -15.539 0.072228 -26.838 0.073743 -25.303

Total Q Loss 0.031317 0.026532 -15.279 0.023041 -26.426 0.023515 -24.912

Fig. 5: Voltage Profiles for the various Load Models

The convergence analysis on this power flow methodshows that its convergence property is neither affectedby the number of buses nor the R/X ratio of thedistribution system. However, the method would takelonger to converge for more heavily loaded systems aswell as for systems whereby there are more than 5levels of laterals. In addition, using different loadmodels also affect the convergence of the method.The more sensitive the load models are to the busvoltage, the slower the rate of convergence.

Table 3: Number of iterations to perform the powerflow analysis for different values of k1 and k2

k1=0.0 k1=1.0 k1=2.0 k1=1.38k2=0.0 2 3 3 -k2=1.0 2 3 3 -k2=2.0 3 3 3 -k2=3.22 - - - 4

Fig. 6: Convergence Patterns for various load models

00.010.020.030.040.050.060.070.080.090.1

0.110.120.130.140.150.16

1 2 3 4

Iteration Number

Max

Vo

ltag

e M

ism

atch

(p

.u.)

Const PQ

Const I

Const Z

Exp

Voltage Magnitude vs Bus Number

0.890000

0.900000

0.910000

0.920000

0.930000

0.940000

0.950000

0.960000

0.970000

0.980000

0.990000

1.000000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33

Bus Number

Vo

ltag

e M

agn

itu

de

Const P&Q

Const Current

Const Z

Exp Model

5.0 CONCLUSION

In this paper, a new method of power flow analysis forsolving balanced and unbalanced radial distributionsystems has been developed and successfullyimplemented. First, the radial distribution system ismodelled as a series of interconnected individualsingle feeders and a set of power flow equations isderived. After that, the solution method is developedfor solving the power flow study. The validity of theresults of RaDisFlow has also been verified bycomparing them with the existing methods of powerflow analysis. The proposed method offers a veryattractive combination of advantages over the existingmethods including the Newton Raphson method,Gauss Seidel method and the Fast-Decoupled methodin terms of convergent characteristic, computationtime, simplicity and memory storage requirements. Inaddition, its performance was also found to be betterthan the other recent methods of power flow analysisdeveloped for distribution systems. A more accurateprediction of the power system performance can alsobe obtained with the incorporation of static loadmodels in the power flow study. The constant-powerload model gave the most conservative results, withthe lowest voltage profile and the highest load demandas opposed to the constant-impedance load model.Hence, it can be seen that accurate modelling of theload is necessary for a good distribution systemdesign.The convergence analysis on this power flow methodshows that its convergence property is neither affectedby the number of buses nor the R/X ratio of thedistribution system. However, the method would takelonger to converge for very heavily loaded systems aswell as for systems whereby there are more than 5levels of laterals. In addition, using different loadmodels also affect the convergence of the method.The more sensitive the load models are to the busvoltage, the slower the rate of convergence.The power flow method proposed in this paper couldbe improved further in several ways. The power flowstudy in RaDisFlow is limited to radial distributionsystems. Therefore, the present power flow studycould be extended to solve weakly meshed distributionsystems using a compensation method similar to thosepresented in [5] and [7]. The load models that areconsidered in RaDisFlow are static load models, theuser has to specify the load modelling constants atevery bus and this type of information might not beeasily available. Therefore, an expert system could bedeveloped to determine the load modelling constantsfor different kinds of loads (ie motor loads, lighting,heating etc). To further improve the performance ofthe power flow solution using RaDisFlow, thepossible use of multi-processors to do parallelcomputations for different laterals of distributionnetwork can be considered as in [8]

6.0 REFERENCES

[1] B.Scott, “Review of Load-Flow CalculationProc, Vol. 2, No.7, Jul 1974, pp 916-

929.

[2] IEEE Task Force on Load Representation forDynamic Performance, “ Bibliography on LoadModels for Power Flow and Dynamic PerformanceSimulation”, IEEE Trans., Power Systems, Vol. 10,No. 1, Feb 1995, pp523-538.

[3] IEEE Distribution Planning Working GroupReport, “Radial Distribution Test Feeders”, IEEEPES, Vol. 6, No. 3, Aug 1991, pp975-985.

[4] M.M.A. Salama, A.Y. Chikhani, “A SimplifiedNetwork Approach to the VAR Control Problem forRadial Distribution Systems”, IEEE Trans. PowerDelivery, Vol. 8, No.3, Jul 1993, pp1529-1535.

[5] G.X. Luo, A. Semlyen, “Efficient Load Flow forLarge Weakly Meshed Networks”, IEEE Trans. PowerSystems, Vol.5, No.4, Nov 1990, pp1309-1316,

[6] C.G. Renato, “New method for the analysis ofdistribution networks, IEEE Trans. Power Delivery,Vol.5, No.1, Jan 1990, pp391-396.

[7] D. Shirmohammadi, H.W. Hong, A. Semlyen,G.X. Luo, “A Compensation-Based Power FlowMethod for Weakly Meshed Distribution andTransmission Networks”, IEEE Trans. PowerSystems, Vol.3, No.2, May 1988, pp753-762.

[8] Y. Fukuyama, Y. Nakanishi, H.D. Chiang, “FastDistribution Power Flow using Multi-Processors”,Electrical Power & Systems, Vol. 18, No.5, June1996, pp331-337.