以電流注入為基礎之多頻三相解耦負載潮流模型 Current-Injection Based Decoupled Three-Phase Load Flow Model

for Multiple-Frequency Analysis

詹東昇 Tung-Sheng Zhan

高苑技術學院電機工程系 Department of Electrical Engineering

Kao-Yuan Institute of Technology

摘 要

本文發展出一套多頻三相負載潮流模型，其可分為基本頻率與諧波頻率兩個子模型。基頻

模型中線路、發電機與負載匯流排之模型皆以等效注入電流的觀念進行推導。諧波頻率模型中，

各種諧波源亦視為等效注入匯流排之諧波電流，其來源可由傅立葉分析或利用基頻模型預先執

行後所得之系統狀態參數，經由各諧波源之諧波模型而求得。經過本文之測試結果得以驗證本

文提出之負載潮流模型快速收斂與求解精確之優越性。

關鍵字:等效電流注入、基頻負載潮流、諧波負載潮流

Abstract

A multiple-frequency three-phase load flow model was developed in this paper. There are two

new sub-models including the fundamental power flow (FPF) and harmonic frequency power flow

(HPF) model. In FPF, models of electrical elements and PV buses were treated in the form of current

injections in a transmission system. The standard Fourier analysis was used to deal with the harmonic

loads to get injection currents. With harmonic currents as equivalent current sources, the HPF can be

derived. Besides, the decoupled fast version of FPF and HPF, called DFPF and DHPF, were also

proposed in this paper. Test results show that the proposed general-purpose methods are better

performers than conventional power flow solutions and are very robust.

Index Terms--Equivalent-Current-Injection (ECI), PV bus, Fundamental Power Flow (FPF), Harmonic

Power Flow (HPF), Decoupled Model.

1. Introduction

A typical study of the interaction between fundamental and harmonic frequencies was proposed by

Xu[1], based on the balanced power flow formulation. Recent studies [2]-[5] proposed many

three-phase harmonic power flow formulations and provided detailed insights in this field. Ref. [6]

further deals with the harmonic power flow using a fast-decoupled model in polar form.

The equivalent current-injection（ECI）concept was proposed to deal with the unbalanced

distribution system, and was extensively tested in [7]-[9], where loads are modeled by PQ buses, and

various formulations could be formed according to network parameters. A phase as well as real and

imaginary part decoupled Jacobian matrix could be obtained with six block-diagonal structured matrix

to yield a great performance [9]. Data requirement is also minimized where only the resistance R is

needed for the distribution network. The method is based on the widely accepted Newton-Raphson

algorithm with state variables in the Cartesian coordinates.

In a like manner, a reduce-sized formulation could be developed for the high-voltage transmission

system where the one-line representation is sufficient to model the balanced three-phase network. If

there is a need to analyze the high-voltage system in unbalanced form, a three-phase model will be

needed. However, the generator PV bus and HVDC transmission will have to be resolved first, where

the unspecified Q should cause problems for ECI convergence. Besides, with the possible scattered

electronic devices over the whole system, the proposed method also needs to resolve the harmonic

flow.

This paper tries to resolve the above-mentioned issues and propose a multiple-frequency

three-phase load flow formulation. The fundamental frequency model has first been developed and

then the harmonic model. The load and harmonic sources are represented by using the concept of ECI,

and the harmonic currents are then approximated by using the standard Fourier series and common

device models. For convenience, the fundamental model is called the fundamental power flow（FPF）

and the harmonic model is called the harmonic power flow（HPF）in this paper. The decoupled models

of the FPF（DFPF） and HPF（DHPF） were also proposed in this paper. All methods were tested to

show the effectiveness and robustness of the proposed algorithm.

2. Three-phase High Voltage Model

According to the standard Fourier analysis, the injection current of harmonic loads can be

formulated by

( )∑∞

=

++

++=

2

)(

1)1()(

sin2

)sin(2)(

mm

m

dc

tmI

tIIti

φω

φω (1)

The specified equivalent current injection in Cartesian plane, Isp-eq, of the fundamental and harmonic

components can be found as

≠+

=+−

=+ −−

1m , sincos

1m , )(

)(

)()(

*)1(

*

)()(

mm

mm

spsp

meqspI

meqspR

jII

VjQP

jIIφφ

(2)

where Psp and Qsp are the active and reactive power injections with fundamental frequency. V(1) is the

bus voltage phasor. For convenience, the proposed three-phase model will be discussed by one-line

representation in this paper, i.e., although the one-line model can be used directly for balanced

three-phase network, models of the unbalanced three-phase network can be developed in a similar form

[9].

Fundamental-Frequency Model A transmission line can be represented by the π-circuit as shown in Figure 1 with admittance

gst+jbst and shunt line charging susceptance bc.

VS

Vt

jbc jbc

gst+jbstIS It

PV Bus

G

Fig. 1 one-line diagram of transmission line and the PV Bus

According to the Newton-Raphson algorithm [9], the ECI mismatch equation can be written as

∆∆

∂∂

∂∂

∂∂

∂∂

=

∆∆

FE

FI

EI

FI

EI

II

II

RR

I

R

(3)

where , , and IR jIII += caleqspIR IIIjII −=∆+∆=∆ − FjEV ∆+∆=∆ at each iteration. The

Jacobian entries for Fig. 1 can be obtained by

'st

s

s,Rst

s

s,R bf

I , g

eI

−=∂∂

=∂∂

stt

s,Rst

t

s,R bf

I , g

eI

=∂∂

−=∂∂

etc.

where bst’=bst+bc. The Jacobian is a state independent constant matrix consisting of only line

conductance in G and susceptance in B, and can be extended to the three-phase formulation by

∆−−

∆

−−−−−−

=

∆−−

∆

abc

abc

abcabc

abcabc

abcI

abcR

F

E

GB

BG

I

I

|

|

(4)

where ∆E and ∆F are the real and imaginary parts of voltage mismatch, respectively.

, and k is the number of iterations. is the three-phase

admittance matrix as stated in [9]. This model also works when there is an absense of one or two

phases.

)Re( abcmatrix

abc YG = )Im( abcmatrix

abc YB = abcmatrixY

PV Bus Modeling

A generator PV node connecting to a transmission line is specified with the injected power Psp and

voltage Vsp. The injected real power Ps of PV bus s in Figure 1 can be calculated to satisfy

spss,Iss,Rssss PIfIeIVP =⋅+⋅== ]Re[ * (5)

Using Taylor’s expansion and substituting Eq. (4) for ΔI, we have

( ) (( ) ( )

sIsstsstssRsstssts

IssRssIssRss

IsIs

sRs

Rs

ss

s

ss

s

ss

fgfbeebfge

fIgfbeeIbfge

IfIeIfIe

IIPI

IPf

fPe

ePP

∆⋅−+∆⋅−−+

∆⋅++−+∆⋅++=

⋅∆+⋅∆+∆⋅+∆⋅=

∆⋅∂∂

+∆⋅∂∂

+∆⋅∂∂

+∆⋅∂∂

=∆

,'

,'

,,,,

,,

,,

)tstsstststssts

(6)

where , and the specified constant bus voltage amplitude is cals

spss PPP −=∆

2222 spssss VfeV =+= (7)

we have

ssss

ss

ss

s

ss

ffee

ff

Ve

eV

V

∆⋅+∆⋅=

∆⋅∂

∂+∆⋅

∂∂

=∆

22

222

(8)

22where cal

ssp

ss VVV −=∆ . Note that Eq. (6) and Eq. (8) are the mismatch equations of the

proposed PV bus model for FPF. We can further integrate (6) and (8) to construct the single phase

mismatch equation to amend Eq. (4) of a N-bus system by

2

∆−−

∆

∂

∂

∂

∂−−−−−

∂∂

∂∂

=

∆−−

∆

F

E

FV

EV

FP

EP

V

P

ss

ss

s

s

222

|

|

(9)

where

=++

≠−−=

∂∂

siIbfgesibfge

eP

Rssissis

sissis

i

s

,

,

,'

=++−

≠−=

∂∂

siIgfbesigfbe

fP

Ississis

sissis

i

s

,

,

,'

=≠

=∂∂

siesi

eV

si

s

, 2 , 0 2

,

=≠

=∂∂

sifsi

fV

si

s

, 2 , 0 2

To compare Eq. (9) with Eq. (4), it is obvious that the state-independent Jacobian becomes

state-dependent. With PV model, FPF sacrifices its super-linearity. Jacobian elements of the PV buses

need to be updated at each iteration. Some assumptions can be made for simplification to improve the

performance.

(i) From Figure 1, we have

)()] ()[()( ,, ststttssIsRs jbgfjejfejII +⋅+−+=+

i.e., )()( 22222,

2, feststIsRs bgII Ψ+Ψ⋅+=+

with . tsftse ffee −=Ψ−=Ψ ,

It can be shown that

)()(

)()(

sstsstsstsst

ststss

ebfgjfbegjbgjfe

⋅−⋅+⋅+⋅=−⋅+

i.e.,

22

222

)()(

)()(

sstsstsstsst

ststspec

s

ebfgfbegbgV

⋅−⋅+⋅+⋅=

+⋅

Since with )()( )()( 2222222festststst

specs bgbgV Ψ+Ψ⋅+>>+⋅ 1.0 , <<ΨΨ fe , for a well-designed

transmission system, we have

)()()( 2,

2,

22IsRssstsstsstsst IIebfgfbeg +>>⋅−⋅+⋅+⋅

That is, Is,R and Is,I, are no more than one tenth’s other components, and are negligible.

(ii) As a result of (i), and with the general assumptions of

ssss VV θθ sincos ⋅>>⋅

0.1≅sV

Eq. (9) can be rewritten by

∆−−

∆

∂

∂−−−−−

∂∂

∂∂

=

∆−−

∆

E

F

EV

EP

FP

V

P

s

ss

s

s

22

|0

|

(10)

with

=≠−

=∂∂

sigsig

eP

si

si

i

s

, ,

,

=−

≠=

∂∂

sibsib

fP

si

si

i

s

,

, '

=≠

=∂∂

sisi

eV

i

s

, 2 , 0 2

, vectorzeroall :0

Using Eq. (3) for example, PV buses can be inserted to form

[ ] [ ][ ] [ ]T,

2,1

T,,1

INsII

RNsRR

IVII

IPII

∆∆∆=∆

∆∆∆=∆

LL

LL

and the three-phase mismatch equation can also be obtained in a compact form as

[ ] [ ] [ ]abcabcabc ∆VJ∆I ⋅= EFRI (11)

where the new Jacobian matrix [Jabc] has included the modifications of PV buses.

This arrangement yields a constant matrix structure, so the execution time could be improved, and the

Jacobian needs to do LU factorization only once. The algorithm of FPF can be summarized as

1) Initialize and identify PV buses.

2) Build Y admittance matrix.

3) Modify elements of Y matrix corresponding to the PV bus positions, and execute LU factorization.

4) Calculate ECI from the specified PQ buses and calculate P and V for PV buses.

5) Calculate the ECI mismatch vector for PQ buses and the real power and squared bus voltage

amplitude for PV buses.

6) Solve Eq. (11) for [ ]abcEFV∆ , and update E and F.

7) If { } toleranceV abcEF ≥∆max , go to step 4).

8) Stop and print out the result.

Harmonic Model

The existence of nonlinear power devices on PQ buses will introduce nonsinusoidal currents into the

power system. They can be treated as equivalent harmonic current sources. In the proposed harmonic

power flow, it assumes that the slack bus and generators do not generate harmonics into the power

system actively. Only the current contribution of PQ buses with harmonic loads is considered. Besides,

models of generators and nonlinear loads will be different from those used in the proposed FPF. The

harmonic Jacobian matrix must be built before the iteration process. We have the mismatch equation of

the mth order in the form of

∆−−

∆

−−−−−−

=

∆−−

∆

)(

)(

)()(

)()(

)(

)(

|

|

mabc

mabc

mabcmabc

mabcmabc

mabcI

mabcR

F

E

GB

BG

I

I (12)

where , )()()()( mcalmspmabcI

mabcR IIIjI −=∆+∆ 1≠m .

3. Decoupled FPF and HPF

The robustness of a decoupled model [10]-[12] very often depends highly upon the relationship

between the conductance matrix [G] and susceptance matrix [B]. The decoupled model proposed in

this paper needs no assumptions and is insensitive to line parameters. Although the idea extended from

[9] is the first time adopted by the following formulation, this is also the first harmonic power flow

without needing r/x assumptions. From (11), we have

)1()1()1()1()1(' ][][ EGFBI R ∆⋅+∆⋅−=∆ (13)

)1()1(')1()1(')1(' ][][ EBFGI I ∆⋅+∆⋅=∆ (14)

Multiplying to both sides of Eq. (13) and multiply to Eq. (14), we can

derive by adding the two to yield

1)1()1(' ]][[ −BG 1)1(')1( ]][[ −− BG

)1()1()1( 1]1[ DEJ =∆⋅ (15)

)1()1()1( 2]2[ DFJ =∆⋅ (16)

where

][]][[][]1[ )1(1)1()1(')1(')1( GBGBJ −+=

][]][[][]2[ GBGBJ −−= )1('1)1(' −

)1(')1('1)1(')1( −

)()()( mmm

)1(')1('1)1()1(' ]][[1 IR IIBGD ∆+∆⋅= −

]][[2 RI IIBGD ∆+∆⋅−=

For HPF, the decoupled model of the mth order can be shown by

)()()( 1]1[ mmm DEJ =∆⋅

2]2[ DFJ =∆⋅

where

][]][[][]1[ )(1)()()()( mmmmm GBGBJ −+=

]1[]2[ )()( mm JJ −=

)()(1)()()( ]][[1 mI

mR

mmm IIBGD ∆+∆⋅= −

)(1)()()()( ]][[2 mI

mmmR

m IBGID ∆⋅−∆= −

We have

)()()( 1]1[ mmm DEJ =∆⋅ (17)

)()()( 2]1[ mmm DFJ =∆⋅− (18)

The three-phase model can thus be developed accordingly. 4. Equipment Modeling

Two-end HVDC System

A two-end HVDC model is shown in Fig 2. The sending end has a three-phase 6-pulse rectifier and a

transformer unit delivering DC power to the receiving end through the DC transmission line with

resistance Rdc. For unbalanced system loading, the converter model can be found in [14] and [20]. In

this paper, the model developed in [14] is used. On the other hand, for balanced loading, general

assumptions will be used in this paper as

(i) DC-side current Idc has no ripple.

(ii)AC-side voltage is balanced.

Rdc

Rectifier InverterIdc

SendingEnd

ReceivingEnd

IsendV dc,send V dc,receive

V send V receive

Fig. 2 The HVDC Transmission System

sendα and are the firing angle and extinction angle of the sending (rectifier) and receiving end

(inverter), respectively. and receiveα

sendµ receiveµ are commutation overlap angles of the rectifier and inverter.

The AC injection current of sending end Isend, receiving end Ireceive, DC voltage Vdc,send and DC current

Idc,send can be found in [13]. Generally, there are three operation modes, the constant power-delivery

mode, constant current-delivery mode and the constant power-receiving mode. The specified and

unknown parameters of each mode are shown in Table 1. Specified parameters are known before

iteration process, and unknown parameters of Table 1 can be determined by the proposed FPF.

Table 1 Parameters of HVDC in each operation mode

Operation Mode Specified Parameters Unknown Parameters

Constant power-delivery αsend, Pdc,send ,Vdc,receive Vdc,send, Pdc,receive, Idc,send, αreceive µsend,

µreceive

Constant current-delivery αsend , Idc,send ,Vdc,receive Vdc,send, Pdc,send , Pdc,receive, αreceiveµsend,

µreceive

Constant power-receiving Pdc,receive ,Vdc,receive αsend , αreceive, Vdc,send, Pdc,send , Idc,send,

µsend, µreceive

Since the thyristors’ commutation of the converter is not instantaneous, the commutation reactance

Xc will be introduced to model this property. The two-end equivalent model of the HVDC system can

be developed as shown in Eq. (19) and Figure 3.

6,int

,,c

senddcsenddcsenddcXIVV ⋅

⋅−=π (19)

where sendsendsenddc VV αcosint, = .

R dc

(R ectifier ) ( Inverter )

Idc

Sending End R eceiving End

int,senddcV

6cX⋅π

6cXπ

int, receivedcVsenddcV , receivedcV ,

Fig. 3 The HVDC Transmission System

Harmonic analysis of AC/DC converter can be found in [14], and can be reduced to a compact

expression with the two assumptions mentioned above, we have

∗+= )( )1()(12

)1()(11

)(i

mi

mmi VyVyI (20)

where

Ii ： Harmonic injection current of the I-th bus

m ： Harmonic order

y ： Harmonic admittance

V ： Fundamental voltage solution

and parameters of Eq. (20) are

[ ]αδδδπ

)1()1(1

)(11 ),(

)1(23 −−−− −−

= mjmjm

c

m eeAXmm

y

[ ]αδδδπ

)1()1(1

)(12 ),(

)1(23 +−+− −+

= mjmjm

c

m eeBXmm

y

with )(6

11),(δδ

π

δδ−

=mj

m eA , )(6

11),(

δδπ

δδ+

=mj

m eB , and

==

=17,... 11, 5,m , 1-

13,... 7, 1,m , 1mδ .

Harmonic Modeling of Generators and Loads

In [15], generators become a harmonic load that can be treated as a series reactance connected to

earth. This reactance is approximated by the generator’s fundamental negative sequence reactance.

Nonlinear loads are treated as injection current sources; other loads can be represented as equivalent

earthing impedance.

Line Section Model

The m-th order unit-length line admittance is

cL

mmm jmB

jmXRjBG +

+=+

)(1

)()()(

And, skin effects were also considered as in [3].

Transformer Model

For fundamental frequency analysis, only the series winding resistance Ri and leakage inductance Li

are considered. [3] shows the harmonic model with resistance Rpi and inductance Lm used to represent

the frequency-dependent short-circuit property. Furthermore, the equivalent T harmonic model is

developed and shown in Figure 4. That is

−−

+−−−−−

+

=

−−

−

)(

)(

1

32

323

331

31

)(

)(

|

|

mj

mi

eqeq

eqeqeq

eqeqeq

eqeq

mj

mi

V

V

YYYY

Y

YYY

YY

I

I

Yeq2Yeq1

Yeq3V1

I1 I2

V2

Fig. 4 Harmonic T-Model of Transformer

5. Test Results

The proposed power flow was implemented by MATLAB on a Pentium III 700-MHz, 256MB

SDRAM personal computer with Windows 2000. The convergence tolerance for F.P.F. is 10-3 pu., and

10-4 pu. is used for smaller flows of HPF and DHPF. For comparison purposes, five existing (old)

methods were used for tests. The Newton-Raphson (NR) model and the conventional Fast decoupled

load flow (FDLF) in polar form [11], represented by NR_old and FDLF_old in this paper. Two newly

developed methods for harmonics were also tested in this paper; one is a unified Newton method with

the full equivalent load flow formulation [2] represented by “[2]_old”, and the other is a

comprehensive hybrid methodology with capabilities of both time-domain and frequency-domain

computations [4], represented by “[4]_old”. The Poincaré acceleration is used to speed up computation.

Both models were extended to three-phase models. The last method for comparison is a HVDC model

presented in [19] called “[19]_old” in this paper. Various tests were conducted to show the

effectiveness of the proposed method. A modified IEEE 14-bus system is shown in Figure 5 with two

harmonic sources, one is a 6-pulse AC/DC converter connecting to a DC230KV, 700A HVDC between

Bus 2 and Bus 4, and another one connects to a 6-pulse converter on Bus 3 supplying a 210KV, 150A

DC Load. The modified IEEE 14-bus system was extended to a three-phase model with unbalanced

load to test the proposed HPF and DHPF.

Convergent Accuracy Test For a newly developed algorithm, the convergent accuracy must be verified first. In Figure 6,

solutions of the algorithms of [2]_old and [4]_old were used as references to verify the accuracy of

HPF/DHPF with the modified IEEE 14-Bus system. Harmonic voltage solutions of only Bus 2 and

Bus 14 are shown in this figure. For the same tolerance, tests show that the proposed algorithms

provide the same accuracy.

G G

G

G18

7

6

5 4

32

12

1110

9

1413

HVDC system

AC/DCConverter

Fig. 5 Single-line diagram of modified IEEE 14-bus system

Bus 2

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

3.50E-03

4.00E-03

5 7 11 13 17 19 23 25 29 31 35 37Harmonic Order

Har

mon

ic V

olta

ge, p

.u.

HPF/DHPF

[4]_old

Bus 14

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

5 7 11 13 17 19 23 25 29 31 35 37

Harmonic Order

Har

mon

ic V

olta

ge, p

.u.

HPF/DHPF

[4]_old

[2]_old [2]_old

Fig. 6 Accuracy Tests for Harmonic Voltages Table 2 also shows the accuracy of the proposed HVDC model compared with that proposed in [19].

An accurate solution can be observed.

Table 2 HVDC System Test Result

[19]_old Mode

αsend /µsend

αreceive /µreceive

Vdc,send/Vdc,receive

(KV) Pdc,send/Pdc,receive

(MW) Idc (A)

1 10.7°/27.7° 128.04°/25.5° 249.6/230.07 110/101.8 440.3 2 15.6°/22.3° 126.64°/22.27° 240.8/229.73 170.4/162.04 705.4 3 20.1°/10.1° 128.74°/9.89° 238.1/230.01 129.3/120.01 521.6

FPF Mode

αsend /µsend

αreceive /µreceive

Vdc,send/Vdc,receive

(KV) Pdc,send/Pdc,receive

(MW) Idc (A)

1 10°*/27.6° 127.4°/26.1° 250.25/230* 110*/101.1 439.6 2 15°*/22.4° 126.76°/22.17° 242.1/230* 170.4/161.92 704* 3 19.2°/9.38° 129.6°/10.21° 238.4/230* 129.3/120* 521.7

ps.1: “*” : specified parameter. ps.2: Mode 1: Constant power-delivery, Mode 2: Constant current-delivery, Mode 3: Constant

power-receiving

Harmonic Distortion Test

The total harmonic distortion, THD%, was tested with different firing angles of HVDC converter as

shown in Figure 7 and 8. Figure 7 shows parts of the results compared with [2]_old and [4]_old. The

effect of various firing angle of HVDC system of the proposed model is also illustrated in Figure 8.

The THD% of each bus increases as the firing angle increases. A workable model with good accuracy

can be observed.

firing angle = 5

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14

System Bus

Vol

tage

TH

D (%

)

HPF/DHPF[2]_old[4]_old

firing angle=15

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14System Bus

Vol

tage

TH

D (%

)

HPF/DHPF[2]_old[4]_old

Fig.7 Harmonic Distortion Test for Voltage THD%

0

0.5

1

1.5

2

2.5

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14System Bus

Vol

tage

TH

D(%

)

angle=0angle=5angle=10angle=15

Fig. 8 Effects of firing angle on Voltage THD%

Performance Test In this test, both the fundamental power flow and the harmonic power flow were tested to show the

effectiveness of the proposed method.

(i)Fundamental Tests

The proposed FPF and DFPF were compared with NR_old and FDLF_old, respectively. The number

of iterations and the normalized execution time of each method are shown in Table 3 and Figure 9,

respectively. Besides the theoretical IEEE test systems, the TPC 33 Bus system, tested in Table 3, is a

practical system of Taiwan Power Company with simplified 14 generators and 16 equivalent load

buses. Real operational data were collected from SCADA at pm 3:00 summer 2000[17]. It can be seen

that the proposed FPF and DFPF yield very close number of iterations, with much less execution time

as shown in Figure 9. In this Figure, the performance 100% (25.75 sec) is set for NR_old algorithm

with IEEE 118 Bus system. Note that DFPF shows a slightly better performance than FDLF_old for

large networks because of the less assumptions on R/X.

Table 3 Number of Iterations

Test System NR_old FPF DFPF FDLF_old TPC 33 Bus 3 4 5 5 IEEE 30 Bus 3 5 5 5 IEEE 57 Bus 4 5 6 6 IEEE 118 Bus 4 6 7 7

100%

35%

12% 12%3%4%

25%

9%

2%

14%

5%2%

15%

5%2%2%0%

20%

40%

60%

80%

100%

120%

TPC 33 Bus IEEE 30 Bus IEEE 57 Bus IEEE 118 Bus

NR_old

DFPF

FPF

FDLF_old

Fig. 9 Comparison of Normalized Execution Time

(ii)Harmonic Tests

Figure 10 and Table 4 compare the performance of HPF and DHPF with [2]_old and [4]_old. The

modified IEEE 14 Bus system, IEEE 57 Bus system and IEEE 118 Bus system with three-phase

numeric were used for tests. The IEEE 57 Bus system was modified with two HVDC systems between

line 18-19 and 30-31. The 118 Bus system was modified with three HVDC systems on line 30-38,

19-34, 69-75, and four AC/DC rectifiers for DC loads on bus 23, 43, 67 and 88. In this Figure, the

performance 100% (55.15 sec) is set for [2]_old with the modified IEEE 118 Bus system. It shows that

besides the reduction of the number of iterations, the proposed methods are much better performer in

terms of the execution time.

Table 4 Number of Iterations Tests System HPF DHPF [2]_old [4]_old

Mod. IEEE 14 Bus 7 11 20 11 Mod. IEEE 57 Bus 10 16 24 16 Mod. IEEE 118 Bus 13 19 30 21

3% 2%10%

22%22%

61%

100%

43%

17%

31%

77%

17%

0%

20%

40%

60%

80%

100%

120%

Mod. IEEE 14 Bus Mod. IEEE 57 Bus Mod. IEEE 118 Bus

HPF

DHPF

[2]_old

[4]_old

Fig. 10 Comparison of Normalized Execution Time for Harmonic Power Flow

Robustness Test Figure 11 shows results of the robustness (R/X ratio) test for the IEEE 57 system. The R/X ratios

were adjusted by multiplying a scaling factor, T, that ranges between 0.1 and 4.5. Various tests were

conducted for the proposed methods to show its robustness. It can be seen that the FPF method is less

sensitive to line parameters than others for all tests. Simplified approaches will need more iterations to

converge. DFPF is almost as robust as FPF and is much more robust than FDLF_old. FDLF_old

diverged when T is greater than 4.5. Note that FDLF_old needs assumptions on R/X ratios while DFPF

doesn’t.

Harmonic Voltage Spectrums

HPF and DHPF were further tested for both the balanced and unbalanced loads for the modified

IEEE 14-Bus system. Both methods converge at the same solution. Only 6-pulse converter is

considered to provide ( 16 ±m )th order components into the system [18]. For unbalanced load, the

converter model in [14] was used. The real and imaginary solutions of the mth harmonic voltage have

been calculated. Figure 12 and 13 illustrate voltage spectrum of Bus 12 under balance and unbalance

loading, respectively. In Figure 13, load demands of the test system were adjusted by increasing 20%

for phase A; increasing 10% for phase B and decreasing 5% for phase C. It shows that the unbalance

loading results in more harmonic distortions for bus voltages.

0

5

10

15

20

25

0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5T

Itera

tion

DFPFFDLF_oldFPF

Fig. 11 R/X Ratio Test for IEEE 57 Bus System

Bus 12

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

1.20E-03

1.40E-03

5 7 11 13 17 19 23 25 29 31 35 37Harmonic order

Har

mon

ic V

olta

ge, p

.u. phase A

phase Bphase C

Fig. 12 Harmonic Voltage Spectrum with Balanced Loads

Bus 12

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

5 7 11 13 17 19 23 25 29 31 35 37

Harmonic order

Har

mon

ic V

olta

ge, p

.u. phase A

phase Bphase C

Fig. 13 Harmonic Voltage Spectrum with Unbalanced Loads

6. Discussions and Conclusion

This paper presents a new formulation for solving the multiple-frequency three-phase

balance/unbalance power flow problem based on the nodal current injection and a sparse Jacobian

matrix. Forms of four models were developed for tests. The advantages of the proposed models can be

delineated below.

The Jacobian matrix can be easily constructed from Y admittance matrix.

A general purpose, very robust good performer can be observed.

Besides the extensive tests of ECI idea for distribution systems [9], FPF and DFPF can work alone

as a general purpose tool for one-line and three-phase systems to solve transmission power flow.

DFPF and DHPF need no assumptions on R/X and are applicable for solving the overall

high-voltage and low-voltage power network.

Although DFPF shows only slightly better performance than FDLF_old, it is a much more robust

algorithm for various networks.

All proposed method work with a constant and sparse Jacobian matrix where sparse technique can

be used to further improve the performance, and the LU factorization has to be done only once.

Reference [1]W. Xu, J. R. Marti, and H. W. Dommel, “Harmonic power flow studies. Part I and II”, IEEE trans. on Power

Apparatus and Systems, Vol. 101, No. 6, Feb. 1982.

[2]G. N. Bathurst, B. C. Smith, N. R. Watson and J. Arrillaga, “A modular approach to the solution of the

three-phase harmonic power-flow”, IEEE transactions on Power Delivery, Vol. 15, No.3, July 2000.

[3]Task Force on Harmonics Modeling and Simulation, “Modeling and Simulation of the Propagation of

Harmonics in Electric Power Networks, Part I and II”, IEEE Transactions on Power Delivery, Vol. 11, No.1,

January 1996.

[4]A. Semlyen and M. Shlash, “Principles of modular harmonic power flow methodology”, IEE

Proc.-Generation, Transmission and Distribution, Vol.147, No. 1, January 2000.

[5]M. Shlash and A. Semlyen, “Efficiency issues of modular harmonic power flow”, IEE Proc.-Generation,

Transmission and Distribution, Vol.148, No. 2, March 2001.

[6]J. Arrillaga and C. D. Callaghan, “Three phase AC-DC load and harmonic flows”, IEEE Trans. on Power

Delivery, Vol. 6, No. 1, January 1991.

[7]W. M. Lin and J. H. Teng, “Phase-Decoupled Load Flow Method for Radial and Weakly-Mesh Distribution

Networks”, IEE Proc.-Generation, Transmission and Distribution, Vol.143, No. 1, January 1996, pp.39-42.

[8]W. M. Lin, J. H. Teng, “Current-Based Power Flow Solutions for Distribution Systems”, IEEE ICPST ’94

Conference Beijing, China, 1994, pp. 414 - 418.

[9]W. M. Lin, Y. S. Su, H. C. Chin, and J. H. Teng, “Three-Phase Unbalanced Distribution Power Flow

Solutions with Minimum Data Preparation”, IEEE Transactions on Power Systems, Vol. 14, No. 3, August

1999, pp. 1178-1183.

[10]B. Stott and O. Alsac, “Fast Decoupled Load Flow”, IEEE Trans. on Power Apparatus and Systems, Vol.

PAS-93, 1974, pp. 859-869.

[11] A. M. Robert van Amerongen, “A General-Purpose Version of the Fast Decoupled Load Flow”, IEEE Trans.

on Power Systems, Vol. 4, No. 2, May 1989, pp. 760-770.

[12]A. Monticelli, A. Garcia and O. R. Saavedra, “ Fast Decoupled Load Flow: Hypothesis, Derivations and

Testing”, IEEE Trans. on Power Systems, Vol. 5, No. 4, Nov. 1990, pp. 1425-1431.

[13]Cornel, H. C. J., “QENS-An Enhanced Version of the Network Simulator Program,” M. Sc. Thesis, Queen’s

University, Kingston, Canada, 1987.

[14]J.G. Mayordomo, L.F. Beites,R. Asensi, etc., ”A Contribution for Modeling Controlled and Uncontrolled

AC/DC Converters in Harmonic Power Flows”, IEEE Transactions on Power Delivery, Vol. 13, No. 4,

October 1998.

[15]Li Gengyin, Xu Chunxia, Liu Guanqi, and Zhao Chengyong, “Calculation of three-phase harmonic power

flow in power systems with traction loads”, IEEE POWERCON '98. International Conference, Vol. 2, pp.

1565 –1570.

[16] Lihua Hu, “Sequence impedance and equivalent circuit of HVDC systems”, IEEE Transactions on Power

Systems, Vol. 13, No. 2, May 1998.

[17]M. S. Kang, “The Study of Load Characteristics in Taipower and Its Effect on Power System Operation”,

Ph.D. Thesis, National Sun Yat-Sen University, Kaohsiung, Taiwan, R.O.C.

[18]Ned Mohan, Tore M. Undeland, and William P. Robbins, “Power Electronics”, John Wiley & Sons, Inc.

[19]Lihua Hu, “A new model for HVDC systems based on model of a regulating transformer”, Power

Electronics and Variable Speed Drives, Seventh International Conference on (IEE Conf. Publ. No. 456),

September 1998.

[20]B.C. Smith, N.R. Watson, A.R. Wood and J. Arrillaga, “A Newton Solution for the Harmonic Phasor

Analysis of AC/DC Converters”, IEEE Transactions on Power Delivery, Vol. 11, No. 2, April 1996.