An Improved Power Flow for an Ill-Conditioned Power System with FACTS

Devices

NAOTO YORINO,1 TOHRU TAKEUCHI,

1 HIROSHI SASAKI,

1 and HIROAKI SUGIHARA

2

1Hiroshima University, Japan

2Chugoku Electric Power Co., Inc., Japan

SUMMARY

This paper addresses the issue of the computation

techniques for assessing the steady-state power flows con-

trolled by Flexible AC Transmission Systems (FACTS),

which contain variable series compensators (VSC), phase

shifters (PS), interphase power controllers (IPC), and

unified power flow controllers (UPFC). An improved

Newton�Raphson load flow program has been developed

based on an analysis of the convergence characteristic of

the conventional method. It is shown that the conventional

method tends to suffer from ill-conditioning problem, re-

sulting in shrinkage of the convergence region. Based on

examinations of the condition number of the Jacobian, a

penalty function method is adopted in order to avoid the

ill-conditioning problem and to guarantee a successful con-

vergence. Although the computational burden is increased

about 2 to 3 times, the proposed method considerably

extends the region of convergence. The effectiveness of the

proposed method is demonstrated through numerical ex-

aminations using IEEE 57 and 118 bus systems. © 2002

Wiley Periodicals, Inc. Electr Eng Jpn, 140(1): 30�37,

2002; Published online in Wiley InterScience (www.

interscience.wiley.com). DOI 10.1002/eej.10027

Key words: power system; power flow calculation;

ill-posed conditions; FACTS; VSC; PS; IPC; UPFC.

1. Introduction

Due to recent increases in power demand and to

power market deregulation, power transmission equipment

is being exploited more intensely; at the same time, laying

new transmission lines is difficult because of geographical,

economic, and environmental limitations. As a result, new

techniques are being developed to obtain the most carrying

capacity from existing transmission systems, while main-

taining an acceptable level of system reliability and safety.

In this context, high-performance control over power sys-

tems is necessary; in particular, techniques based on power

electronics have been attracting increasing attention [1, 2].

FACTS includes devices controlling power, voltage,

phase, impedance, and other parameters of ac networks

using power electronics. This makes it possible to improve

both the steady-state and transient characteristics as well as

the stability of power systems, while making the most of

the capacity provided by existing transmission lines, and

thus resulting in efficient and flexible networks [3]. FACTS

controllers have been increasingly employed in power sys-

tems, drawing much attention in the field of power flow

control [4�6]. In this context, a practicable efficient ap-

proach is proposed in Refs. 7 and 8 by extending the

conventional power flow calculation based on the Newton�

Raphson method to power systems equipped with FACTS

devices. The control specifications employed in conven-

tional power flow calculation (active and reactive power)

are supplemented with active line power flow (in the case

of UPFC, active line power flow and reactive line power

flow) for lines provided with FACTS devices. This ap-

proach involves finding solutions by secondary conver-

gence.

The authors have examined the approach (below

referred to as the conventional method) proposed in Refs.

7 and 8 to prove that achieving convergence is often prob-

lematical for certain FACTS devices. Thus, a new technique

offering better convergence performance is necessary. This

paper aims at clarifying the problem, and improving con-

vergence performance in power flow calculation. The

FACTS devices dealt with in this study are the VSC (Vari-

able Series Compensator), PS (Phase Shifter), IPC (Inter-

phase Power Controller), and UPFC (Unified Power Flow

Controller).

© 2002 Wiley Periodicals, Inc.

Electrical Engineering in Japan, Vol. 140, No. 1, 2002Translated from Denki Gakkai Ronbunshi, Vol. 121-B, No. 8, August 2001, pp. 967�972

30

2. The Conventional Method

The method proposed in Refs. 7 and 8 is outlined

below.

2.1 VSC (Variable Series Compensator)

The active line power flow Pkm from bus k to bus m

is expressed as

Here,

The asterisk denotes the complex conjugate.

A block diagram of the VSC is shown in Fig. 1. With

the VSC, the Pkm can be controlled by varying the line

reactance Xkm. The VSC is calculated by Newton�s method.

The correction at each iteration is found by solving the

following:

2.2 PS (Phase Shifter)

The block diagram of the PS is shown in Fig. 2. With

the PS, the Pkm can be controlled by varying the phase angle

φkm. The PS is calculated by the Newton�Raphson (N-R)

method. The correction at each iteration is found by solving

the following:

2.3 IPC (Interphase Power Controller)

A block diagram of the IPC is shown in Fig. 3. With

the IPC, the Pkm can be controlled by varying the phase

angle φkm or line reactance Xkm. The IPC can be represented

equivalently as a combination of the VSC and PS models.

When both branches are controlled via the VSC, Eq. (3) is

applied. When both branches are controlled via the PS, Eq.

(4) is applied. If one branch is controlled via the VSC and

the other via the PS, then the correction at each iteration is

found by solving the following:

Fig. 1. Variable series compensator.

(1)

(2)

(3)

(4)

Fig. 2. Phase shifter.

Fig. 3. Interphase power controller.

31

2.4 UPFC (Unified Power Flow Controller)

When the UPFC is used as a line compensator, it can

be represented by an SVS (Synchronous Voltage Source) as

shown in Fig. 4. With the UPFC, the Pkm and reactive line

power flow Qkm can be controlled by varying the magnitude

|Vs| and phase angle θs of the SVS voltage. The UPFC is

calculated by the N-R method, and the correction at each

iteration is found by solving the following:

3. Problems of the Conventional Method

In the conventional method, the N-R algorithm is

applied directly to a power system including FACTS de-

vices; here, the N-R convergence region proves to be rather

narrow for the VSC and IPC, that is, for devices that control

power flow by means of reactance. This problem was

confirmed for all systems examined (6-, 14-, 30-, 57-, and

118-bus systems) irrespective of the system scale and op-

erating conditions. On the other hand, this problem was not

detected in the case of devices that control power flow by

the phase angle, such as PS and UPFC.

For example, consider a VSC installed on line 40 of

an IEEE 57-bus system with specifications as given in Table

1. When conventional power flow control is performed by

varying the reactance Xkm of line 40 (k = 24, m = 26), the

power flow converges to Pkm < 0.24 (p.u.) as shown in Fig.

5. On the other hand, when using Pkm as a control specifi-

cation with the method proposed in Refs. 7 and 8, the

convergence limit is about Pkm < 0.16 (p.u.) as shown in Fig.

12. The mismatch in the conventional method with Pkm set

to 0.08 (p.u.) (convergence) and to 0.2 (p.u.) (divergence)

is shown in Fig. 6. Thus, with the conventional method,

convergence is not guaranteed even though a solution does

exist.

To identify the reason for this problem, the Jacobian

elements related to Pkm of the line provided with the VSC

were examined. These elements are as shown below:

Here,

(5)

Fig. 4. Unified power flow controller.

(6)

Table 1. Power control specifications (Case 1)

Line with de-

vice installed

(connected

buses)

Installed

device

FACTS

specification

(p.u.)

Before

control (for

reference)

40 (24�26) VSC Active line

power flow:

variable

0.0756

(7)

(8)

(9)

(10)

(11)

(12)

32

The above Jacobian elements were examined by repeating

the conventional power flow calculation while varying the

Xkm as a parameter (Figs. 7 and 8). As is evident from the

diagrams, all of the elements show strong nonlinear vari-

ation with respect to Xkm when Xkm is near zero. Therefore,

convergence of the N-R method cannot be expected here.

Next the condition number of the Jacobian was ex-

amined. Figure 9 shows the condition number with every

iteration when Pkm is set to 0.08 (p.u.) (convergence) and to

0.2 (p.u.) (divergence). As is evident from the diagram, the

condition number is nearly unchanged in the case of con-

vergence, while growing extremely large in the case of

divergence. In the equation

Fig. 6. Convergence characteristic of conventional

method (Case 1).

Fig. 7. Element of Jacobian (Case 1).

Fig. 8. Elements of Jacobian (Case 1).(13)

Fig. 5. Region of existence of solution for power flow

(Case 1).

Fig. 9. Number of conditions for conventional method

(Case 1).

33

it is possible that the minor noise ε included in ∆f increases

due to the enormous singular value of J�1, the original

vector ∆x0 is overridden, and the equation becomes numeri-

cally unstable. Examination of the N-R convergence proc-

ess has shown that the noise component J�1ε fluctuates

greatly.

The above discussion pertains to the VSC shown in

Fig. 1, but the same situation was also confirmed for the

VSC in the internal branch of the IPC in Fig. 3. Thus, the

above holds true for the IPC as well.

4. Improvement of Convergence Performance

(Proposed Method)

In this study, emphasis is placed on the oscillating

noise component, and the instability is removed by suppres-

sion of the oscillation. A penalty is imposed on the oscillat-

ing component to be suppressed by minimizing a penalized

function. Specifically, the penalty function p(∆x), which

increases with the oscillation of vector ∆x, is introduced,

and ∆x is chosen so as to minimize the objective function

defined as

In this study, the following quadratic form of ∆x is em-

ployed as penalty function p(∆x):

Here P is a regular symmetrical matrix. In this paper, the

unit matrix E is used for P for simplicity. In this case, Eq.

(14) may be rewritten as

The following expression is used to find ∆x so as to mini-

mize the objective function S(∆x):

Here λ is zero or positive. With greater λ, suppression of

the oscillating component provides a stronger effect but, on

the other hand, more time is required for convergence. In

this study, the parameter λ is adjusted in the following way.

1. The initial value of λ is set large enough, and the

convergence calculation is performed while suppressing the

oscillating component.

2. As the oscillating component becomes smaller, the

convergence calculation is continued at gradually de-

creased λ.

3. As the oscillating component becomes small

enough, the solution is found by the N-R algorithm as in

the conventional method.

Specifically, the initial value of λ was set to 1, and

then λ was adjusted as in Eq. (18), using the evaluation

function S(∆x) as in Eq. (16):

Once S ≤ 10�5 was reached, the solution was found

by the N-R algorithm.

Equation (18) was obtained empirically using more

than 100 random operating conditions for 6-, 14-, 20-, 57-,

and 118-bus systems. That is, Eq. (18) works for all of these

systems; on the other hand, Eq. (18) does not necessarily

provide the best convergence in all cases.

With the proposed method, Eq. (17) is solved itera-

tively using LU decomposition; however, as is evident from

the format of Eq. (17), it is very similar to the usual method

for the state estimation. Thus, since the bracketed term on

the right-hand side of Eq. (17) is a sparse matrix, the

computational burden is not dependent on system size (as

with the state estimation), but is 2 to 3 times heavier than

the conventional power flow calculation.

5. Numerical Examples

The proposed method was verified for a variety of

power systems and operating conditions, and good results

were obtained in all cases. Below, the effectiveness of the

proposed method is proved by two examples. In Case 1,

only the VSC is installed in an IEEE 57-bus system (see

Table 1) to reveal the problems with the conventional

method. In Case 2, various FACTS controllers are installed

in an IEEE 118-bus system as shown in Table 2. The authors

examined many other cases as well, while increasing the

number of FACTS devices as well as varying other parame-

ters, and stable convergence was verified in every case. For

both the proposed and conventional methods, the conver-

gence condition was set as

If the number of iterations exceeded 20, the calculation was

terminated.

5.1 Case 1

The convergence performance of the proposed

method with active line power flow set to Pkm = 0.08 (p.u.)

and Pkm = 0.2 (p.u.) is shown in Fig. 10. This convergence

pattern is typical of the proposed method. In contrast to the

conventional method, here convergence is obtained for both

settings. The variation of the number of conditions for the

(17)

(18)

(15)

(16)

(14)

(19)

34

Jacobian is presented in Fig. 11; as is evident from the

diagram, the number remains nearly unchanged.

In addition, the range of convergence was examined

by counting the number of iterations until convergence,

while varying the active line power flow as a parameter; the

results are presented in Fig. 12. The conventional method

ensured convergence with the power flow up to 0.16 p.u.;

on the other hand, the proposed method had a significantly

wider range up to 0.24 p.u. This coincides with the region

of existence of the power flow solution found by the usual

power flow calculation. Quite similar results were also

obtained for a large variety of systems.

5.2 Case 2

Here the FACTS controllers were installed as shown

in Table 2, and the comparison between the proposed and

conventional methods was made while varying the power

flow on line 16. The results are shown in Figs. 13 to 15.

Comparison shows that the proposed method (Fig. 14)

gives better convergence performance than the conven-

tional method (Fig. 13). The number of iterations until

convergence for both methods is shown in Fig. 15. As is

evident from the diagram, the proposed method achieves

convergence in a wider range, while the number of itera-

tions is nearly unchanged. Similar good results were ob-

tained for other systems.

Table 2. Power control specifications (Case 2)

Line with de-

vice installed

(connected

buses)

Installed

device

FACTS

specification (p.u.)

Before

control

(for

reference)

16 (7�12) VSC Active line power

flow: variable

0.3509

51 (33�37) IPC Active line power

flow: 0.6

0.5488

81 (55�56) PS Active line power

flow: 0.2

0.1878

140 (85�89) UPFC Active line power

flow: 0.45

0.4088

Reactive line power

flow: 0.1

0.0955

Fig. 10. Convergence characteristic of proposed method

(Case 1).

Fig. 11. Number of conditions for proposed method

(Case 1).

Fig. 12. Region of convergence (Case 1).

35

5.3 Computational burden

The computation time per iteration of the proposed

method was examined with reference to the conventional

method as unity. Figure 16 shows results for Ward & Hale

6, IEEE 14, IEEE 30, IEEE 57, and IEEE 118 power

systems, with the number of FACTS devices being 5, 10,

and 15% of the total number of lines. As stated above, the

proposed method required 2 to 3 times as much computa-

tion, irrespective of system size or the number of FACTS

controllers.

6. Conclusions

This paper has proposed a method for power flow

calculation with regard to FACTS controllers, namely,

VSC, PS, IPC, and UPFC. The proposed method gives

better convergence in a considerably wider range than the

conventional method. The computing time is 2 to 3 times

as long irrespective of system scale. The effectiveness of

the proposed method is shown by numerical examples

using IEEE 57 and IEEE 118 systems.

REFERENCES

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4. Han ZX. Phase shifter and power flow control. IEEE

Trans Power Apparatus Syst 1982;101:3790�3795.

Fig. 13. Convergence characteristic of conventional

method (Case 2).

Fig. 14. Convergence characteristics of proposed

method (Case 2).

Fig. 15. Region of convergence (Case 2).

Fig. 16. Computation time of proposed method versus

conventional method.

36

5. Noroozain M, Andersson G. Power flow control by

use of controllable series components. IEEE Trans

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AUTHORS (from left to right)

Naoto Yorino (member) completed the M.E. program at Waseda University in 1983 and joined Fuji Electric. He obtained

his D.Eng. degree from Waseda University in 1984, and joined the faculty as a research associate in 1985. He moved to Hiroshima

University in 1987, and has been an associate professor since 1991. His research interests are power system stability and

economical operation. He received the 1985 George Montefiore Award.

Tohru Takeuchi (student member) graduated from Hiroshima University (electrical engineering) in 1999 and is now a

doctoral candidate (systems engineering).

Hiroshi Sasaki (member) received his D.Eng. degree from Waseda University (electrical engineering) in 1968 and joined

the faculty of Hiroshima University as a lecturer, becoming an associate professor in 1980 and a professor in 1989. His research

interests are transient stability of power systems, state estimation, optimal flow calculation, and expert systems. He is a member

of IEEE, CIGRE, the Japan Solar Energy Society, the Information Processing Society of Japan, and the Japan Society of Energy

and Resources.

Hiroaki Sugihara (member) graduated from Kyoto University (electrical engineering) in 1981 and joined Chugoku

Electric Power Co. His research interests are power systems.

37