A non-iterative and exact linearization load flow technique for circuit contingency effects in power systems

Electric Power Systems Research, 19 (1990) 213 - 218

213

A Non-iterative and Exact Linearization Load Flow Technique for Circuit Contingency Effects in Power S y s t e m s ADEL ALI ABOU EL-ELA Faculty of Engineering and Technology, Menoufia University, Shebin El-Kom (Egypt) (Received March 2, 1990)

ABSTRACT

This paper presents an exact, fast and noniterative linearization technique to evaluate the effect of line contingencies upon the system bus voltages. This technique linearizes the load flow equations in Cartesian form with proper compensation to the second-order terms where the voltages are represented in rectangular coordinates. This method uses the complete Taylor series expansion of the nodal equations while retaining the non-linearity of the load flow equations. Only the non-zero elements of the second-order load flow matrix related to the real voltage deviations are stored in the digital computer. The proposed method is highly efficient for the solution of a transmission network, but also has particular advantages for lower voltage networks and for difficult or ill-conditioned load flow problems. Using the proposed method, the effect of different line outages is studied to reach a steady-state solution. The proposed method has been validated by its application on a standard test system.

1. INTRODUCTION The need for a fast and reliable load flow technique to evaluate a large number of single and multiple line outages in a power network has long been appreciated. It is particularly necessary for the proper planning and operation of power systems. Conventional AC load flow techniques have been widely used in the past but the length of the computation time has always been a problem. Although high speed computers and fast AC load flow solution techniques are required, present analyses 0378-7796/90/$3.50

of system outages for on-line security assessment and/or off-line planning studies are very tedious and uneconomical. In order to overcome the computation burden involved in contingency analysis, a considerable effort has been devoted to develop fast and approximate solution techniques so that a large number of system contingencies can be analysed quickly for on- and off-line calculation purposes. A sacrifice in accuracy for speed has become a common practice, so that a large number of line outages can be studied in a reasonable time. In some cases, linearization techniques have to be used iteratively to get reasonable results. Khu et al. [1] have extended the DC load flow approach to evaluate the effect of line outages upon load bus voltages. This method, although providing a good estimation when changes in voltage magnitudes are reasonably small, yields appreciable errors in the case of large changes. Peterson et al. [2] presented an iterative linear procedure to determine the change in bus voltages and angles due to line outage. In this procedure, the elements of the sensitivity matrix (inverse of the Jacobian matrix) are calculated using the precontingency bus voltages and are kept constant for different line outage simulations. A power flow technique retaining only voltage control busbars in the iterative process was presented in ref. 3. This method did not consider the effects of line outage on the power system. In ref. 4 the performance of two exact load flow and outage simulation techniques [3, 5] was compared. The line outages were simulated using the current injection principle for both techniques. Long computation times and large computer storage were required. © Elsevier Sequoia/Printed in The Netherlands

214

An algorithm for load flow solutions in electrical power systems has been presented in ref. 6. The algorithm is based on the NewtonRaphson process and uses a partitioned matrix approach to the Jacobian equation. This algorithm takes a lot of computation time and large computer storage, therefore it is not suitable for on-line application. A small modification to the Peterson technique [5] was presented in ref. 7 to study the effect of line outages upon the system bus voltages. This method depends on the complete Jacobian matrix which takes considerable computer storage [3]. This paper presents an exact, fast and noniterative linearization method to evaluate the effect of line contingencies upon the system bus voltages. This method uses the complete Taylor series expansion of the nodal equations while retaining the non-linearity of the load flow equations. Only the non-zero elements of the sensitivity matrices are stored in the digital computer.

The bus voltages can be represented as a function of the real part of the voltage. Hence, eqns. (3) and (4) can be rewritten as N

Pi = ~ eiej[Vij(l + J~i~j) j=l

- Bi~(,~i - 2i)1

(5)

N

Qi = ~ eiej[Bij(l + ~.i,~j) j--1

+ V,j(,~ - 2~)1

(6)

where

~, = fi ~el = tan Oi,

~j -= fj/ej = tan Oj

and 0 i is the phase angle of the ith bus voltage. The incremental injection power can be written in Cartesian coordinates using the Taylor series expansion of the active and reactive power equations (5) and (6) around the base-case load flow solution as AP = J1 Ae + 0.5H1 Ae he

(7)

AQ = J2 Ae + 0.5H 2 Ae Ae

(8)

2. PROPOSED MATHEMATICALFORMULATION

2.1. Exact linearization of load flow equation The power injected at bus i is expressed as P~ - JQi = Ei*Ii

i = 1. . . . .

N

(1)

or N

P~-jQ~=Ei*

~

YijEj

(2)

j=l

where N is the total number of buses, Pi and Q~ are the active and reactive powers injected at bus i, E~ is the complex voltage at bus i, * denotes the conjugate, and Yij is the admittance between buses i and j. From eqn. (2), the power injected, in Cartesian coordinates, for active and reactive nodal power at bus i can be expressed as N

P~ = ~ [ei(ejGij -}-fjBij) j=l

+ fi(f~V,~

-

ejB,j)]

(3)

N

Q, = ~ [f,(ejG~j + fjB~j) j=l

- ei (fjGij -}- ejBij)]

where AP and AQ are vectors of incremental changes in the injected active and reactive power, J~ and J2 are N × N Jacobian matrices, and H~ and H2 are N × N 2 Hessian matrices composed of the second-order derivatives. The Jacobian and Hessian matrices are calculated in the Appendix. The Taylor series expansion represented by eqns. (7) and (8) has no terms higher than the second-order derivatives as the original equations (5) and (6) are quadratic in the real component of bus voltages e. The two matrices H1 and H2 are independent of e and are only related to the conductance G and susceptance B of the nodal admittance matrix and hence they are constants during any change in the active and reactive powers (see Appendix). Thus, the solution of eqns. (7) and (8) is an exact formulation of the load flow equations. The second order of the real component of the bus voltage can be calculated, using eqn.

(7): (4)

where ei and fi are the real and imaginary components of the voltage at bus i, and Y i j = G i j - - jBij.

Ae Ae = 2H~-I(AP -- J 1 A e )

(9)

where H1 -~ can be obtained using a numerical technique based on the concept of the

215

N

pseudo-inverse [8]:

YL.L = ~

H1 - 1 = HT( H1 H1T) - 1 or

R = H1T(HIH1 T) -1

(11)

where R is an N 2 × N matrix. The matrix H1H1T contains many zero elements. It has been stressed several times t h a t only the non-zero elements of a sparce matrix need to be stored in a digital computer. Since the storage array will not be identical to the original matrix, both the values of the nonzero elements and a means of identifying these elements must be incorporated in the storage technique. Hence, a direct solution for the inverse matrix can be obtained with a minimum amount of storage and computation time. Substituting the second order of the real part of the bus voltage, Ae Ae, from eqns. (9) and (11) in eqn. (8), the change in reactive power as a function of the change in the real part of the bus voltage can be expressed as A Q = J2 A e + H z R ( A P

- J1 A e )

(12)

or

AQ - H2R AP = (J2 - H2RJ1)

Ae

(13)

The real part of the bus voltage can be obtained as a function of the injection powers as Ae = D-X8

= (J2 - H2RJ1)-I

(17)

YK.,

(18)

I#K N

YX, K = •

I~l I#L

It is possible to use the compensation technique [9l to re-solve the system when a few lines have been removed from the configuration. A change in the susceptance of branch h, A?h, between buses K and L produces a variation in the bus voltage angle A0: A0 -

- A~h~h ZCh 1 + ChtZCh A~h

(19)

where ~h is the angle across branch h before the change, Z = B-1 and the vector Ch is Cht--['''+

K 1 ....

L 1 '''1

The vector ZCh in (19) is the solution S of the system BS = Ch obtained using the triangular factor of B. Expression (19) can therefore be used to evaluate the state of the network after the change (0"= 0 + A0) without any recalculation of the triangular factors of B; only system BS = Ch needs to be solved. The changes in the injection of active and reactive powers at bus i in eqn. (15) can be calculated as

(13)

N

hPi = ~ el(O) ej(O)[V~jew(1 + ~ew~?ew)

where D-~

YL,,

I=1

(10)

j=l (14)

B~j~w(~w _ ~?~w)] _ Pi (0)

_

and

(20)

N

AQi = ~, e,(O) ej(O)[B~W(1 + ,tr°"X?ew) S = AQ - H2R AP

(15)

Equation (13) is a linearized relation of Ae in terms of the injection powers while the nonlinearity of the load flow equation is retained.

j=l

+ V~W(~}~w _ ~?~w)] _ Qi (0)

(21)

where (0) denotes the initial condition,

G~ow = Gij(O ) + AGij 2.2. Effect of a circuit contingency on bus voltages The exact linearized formula of the load flow equations, eqn. (13), is used to determine the accurate solution of voltage changes following a circuit contingency. At a line outage between buses L and K, the off-diagonal and diagonal elements of the admittance matrix Y will be changed: YL.K -~ YK, L = 0 . 0

(16)

B~j ew =

Bij(O ) -~ ABij

A new = h i ( 0 ) "~ A ~ i

The Jacobian matrices J1 and J2 and the Hessian matrices H1 and H2 in eqns. (14) and (15) will be modified according to the circuit contingency between buses L and K:

J(L, K) = J(K, L) = 0

(22)

216

and

TABLE 2

H(L, K ) = H(K, L) = 0

(23)

Since the changes in the real part of the voltage following a circuit contingency can be computed easily from eqn. (13), the changes in the imaginary part, Af, will be obtained and the new complex bus voltage can be computed: E = E(0) + AE

(24)

Generation and load data Bus No.

Generation

Load

(MW)

(MW)

(MVAR)

(MVAR)

1

0

0

0

0

2 3 4 5

40 0 0 0

30 0 0 0

20 45 40 60

10 15 5 15

3. APPLICATION

3.1. Test systems

Two standard test systems have been used for an extensive study of the proposed linearization technique to evaluate the effect of line contingencies upon the system bus voltages. The first test system [10] contains 5 buses and 7 transmission lines. The line and busbar data of this system are given in Tables 1 and 2, respectively. The second test system is the IEEE 14-bus system which contains 20 transmission lines. The complete data of this system are given in ref. 11. TABLE 1 Transmission line data Line No.

Bus code I- J

Impedance (p.u.)

Line charging

1 2 3 4 5 6 7

1122234-

0.02 + 0.08 + 0.06 + 0.06 + 0.04 + 0.01 + 0.08 +

j0.030 j0.025 j0.020 j0.020 j0.015 j0.010 j0.025

2 3 3 4 5 4 5

j0.06 j0.24 j0.18 j0.18 j0.12 j0.03 j0.24

Y'/2

3.2. Results and comments

Table 3 shows a comparison between the proposed voltage technique and the NewtonRaphson load flow technique to calculate the bus voltage after the occurrence of circuit contingencies in lines 1, 3, 4 and 5 of the first test system. Table 4 shows a comparison between the proposed linearization voltage technique and the Newton-Raphson load flow technique to calculate the bus voltage for the second test system following outages of lines 3, 7, 12 and 14. The solution of the proposed technique is a direct procedure and it takes only one iteration. Furthermore, the proposed technique adopts the Newton-Raphson technique and hence convergence is assured for any line outage. Table 5 shows a comparison between the computation time and the number of iterations for the proposed technique and the Newton-Raphson load flow technique for the two test systems. A microprocessor compatible with the IBM 8088 computer was used for these calculations.

TABLE 3 Comparison between the proposed voltage technique and the Newton-Raphson technique for line outages for the first system Bus voltage

Line outage

(v) No. 1

E1 E2 E3 E4 E5

No. 3

No. 4

No. 5

NR

Proposed

NR

Proposed

NR

Proposed

NR

Proposed

1.06(O 1.0000 0.9760 0.9710 0.9660

1.0600 1.0001 0.9780 0.9733 0.9683

1.06(X) 1.0000 0.9790 0.9770 0.9690

1.0600 1.0001 0.9730 0.9779 0.9699

1.0600 1.0000 0.9790 0.9730 0.9680

1.06(O 1.0000 0.9795 0.9744 0.9696

1.0600 1.0O00 0.9630 0.9570 0.8700

1.0600 1.0000 0.9643 0.9583 0.8715

217 TABLE 4 Comparison between the proposed voltage technique and the Newton-Raphson technique for line outages for the second system Bus voltage (V)

Line outage No. 3

E1 E2 E3 E4 E~ E~ E7 Es E9 Elo El, E12 E13 E14

No. 7

No. 12

No. 14

NR

Proposed

NR

Proposed

NR

Proposed

NR

Proposed

1.060 1.000 0.912 0.918 0.940 0.918 0.888 0.888 0.874 0.873 0.891 0.899 0.891 0.860

1.060 1.000 0.914 0.921 0.944 0.921 0.889 0.889 0.875 0.875 0.894 0.900 0.899 0.863

1.060 1.000 0.911 0.916 0.990 0.953 0.893 0.893 0.881 0.885 0.913 0.832 0.922 0.877

1.060 1.000 0.913 0.917 0.994 0.955 0.894 0.894 0.885 0.888 0.915 0.834 0.928 0.871

1.060 1.000 0.929 0.949 0.959 0.917 0.949 0.949 0.840 0.843 0.874 0.895 0.884 0.836

1.060 1.001 0.923 0.944 0.954 0.910 0.942 0.942 0.843 0.846 0.876 0.896 0.886 0.839

1.060 1.000 0.930 0.956 0.967 0.964 0.911 0.911 0.892 0.880 0.873 0.943 0.933 0.889

1.060 1.001 0.933 0.954 0.961 0.966 0.914 0.914 0.894 0.883 0.875 0.945 0.936 0.883

TABLE 5

REFERENCES

Computation time and number of iterations for the proposed voltage technique and the Newton-Raphson technique for the two test systems

1 K. T. Khu, M. G. Lauby and D. W. Bowen, A fast linearization method to evaluate the effects of circuit contingencies upon system load-bus voltages, IEEE Trans., PAS.101 (1982) 3926- 3932. 2 N. M. Peterson, W. F. Tinney and D. W. Bree, Iterative linear AC power flow solution for fast approximate outage studies, IEEE Trans., PAS-91 (1972) 2048 - 2053. 3 T. Krishnaparandhama, S. Elangovan, A. Kuppurajulu and V. Sankaranarayanan, Fast power-flow solution by the method of reduction and restoration, Proc. Inst. Electr. Eng., Part C, 127 (1980) 90-93. 4 G. B. Jasmon, R. M. Amin and C. Y. Chuan, Performance comparison of two exact outage simulation techniques, Proc. Inst. Electr. Eng., Part C, 132 (1985) 285-293. 5 S. Iwamoto and Y. Tamura, A fast load-flow method retaining nonlinearity, IEEE Trans., PAS-97 (1978) 1586- 1597. 6 M. R. Irving and M. J. H. Sterling, Efficient NewtonRaphson algorithm for load-flow calculation in transmission and distribution network, Proc. Inst. Electr. Eng., Part C, 134 (1987) 325 - 328. 7 M. A. Farrag and H. A. Attia, Fast and exact method for line outage studies in power systems, 13th Int. Congr. Statistics, Computer Science, Social and Demographic Research, Ain Shams University, Cairo, Egypt, March 1988, pp. 143 - 158. 8 T. K. P. Medicheria, R. Billinton and M. S. Sachdev, Generation rescheduling and load shedding to alleviate line overloads--analysis, IEEE Trans., PAS-98 (1979) 1876- 1883. 9 W. F. Tinney, Compensation methods for network solution by optimally ordered triangular factorization, Proc. PICA Conf., Boston, MA, U.S.A., 1971, IEEE, New York, pp. 377 - 382. 10 G. W. Stagg and A. H. E1-Abiad, Computer Method in Power System Analysis, McGraw-Hill, New York, 1968. 11 M. A. Pal, Computer Techniques in Power System Analysis, McGraw-Hill, New York, 1979.

Variables

Time (s) No. of iterations

5-bus system

14-bus system

NR

Proposed

NR

Proposed

17 4

4 1

32 5

7 1

4. CONCLUSIONS This paper presents a proposed linearization method to evaluate the effects of one or more line contingencies upon the system bus voltages e v e n if n o n - c o n v e r g e n c e exists in the i t e r a t i v e techniques. The complete Taylor expansion of t h e n o d a l l o a d f l o w e q u a t i o n s is c o n s i d e r e d . T h e c o m p l i c a t e d load flow e q u a t i o n s h a v e b e e n simplified by e l i m i n a t i n g the i m a g i n a r y p a r t of the bus v o l t a g e and c o n s i d e r i n g only its real part. T h e c o m p u t a t i o n t i m e and c o m p u t e r storage are thus reduced. To obtain a greater r e d u c t i o n in the c o m p u t a t i o n time, the final s e n s i t i v i t y m a t r i x D - 1 is c o m p u t e d a n d s a v e d f o r a l l c i r c u i t c o n t i n g e n c i e s . T h e n o n - z e r o elements only of the Jacobian and Hessian modified matrices are stored to obtain a further r e d u c t i o n in c o m p u t e r storage.

218

APPENDIX

~Q~/~el = e, [ B u ( 1 + 4,4j) + Gu(4 , - 4 i)]

The elements of the Jacobian matrix are obtained by taking the first partial derivatives of Pi = f(e) and Qi = f(e) of eqns. (5) and (6): t~Pi /t~e i = 2 e i G i i ( 1

-+- 4i 2)

N

+ ~. e j [ G u ( l + 4 ~ 4 j ) - B i j ( 4 ~ - 4 1 ) 1

j=l j~i

OPi/Oej = ei [Gu(1 + 4iitj) - Bo(2i - 42)1

(A-l) (A-2)

and cOQi/(Oei = 2eiBii(1 + )~i2) N

+ ~ ej [Su(1 + 4,4i) + V u ( 4 ~ - 4i11 j=l

(A-4)

The elements of the Hessian matrix are obtained by taking the second partial derivatives of P(e) and Q(e) of eqns. (5) and (6): ~2p~/~ei2 = 2Gii(1 + 42)

(A-5)

(~2p,/~eje, = G u ( 1 + ).,)~j ) - B,j(4, - )~j )

(A-6)

and c~2Qi/~ei 2 = 2Bii(1 + 4i 2)

(A-7)

~2Q,/~eje, = Gu(4 , - 4j) + Bij(1 --F 4i4j)

(A-S)

where hi and ~j are tan 0~ and tan 0j, respectively, 0~ and 0j are the voltage bus phase angles after the circuit contingency, and YIj = Gij - j B i j is the admittance after the circuit contingency.