development of constrained-genetic-algorithm load-flow method

Development of constrained-genetic-algorithm load- flow method

K.P.Wong A. Li M .Y. Law

Indexing terms: Genetic algorithms

Abstract: A genetic-algorithm load-flow algorithm is developed. Methods for satisfying the power balance requirement and the voltage magnitude constraint are then developed and incorporated into the genetic-algorithm method to form a constrained-genetic-algorithm load-flow algorithm for solving the load-flow problem. The robustness of the new load-flow algorithm is enhanced by the dynamic population method, the technique for accelerating the convergence of the optimisation process and the network node sequencing procedure described in the paper. The paper presents study results obtained by applying the new algorithm to study the practical Klos- Kerner 1 1-busbar system under light-load and heavy-load conditions.

List of principal symbols

P p , Pi = specified and calculated active powers at node i

QfP, Qi = specified and calculated reactive powers at node i

V[p, Vi = specified and calculated voltage magnitude at node i

vk = nodal voltage vector at node k Gij, By = real and imaginary parts of the (i, j)th ele-

ment of the admittance matrix Ei, Fi = real and imaginary parts of the nodal volt-

age at node i Eid, Fid = updated real and imaginary parts of the

nodal voltage at node d F = fitness function H = total squared mismatch

NP4

NP"

= total number of PQ nodes = total number of PV nodes

0 IEE, 1997 IEE Proceedings online no. 19970847 Paper first received 3rd April and in revised form 19th August 1996 K.P. Wong and A. Li are with the Artficid Intelligence and Power Systems Research Group, Department of Electrical and Electronic Engineering, University of Western Australia, Nedlands, Western Australia 6907 M.Y. Law is with the Planning and Contracts Section, Western Power Corporation, 363 Wellington Street, Perth, Western Australia 6000

1 Introduction

When the load in a power system is heavy, the system becomes stressed and it will be difficult for the New- ton-Raphson load flow (NRLF) method [l] to converge. This is because when the power system is operating very close to its ceiling load point, the Jaco- bian of the load-flow equation set tends to be singular. However, it is of practical importance to be able to determine the load-flow solutions of the system under highly stressed conditions.

It is also well known that generally the load-flow equation set has multiple solutions [2, 31. In the multiple solutions, one is the normal solution and the others are abnormal solutions. However, if the system is operating at its steady-state ceiling point there is only one solution. A limiting condition of the system is indicated when the existence of near solutions is found. The esti- mation of the steady-state operating limit is important since any extra loading imposed on the system beyond the limit can lead to voltage collapse. To achieve this, however, it requires a load-flow algorithm which can determine multiple solutions. Although an optimal multiplier method [3] has been developed and incorporated in the NRLF method to find a pair of near solutions, it is not general enough to find all the multiple solutions. Moreover, since this method is still based on NRLF, it will not be able to determine the solution at the steady-state ceiling point of the system.

The advent of power electronic devices provides new ways of improving and expanding the performance and capacity of power transmission systems. Transmission systems which adopt power electronic devices for the control of power transmission have been referred to as flexible AC transmission systems (FACTS). The load flow equations of a power system containing FACTS will be much more nonlinear and the equation set may be nonconvex. The conventional NRLF method may not solve the equations satisfactorily.

Recently, there has been a great deal of interest in modelling nonlinear loads and in investigating the effects of these loads on the stability of power systems using chaos and bifurcation theory [4, 51. Before the stability can be examined, the steady-state operating point of the system is sought by solving the load-flow equations. However, the inclusion of nonlinear load models in the load-flow equations can cause gradient- based load flow methods such as the NRLF method to fail.

From the above discussions, there is a need to develop new load-flow algorithms which do not con-

91 IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997

tain the limitations and difficulties of the NRLF method. This paper reports the work on the development of a new load-flow algorithm. It proposes to treat the load-flow problem as an optimisation problem. The squared mismatches of the active and reactive powers at the network nodes and the squared mismatches of the nodal voltages are to be minimised subject to the power balance requirements at the nodes of the power system and the constraints on the voltage magnitudes at the generator nodes. The load-flow problem is then solved by an algorithm based on the concept of the genetic algorithm (GA) [6]. This concept is adopted here for the development of a load-flow method because the GA optimisation process (a) does not require the formation of the Jacobian matrix; (b) is almost insensitive to the initial settings of the solution variables and (c) has the ability to find multiple load- flow solutions.

A GA load-flow algorithm is first developed in the paper. Methods for satisfying the power balance requirement and the voltage magnitude constraint are then developed and incorporated into the GA method to form a constrained-GA load-flow (CGALF) algorithm. The robustness of the new load-flow algorithm is enhanced by the dynamic population method, the technique for accelerating the convergence of the optimisation process and the network node sequencing procedure developed in the paper. Whilst the performances of the developed CGALF algorithm have been tested using the Klos-Kerner 3-node system [2] and the Ward- Hale 6-node system [7], the paper presents study results obtained by applying the CGALF algorithm to study the Klos-Kerner 1 1-node system [2] under light-load and heavy-load conditions. The CGALF algorithm can find multiple solutions and it can determine the load- flow solution at the ceiling load point. In the application example, it is also shown how by voltage relaxa- tion at the generator nodes, the new algorithm can move the heavily loaded system back to the solvable state when the system lies just outside the solvable region.

2 Load-flow problem

The load-flow problem can be described mathemati- cally as follows. Consider there are a total number of N nodes in a power system. At any node i, the nodal active power Pi and reactive power Qi, are given by:

N N

P, = E, (G;, E, - B,, F ~ ) + F, ( G ~ ~ F~ + B,, E, ) j=1 j=1

i = 1 , 2 , 3 , . . . , N (1) N N

j=1 3=1

2 = 1 , 2 , 3 , . . , N (2) where G, and B, are the (i, j)th element of the admittance matrix. E, and F, are real and imaginary parts of the voltage at node i. If node i is a PQ node where the load demand is specified, then the mismatches in active and reactive powers, AP, and AQ,, respectively, are given by

n P, = I P,~P - P, 1 AQt = IS$ - Qz1

( 3 )

(4) in which P,“P and Q:p are the specified active and reac-

92

tive powers at node i. When node i is a PV node, the magnitude of the voltage VfP and the active power generation at i are specified. The mismatch in voltage magnitude at node i can be defined as

nv, = - v,l (5) and the active power mismatch is given in eqn. 3. In eqn. 5 , V , is the calculated nodal voltage at PV node i and is given by

& = d m (6)

The unknown variables in this problem are (i) the voltages at the PQ nodes and (ii) the real and imaginary parts of the voltages at the PV nodes and the reactive powers of the generators connected to the PV nodes. It is required to determine the values of the variables such that the mismatches in eqns. 3-5 are zero.

Apart from solving the load-flow problem by conventional methods, the problem can be viewed as an optimisation problem, in which an objective function H is to be minimised. The objective function can be defined as the sum of the square of the power mismatches and the voltage mismatches as in

%E Np q$ N p v

+ jv,”-v,12 (7 ) 2ENpv

where Npq, Ne” are the total numbers of PQ and PV nodes, respectively.

3 Simple genetic algorithm for solving the problem

GAS are search techniques based on an analogy with biology in which a group of solutions evolves through natural selection. In their implementation, a population of randomly generated candidate solutions evolves to an optimum solution through the operations of genetic operators consisting of reproduction, crossover and mutation. These techniques have recently been applied to some power system operation problems [8-10].

As an optimisation technique, GAS are much less dependent on the start values of the variables in the optimisation problem when compared with the widely used Newton-Raphson method. In addition, GAS in their solution process do not rely on the guidance of the gradient such as the Jacobian matrix in the case of the Newton-Raphson method and hence they are more capable of determining the global optimum solution.

An earlier method based on genetic algorithms (GAS) [6, 111 has been reported in [8]. In that genetic-based method, there is no measure developed to handle the power balance equation set in the load-flow problem. Because of this, the earlier method in [8] does not produce accurate load-flow solutions in that the power balance requirement is not met closely. The accuracy of the solutions obtained by the earlier method is not high also because of the adoption of the binary-bit coding method to code the voltage values which are continu- ous. For these reasons, the repeatability of the earlier genetic-based method is poor.

From the above discussions, this paper develops a constrained-GA load-flow (CGALF) algorithm based on GAS and a constraint satisfaction method developed in Section 4. A genetic-algorithm load-flow (GALF) method is first developed and its principal components

IEE Proc -Gener Transm Distrib , Vol 144, No 2, March 1997

are presented in the following Section. An introduction to genetic algorithms and genetic operators can be found in [6, 101.

3.1 Components in genetic approach To solve the load-flow problem, the following GA components are designed: (i) Chromosomes: The real and imaginary parts of the voltages of the nodes in the power system are encoded using floating-point numbers [lo, 121 and are set as elements in the chromosomes. By this coding method, no discretisation error [IO] will be introduced. (ii) Fitness function: The following fitness function is proposed:

M

where M is a constant for amplifying the fitness value. The value of H approaches zero towards convergence. To avoid any numerical difficulty that may occur in calculating F, H is augmented by In the present work, this fitness function is used for determining both the normal and abnormal load-flow solutions. (iii) Crossover operation: In the present work, the 2- point crossover method [IO, 131 is adopted so that more diversity in the population of chromosomes can be achieved. (iv) Mutation operation: An element of a chromosome is selected randomly. The voltage value of the element is replaced by a value arbitrarily chosen within a range of voltage values.

3.2 GA load-flow algorithm With the components in items (i) to (iv) in Section (3. I), the procedure for solving the load-flow problem using the basic GA approach is summarised below: Step 1 Initialise s chromosomes in the population. Step 2 Generate the next generation of s chromosomes in the following way:

Step 2.1 Evaluate the fitness of the chromosomes in the current generation using eqn. 8. In the present work, the fittest chromosome in the current generation is always retained in the next generation. Step 2.2 Reproduction: select two chromosomes as the parent by the 'roulette wheel' method [6] for reproduction. Step 2.3 Crossover: with the specified crossover probability, P,,, apply crossover to the two selected parents in the current generation when the value of a random number generated between 0 and 1, rand[O, 11, is less than Prc. Otherwise, the two parents are retained and are taken as the child chromosomes in the next generation. Repeat the selection step in Step 2.2 and the present step until s child chromosomes are formed in the next generation. Step 2.4 Mutation: for each chromosome in the next generation, apply mutation to the elements of the chromosome when the mutation probability, P,,, is greater than rand[O, 11. Otherwise, the chromosome will remain intact.

Step 3 The next generation formed in Step 2 is now taken to be the current generation. New generations are produced by repeating the solution process starting from Step 2 until the specified maximum number of generations is reached.

( 8 ) F = 10-5 + H

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997

4 load flow

Method to satisfy the equality constraints in

The GALF algorithm in Section 3.2 does not guarantee the complete satisfaction of the equality constraints expressed in the form of the equations of power mismatch and voltage mismatch in eqns. 3-5. This means that, although GALF minimises the total squared mismatch H in eqn. 7, it does not necessarily produce a solution which leads to a zero or a near zero value for H. Some constraint handling techniques have previ- ously been developed and can be found in [14]. How- ever, these techniques are not applicable to the present problem, and new constraint satisfaction techniques are developed in Sections 4.1 - 4.3.

4. I Constraint satisfaction technique for updating candidate nodal voltages To assist the satisfaction of the equality constraints in eqns. 3-5, it is proposed here to process and to update the candidate solution voltages in a new chromosome formed by either crossover or mutation such that the revised load-flow solution in the chromosome will have a zero H value or an H value smaller than that of the original solution, based on the following technique: (a) satisfying the powers at a PQ node i by updating a PQ node d. Consider a PQ node i in a given network. Its specified active and reactive powers can be satisfied by updating the voltage at a PQ node d in the same network. The method to satisfy the specified powers will be developed in Section 4.2. By considering all the PQ nodes, that is, i = 2, ..., N, where N is the total number of network nodes, a complete set of possible voltages for d can be found. From this set of voltages, the updated voltage of node d is the voltage which gives the maximum reduction to the total power mismatch. Node 1 is reserved for the slack node

(b) updating the voltage at a PV node to satisfy its voltage and active power requirements. For any PV node, the real and imaginary part of its voltage are updated so that the voltage magnitude constraint and the active power constraint can be met. The constraint satisfaction method for the PV nodes will be detailed in Section 4.3.

4.2 Constraint satisfaction for PO nodes The constraint equations for the satisfaction of the specified active and reactive powers at any PQ node i by updating the nodal voltage of a PQ node d are developed in the following. Let the real and imaginary voltages of node d be Eid and Fid. The power mismatches AP, in eqn. 3 and AQi in eqn. 4 for node i are now set to zero. From eqns. 1-4, when d z i, Eid and Fid can be calculated according to

(9)

93

where: N N

j=1 3 # d

When d = i, the power constraints at PQ node d itself are required to be met. The constraint equations for calculating Ezd and Fzd of node d can be derived from eqns. 1 4 by the same procedure above and by setting the subscript i in eqns. 1-4 to d. The two resultant expressions are lengthy and are therefore omitted here for clarity.

As explained in Section 4.1, the new values of Ezd and Fld which lead to the lowest value of H will replace the old values of Eld and Fzd held in the chromosome. Oth- erwise, the original solution is retained.

4.3 Constraint satisfaction for PV nodes The constraint equations for the satisfaction of the voltage magnitude constraint at any PV node d by updating its own nodal voltage are developed in the following. Let the real and imaginary voltages of the PV node d in the chromosome be Edd and Fdd. The mismatches APd in eqn. 3 and AVd in eqn. 5 for node d can now be set to zero. From eqns. 1, 3, 5 and 6, the expressions for Edd and Fdd are:

solution process to the global optimum point where the value of H is zero, a method is developed in Section 5.3 for deciding the sequence of the network nodes according to which the constraint satisfaction methods in Sec- tions 4.2 and 4.3 are applied to the nodes.

5. I Dynamic population technique To assist CGALF to escape from local minimum points, the diversity of the chromosomes, that is, the dissimilarity of the candidate solutions in the chromosomes must be increased. This can be achieved by introducing new chromosomes into a population dynamically throughout the solution process. In the solution process, a percentage of existing weaker chromosomes in a population is replaced by randomly generated chromosomes when the values of objective function H are identical for a specified number of generations or iterations. Similar to the other chromosomes in the population, the randomly generated chromosomes are also subjected to the constraint satisfaction process. This technique is here referred to as the dynamic population technique. It is particularly useful when the CGALF algorithm is used to find the minimum total squared mismatch H for loading conditions under which the load flow problem is insolvable.

It has been found experimentally that the range of the percentage of chromosomes to be replaced dynamically is from 30 to 50%. In general, if the specified population size is low, then the percentage value should be high in the given range.

X d d d V l p 2 ( X & + Z&) - (P;l"" - V,"P2Gdd)2 F

X2d + 22, (13)

where:

X d d E C ( G d j E I -Bd,FJ) N N

Z d d = X ( G d j F J +BdjEj) 3 = 1 3 = 1 i # d 3 f d

(14) With the two possible sets of values of Edd and Fdd calculated according to eqns. 12-14, the total squared mismatch H associated with the revised solutions of the chromosome can be evaluated respectively. The solution which gives the lower value of H is taken as the solution for the chromosome under consideration.

The developed constraint satisfaction techniques are incorporated into the GALF algorithm after the crossover operation in step 2.3 and the mutation operation in step 2.4 in Section 3.2. The modified algorithm is here referred to as the constrained-genetic-algorithm- load-flow (CGALF) algorithm.

5

The performance of the CGALF algorithm can be enhanced by incorporating in it a dynamic population technique and a solution acceleration technique. These techniques are developed in Sections 5.1 and 5.2. More- over, to greatly increase the chance for directing the

Methods for enhancing the CGALF algorithm

94

5.2 Solution acceleration technique Whilst the dynamic population technique in Section 5.1 provides a mechanism for CGALF to jump out of the local optimum points, a solution acceleration technique can be developed to assist CGALF to determine the optimum solution in a fast manner. The idea in this technique is to modify the constrained candidate solutions in the chromosomes of the population during the solution process such that the revised solutions in the chromosomes are closer to the candidate solution in the best or fittest chromosome found so far. This can be implemented in the following way. For any node k the nodal voltage vector difference, AVk, between the nodal voltage of k, v k , in a chromosome and the corresponding voltage, Vk,best, in the best chromosome is first calculated. The modified voltage at node k, Vk', in the chromosome is then given by the sum of AVk and the nodal voltage of node k in the best chromosome, Vk,best . Equivalently, Vk' is given by

vk = 2 V k , b e s t - vk (15) The solution acceleration technique is performed before step 2.1 in step 2 of Section 3.2. As mentioned in item (i) of Section 3.1, the chromosomes are encoded using floating-point numbers. The floating-point numbers are therefore utilised in the developed solution acceleration technique above.

5.3 Nodal voltage updating sequence The simplest sequence by which the nodal voltages of the nodes can be updated is to follow the sequence of the numbers assigned to the nodes. However, this is not the most efficient way. A better alternative is given in the following procedure: (i) Update the voltages of the PV nodes in the sequence of the node numbers using eqns. 12-14.


(ii) Then, the PQ node, which has the largest total mismatch, is updated first using the constraint satisfaction methods in Section 4.2. (iii) Repeat step (ii) until all the PQ nodes are proc- essed. In step (i) above, the update operation attempts to meet the voltage magnitude constraints and active power requirements of the PV nodes. This is an important step as it enables better voltage values to be found during the update of the voltages at the PQ nodes in step (ii). The strategy employed in step (ii) guarantees a reduction of the mismatch at the node with the largest total mismatch. The strategy is applied dynamically during the processing of the nodes as indicated in step (iii).

6 Application examples

The CGALF algorithm developed has been implemented using the C programming language and the software system is run on a Pentium 90 personal computer. It has been validated through its applications to determine the normal and abnormal load-flow solutions of the Klos-Kerner 3-node (KK3) and 11-node (KKL11) systems [2], and the Ward-Hale 6-node system [7]. To illustrate the power and the performance of CGALF, studies performed on the KK11 system are presented in the paper. The Klos-Kerner test systems are adopted in the present work since they are the first systems used to demonstrate the nonuniqueness of the load-flow solutions. Furthermore, the KKl l system is a practical system and is of particular interest when it is under the heavy-load condition. This system has also been studied extensively by other researchers [2, 31.

U

Fig. 1 Klos-Kerner 11-node test system

Fig. 1 shows the KKl l test system. Node 1 is the slack node at a voltage level of 1.05p.u. and nodes 5 and 9 are PV nodes with target voltages of 1.05p.u. and 1.0375p.u., respectively. The network data of KKl l are given in Table 1. Two loading conditions are considered. In Table 2, the load demands at the nodes under a light-load condition and a heavy-load condition are tabulated. The base power for the system is 100MVA.

The parameter settings for executing CGALF are as follows. The value of A4 in the fitness function in eqn. 8 is 100, the probability of crossover is 0.9, the probability of mutation is 0.01, the population size is 100 and the maximum number of generations or iterations is 100. The initial candidate solution chromosomes are formed randomly in the voltage range from 0.0p.u. to


1.2p.u., and in the voltage phase angles range between 0" and -180". These ranges are also employed in the mutation operation. 50% of the weaker chromosomes in the population are replaced by the new chromosomes using the dynamic population technique. This technique is imtiated in the CGALF optimisation process when the value of the total squared mismatch H is unchanged for five successive iterations.

Table 1: Network data of Klos-Kerner 11-node system

Shunt admittance Series impedance (P.u.)

R X B

Node (P.U.) connection

1 2 0.00250 0.0188 0.640

1 10 0.00500 0.0375 1.280

2 3 0.001 25 0.0125 0.258

2 11 0.00625 0.0625 1.600

3 4 0.00375 0.0312 0.640

3 6 0.00438 0.0375 1.280

4 5 0.00312 0.0312 0.960

4 6 0.00312 0.0312 0.960

4 11 0.00188 0.0123 0.320

5 7 0.00625 0.0625 1.600

6 7 0.00625 0.0625 1.600

7 8 0.00125 0.0125 0.256

8 9 0.00188 0.0188 0.480

9 10 0.00125 0.0125 0.320

10 11 0.00312 0.0312 0.960

System power base is 100 MVA

Table 2: Specified generation, loads and nodal voltages for Klos-Kerner 1 I-node system

Node Light-load case Heavy-load case

no. VSP p q o ) p s p QSP VSP ev(0) p s p QSP

1 1.05 0.0 2 - - 3 - - 4 - -

5 1.05 - 6 - - 7 - - 8 - -

9 1.0375 - 10 - 11 -

- -

1.05 0.0

-0.10 -0.10 - - -0.15 -0.05 - - -0.20 -0.05 - -

0.20 - 1.05 - 0.00 0.00 - -

-0.12 -0.10 - - -0.05 -0.15 - -

0.50 - 1.0375 - -0.20 -0.10 - -

0.08 0.06 - -

_ _ - -

-9.00 -2.00

-16.5095 -1 .OO

-25.00 -2.00 10.00 -

0.00 0.00

-12.00 -0.80

-4.00 -0.50

25.00 - -8.00 -2.00

9.00 6.30 ~~

Voltage and powers are in p.u. on 100 MVA base - identifies load demand

6.1 Light-load condition Under the light-load condition, the KK11 system has one normal solution and multiple abnormal solutions [2]. In 100 separate executions, CGALF found 41 solutions, one of which is the normal solution. Table 3 summarises the best and the worst normal solutions found by CGALF and that in [2]. The best and worst solutions by CGALF have zero total squared mismatch H while that from [2] has 4.503508. This shows that the CGALF solutions are very accurate. The best solution is obtained at the 7th iteration and the worst solution at the 9th iteration. Fig. 2 shows the best and the worst

95

convergence characteristics in finding the normal solution. The computing time is about 1.7 s per iteration.

number of iterations Fi .2 Convergence characteristics of CG.4LF in finding the nornuzl s o & m f o r the light-load case (i) Best convergence characteristic (ii) Worst convergence characteristic

In [2 ] , Klos and Kerner summarised four abnormal solutions. CGALF finds an extra 36 abnormal solutions within 100 executions. One of the abnormal solutions is given in Table 4 and is compared bith the corresponding solution in [2]. In this solution, the voltage of node 10 approaches zero The best and the worst CGALF solutions have zero total squared mismatch, while that from [2] has 0.117677. This denion-

strates the large capacity of CGALF to find the abnormal solutions. Fig. 3 shows the best and the worst convergence characteristics for this case.

1 5 9 13 17 21 25 29 33 number of iterations

Fip.3 C,,vergen: characteri.stics of CGALF in jinding an abnormal

(i) Best convergence characteristic (ii) Worst convergence characteristic

$0 uliori,/oi i / ~ l&i loud L U A ~

6.2 Hea vy-loa d condition The loadings in the KK11 system under the heavy-load condition have been given in Table 2. Tn [2], Klos and Kerner presented two very near normal solutions, the nodal voltage solutions of which only differ in the third or fourth decimal place. However, the total squared mismakhes of the two solutions are 0.058745 and 0.058730. If 0.001p.u. on l00MVA base is assumed to

Table 3: Best and worst results of the normal solution in the light-load case by CGALF

Node Result from reference 2 Best result by CGALF Worst result by CGALF

1 1.050 0.0 1.050000 0.000000 1.050000 0.000000 2 1.117 -0.05 1.107692 -0.531757 1.107692 -0.531344 3 1.127 -0.7 1 .I27545 -0.666285 1.127544 -0.665722 4 1.124 -0.6 1.122848 -0.569190 1.122849 -0.568442 5 1.05 0.0 1.050000 0.048732 1.050000 0.049576 6 1.15 -0.3 4.503509 1.148930 -0.71 1141 0.000000 1.148931 -0.710421 0.000000 7 1.10 -0.3 1.098938 -0.328582 1.098940 -0.327750 8 1.077 -0.1 1,076280 -0.127553 1.076282 -0.126719 9 1.0375 0.3 1.037500 0.267623 1.037500 0.268313 10 1.066 -0.1 1.068077 -0.096143 1,068078 -0.095579 11 1.12 -0.5 1.120351 -0.484420 1.120352 -0.483713

Table 4: Best and worst results of an abnormal solution in the light-load case by CGALF

Node Result from reference 2 Best result by CGALF Worst result by CGALF

1 1.050 0.0 1.050000 0.000000 1.050000 0.000000 2 0.865 -12.8 0.863752 -12.814797 0.863756 -12.814409 3 0.822 -21.4 0.821618 -21.439325 0.821662 -21.438629 4 0.750 -32.9 0.748880 -33.021568 0.748883 -33.020603 5 1.050 -44.3 1.050000 -44.392467 1.050000 -44.391 399 6 0.818 -35.3 0.1 17677 0.816430 -35.338360 0.000000 0.816433 -35.337257 0.000000 7 0.972 -58.3 0.968884 -58.341 721 0.968884 -58.340832 8 0.990 -64.7 0.989085 -64.780800 0.989085 -64.779991 9 1.0375 -73.7 1.037500 -73.767937 1.037500 -73,767250 10 0.002 -110.0 0.002035 -1 10.058594 0.002035 -1 10.077156 11 0.580 -28.7 0.578786 -28.744135 0.578788 -28.743256

96 IEE t-'voc.-Gener. 7 i . r " . Llrsfvih. Vol. 144, No. 2, Mavch 1997

be the solution tolerance, the active and reactive power mismatches at the nodes are greater than this tolerance. To the authors, the power mismatches are high and the two near normal solutions found by Klos and Kerner under the specified loading condition may not be valid.

The authors have presented the system to a Newton- Raphson load flow (NRLF) program. Using 0.001 p.u. on a 100MVA base as the tolerance for the power mismatch and 0.001p.u. voltage as the tolerance for the voltage mismatch, the NRLF fails to converge to a solution after 20 iterations and the total squared mismatch H is 0.005687. Corresponding to the adopted tolerance, the value of H should be 0.000020. One possibility here is that the given loading condition leads to the situation that the load-flow equation set is unsolvable. The second possibility is that the given loading condition is at the steady-state ceiling point of the system, that is at the saddle-nose bifurcation point, at which the Jacobian matrix in the NRLF is singular. The third possibility is that this loading condition is near the saddle-nose point and the Jacobian is nearly singular, causing NRLF to diverge.

The developed CGALF algorithm is applied to the system under the specified heavy-load condition and the solution is given in Table 5. The total squared mis-

match is 0.000131 due to the voltage magnitude mismatch at nodes 5 and the active and reactive power mismatches at the PQ nodes. As mentioned in Section 5.3, the voltage magnitude constraints on the PV nodes are always observed by CGALF. The fact that the voltage magnitude mismatch at node 5 is nonzero indicates that the present load flow problem is unsolvable. The nonzero voltage mismatch arises from a nonzero value of AV5 the nodal voltage vector difference defined in Section 5.2. The value of AV5 should be zero at convergence if the load flow problem is solvable and the value of H is zero. Owing to the insolvability of the problem, NRLF does not converge to a solution. It follows that the two very near solutions given by Klos and Kerner [2] are not valid solutions.

One way to determine the solution which meets the specified load demands is to relax the voltage constraints on PV nodes 5 and 9. This can be achieved eas- ily by modifying CGALF slightly. The modification is in step (ii) of the node sequencing procedure in Section 5.3. In addition to updating the PQ nodes in step (ii), the voltages of the PV nodes are also updated. The constraint equations for calculating the possible new voltages for the PV nodes are given in eqns. 12 and 13 in Section 4.3 for satisfying the required voltage magni-

Table 5: Best solution for the heavy-load case by CGALF

Mismatches

H Node no. vi (P.u.) e, ( 0 ) Pi“ - P, Q,’P - 0, v i s p - vi

(P.U.) (P.U.) (P.U.) - - 1 1.050000 0.000000 -

2 0.722943 -36.733223 -0.001387 -0.004189 - 3 0.712220 -57.940029 -0.005795 -0.002203 - 4

5

6

7

8

9

10

11

0.816123

1.050 146

0.816222

0.904489

0.920475

1.037502

0.964997

0.887599

-56.339577

-40.2 5989 5

-54.381 390

-45.357384

-34.486439

-15.482958

-20.607037

-39.122875

-0.003992

-0.004489

-0.003340

-0.00236

-0.001586

-0.000944

-0.000549

-0.001139

-0.002477 -

-0.002306

-0.000367

0.000821 -

-0.000544

-0.001696

-

-0.000146

- 0.000131 - -

-0.000002 - -

Superscript ’sp’ denotes specified value

Table 6: Best solution for the heavy-load case by CGALF with PV nodes relaxed

Mismatches

H Node no. v, (PA.) e, P,’P - Pi QisP - Q, V:P - V,

(P.U.) (P.U.) (P.U.) 0.000000 - - -

2 0.725584 -36.619987 -0.00001 1 -0.000037 - 3 0.715226 -57.682549 -0.000048 -0.000049 - 4 0.818657 -56.1 17172 -0.000036 -0.000021 -

5 1.050887 -40.106743 -0.000026 - -0.000887

6 0.818812 -54.173855 -0.000058 -0.000016 - 0.000001

7 0.906194 -45.213337 -0.000008 -0.000007 - 8 0.921990 -34.386974 -0.000031 -0.000019 - 9 1.038254 -15.436056 -0.000023 - -0.000754

10 0.9661 13 -20.558481 -0.000001 0.000000 - 11 0.889895 -39.001202 -0.000018 -0.000031 -

1 1.050000

Superscript ’sp’ denotes specified value

IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997 91

tude and in eqns. 9 and 10 in Section 4.2 for the satisfaction of the specified active and reactive powers at the PQ nodes in the network. The voltage which leads to a lower total squared mismatch value will be adopted. By this method, the PV node voltage magnitudes fixed by step (i) in the same procedure can be relaxed in a controlled manner.

Table 6 summarises the solution found by CGALF when PV nodes 5 and 9 are relaxed. The voltage at node 5 is slightly increased beyond its target voltage of 1.05p.u. but the increment is < 0.001p.u. The slight increment in the voltage at node 9 is also less than the 0.001p.u. tolerance. From the result in Table 6, it can be seen that the power mismatches are all within the tolerance. This result shows that the specified load demands can be supported by raising the voltage at nodes 5 and 9 slightly. It is interesting to observe that the solution is valid subject to the tolerance of 0.001p.u. This confirms that when the voltages at nodes 5 and 9 are kept strictly at their targeted levels, the specified load demand will drive the system to oper- ate at a point which lies just beyond the ceiling point.

g loo

-2 E 10 0 3 U CO

I lo'/

:I l o 1 11 21 31 41 51 61 71 81 91

number of iterations Convergence characteristics of CGALF in finding the solution for Fig. 4

the heavy-load case with 0.04% load reduction

The fact that when the voltages at nodes 5 and 9 are slightly raised a valid solution can be found by CGALF indicates that a slight reduction in the power demand will drive the system back into the solvable region with nodes 5 and 9 kept at the target voltage levels. By reducing the original system load demand by

0.04% and by sharing the reduction evenly throughout all the PQ nodes in the system, a ceiling point is found and the solution is unique subject to the 0.001p.u. tolerance. This solution is summarised in Table 7. It is obtained at the 12th iteration where the total squared mismatch is 0.0000048. The execution time is - 55s. Fig. 4 shows the convergence characteristic of CGALF in 100 iterations for this case. In comparison, NRLF fails to converge to a solution at this revised loading condition. In addition, it fails to converge until the original total loading of the system is reduced by 0.05%. This confirms that the NRLF will fail to pro- vide a solution when the loading is very near or at the ceiling point. Whilst at 0.05% load reduction, NRLF successfully determines a solution, and CGALF determines successfully two very near solutions, one of which is the same as that obtained by NRLF. The two solutions are omitted here due to lack of space.

7 Conclusion

A constrained-genetic-algorithm-based algorithm, CGALF, has been developed to solve the load flow problem. It is based on the concept of genetic algorithms and the methods of constraint satisfaction developed in the paper for satisfying the specified powers of the PQ nodes and the specified voltage magnitudes of the PV nodes. Three mechanisms, the dynamic population technique, the solution acceleration technique and the sequencing method for nodal voltage updating, have also been developed and incorporated in CGALF to enhance its performance and computational speed.

The different applications of the developed CGALF to the practical Klos-Kerner 11-node test system have shown that the algorithm has a very good convergence property and the ability to determine the multiple load flow solutions accurately. When the loading condition is such that the load-flow equation set is unsolvable, the insolvability is indicated by CGALF in the form of voltage magnitude violation at the PV nodes. For the original specified heavy loading condition on the KK11 system, this paper has shown through the application of CGALF that the two near solutions obtained by Klos and Kerner [2] are not valid solutions and that the load-flow problem under the specified loading condition is unsolvable.

Table 7: Best solution for the heavy-load case by CGALF with 0.04% load reduction

Mismatches

vi (P.U.) 8, (") P,"- P, 0,"- 0, V,'P- V, H Node no.

1 1.050000 0.000000 -

2 0.724861 -36.629562 -0.000299 -0.000759 - 3 0.714264 -57.727352 -0.000879 -0.000838 -

4 0.817576 -56.150093 -0.000944 -0.000234 - 5 1.050000 -40.104637 -0.000324 - 0.000000

6 0.817744 -54.203972 -0.000633 -0.000602 - 0.000048

(P.U.) (P.U.) (P.U.)

- -

7 0.9051 11 -45.222538 -0.000387 -0,000029 - 8 0.920962 -34.376129 -0.000500 -0.000191 - 9 1.037500 -15.398665 -0.000324 - 0.000000

10 0.965454 -20.534266 -0.000017 -0.000024 - 11 0.888948 -38.999729 -0.000476 -0.000302 - Superscript 'sp' denotes specified value

98 IEE Proc.-Gener. Transm. Distrib., Vol. 144, No. 2, March 1997

It has been found that the solvability can be achieved in this case study by relaxing the voltage constraints on the PV nodes. A method to relax the voltage constraints in CGALF has been developed and explained in Section 6.2. A valid solution has been found by CGALF for the KK11 system under the specified heavy loading. Using CGALF, it has also been found that the solvability of this load flow problem can be achieved without sacrificing the voltage levels at the PV nodes if the total load demand is allowed to be reduced by 0.04%. CGALF has a much better performance than the Newton Raphson method as the latter method fails to converge at the vicinity of the ceiling load point.

8 Acknowledgments

The authors are grateful to Western Power Corpora- tion, Western Australia, and to the Australian Research Council for their generous support. An Li acknowledges the award of an Australian Postgraduate Award (Industry) from the Department of Employ- ment Education and Training and the Australian Research Council.

9 References

1 GRAINGER, J.J., and STEVENSON, W.D. Jr.: ‘Power system analysis’ (McGraw-Hill, 1994)

10 WONG, K.P., and WONG, Y.W.: ‘Genetic and genetickimu- lated-annealing approaches to economic dispatch’, ZEE Proc. C,

11 HOLLAND, J.H.: ‘Adaptation in natural and artificial system’ (The University of Michigan Press, Ann Arbor)

12 WONG, K.P., and WONG, Y.W.: ‘Floating-point number-coding method for genetic algorithms’. Proc. ANZIIS-93, Perth, Western Australia, 1-3 December 1993, pp. 512-516

13 BOOKER, L.: ‘Improving search in genetic algorithms’ in DAVIS, L.: ‘Genetic algorithms and simulated annealing’ (Pitman, London, 1987), pp. 61-73

14 MICHALEWICZ, Z.: ‘Genetic algorithms + data structures = evolution programs’ (Springer-Verlag, 1996), pp. 121-153

1994, 141, (5), pp. 507-513

KLOS, A., and KERNER, A.: ‘The non-uniqueness of load-flow solutions’. Proceedings of 5th Power system computation confer- ence (PSCC), Cambridge, UK, July 1975, 01.3.118) IBA, K., SUZUKI, H., EGAWA, M., and WATANABE, T.: ‘A method for finding a pair of multiple load flow solutions in bulk power systems’, IEEE Trans. Power Syst., 1990, 5, (2), pp. 582- 59 1 CHIANG, H., HIU, C., VARAIYA, P., WU, F., and LAUBY, M.: ‘Chaos in a simple power system’, IEEE Trans. Power Syst., 1993, 4, (4), pp. 1407-1417 AJJARAPN, V., and LEE, B.: ‘Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system’, IEEE Trans. Power Syst., 1992, 7, (l), pp. 424431 GOLDBERG, D.E.: ‘Genetic algorithms in search, optimisation, and machine learning’ (Addison-Wesley, 1989) WARD, J.B., and HALE, H.W.: ‘Digital computer solution of power-flow problems’, Trans. AIEE, 1956, PAS-75, pp. 398404 YIN, X.D., and GERMAY, N.: ‘Investigations on solving the load-flow problem by genetic algorithms’, Electr. Power Syst. Res., 1991, 22, pp. 151-163 IBA, K.: ‘Reactive power optimization by genetic algorithm’, IEEE Trans. Power Svst., 1994, 9. (2). vv. 685-692

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development of constrained-genetic-algorithm load-flow method

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