28

CHAPTER 3

PARTICLE SWARM OPTIMIZATION APPROACH FOR

REACTIVE POWER PLANNING

3.1 INTRODUCTION

The reactive power planning problem refers to the decision for the

future locations, types, sizes and times of installations of reactive power

sources like capacitors which guarantee a satisfactory system operation,

particularly, adequate voltage levels throughout the system at a minimum

cost. The reductions of the transmission losses as well as the consideration of

the system security and adequacy are other aspects that may also be included

in the statement of the problem. Usually the planning problem is divided into

operational and investment planning sub problems. In the operational

planning problem, the available shunt reactive sources and transformer tap-

settings are optimally dispatched at minimal operation cost. In the investment

planning problem, new reactive sources are optimally allocated over a

planning horizon at a minimal total cost (operational and investment). This

chapter proposes a particle swarm optimization approach for solving the

reactive power planning problem.

3.2 IMPORTANCE OF REACTIVE POWER PLANNING IN

POWER SYSTEMS

Reactive Power Planning (RPP) is concerned with optimal siting

and sizing of new shunt capacitors in a power system. By RPP, the capital

29

costs involved in providing the requisite shunt compensation will be

minimized and a satisfactory voltage profile is achieved. Further, optimal

shunt compensation inherently aids in reducing transmission loss. By

systematic and proper shunt capacitor planning, the real power transfers will

be maximized and there by maximally utilizing the existing transmission

system capacities. This allows reduction on the huge capital investment

required for transmission system augmentation to some extent. The obvious

financial and technical implication of RPP puts the onus on the power system

engineer to develop the best strategies to plan for new shunt compensation

such that the costs are minimum; they aid in the best system performance and

assist in the maximum usage of the existing transmission system capacities.

3.3 PROBLEM FORMULATION

The RPP can be formulated as an optimization problem subject to

equality and inequality constraints. The flow constraints are used as equality

constraints; reactive power generation restrictions, transformer tap setting

restrictions and bus voltage restrictions are used as inequality constraints. The

typical objective function used in RPP is the total cost which includes cost of

energy loss and reactive power source installation cost. This is achieved by

adjusting the generator bus voltage magnitude, transformer tap position and

the reactive power generation of capacitor bank. Mathematically the RPP

problem is stated as:

Minimize F = Wc+Ic (3.1)

where ‘Wc’ represents the total cost of energy loss given by

i l jikENl Nl nk

ijjijikl

l

losslc CosVVVVgdhPdhW

),(

2( 22 (3.2)

30

‘Ic’ represents the cost of reactive power source installation which has two

components, a fixed installation cost and purchase cost:

( )c

c i ci ci

i N

I e C Q (3.3)

3.4 CONSTRAINTS OF THE RPP PROBLEM

For reliable operation of the power system, the system has to satisfy

the equality constraint corresponding to the power flow model and a large

number of operational and system constraints. Mathematical formulations of

these constraints are presented here.

3.4.1 Power Flow Model as Equality Constraints

In a power system with Nb buses, at each bus i, the sum of the total

injected real and reactive power and the specified power must be equal to

zero. The specified power is equal to the difference between the power

generation and the load. These constraints are mathematically represented as,

BN

j

Bijijijijjii NiSinBCosGVVP1

1..,.........2,1,0)( (3.4)

BN

j

PQijijijijjii NiCosBSinGVVQ1

..,.........2,1,0)( (3.5)

The set of equations formed by equation (3.4) for all system buses

except the slack bus and equation (3.5) for all load buses constitute the power

flow equation. The power flow equations determine the steady-state

conditions of the power system network for specified generations and load

patterns, calculate voltages, phase angles, and flows across the entire system.

When solving the power flow equations iteratively, successive solutions

will have a mismatch between the specified and the injected power. So

31

equation (3.4) and (3.5) will not be satisfied. Hence, a tolerance is specified

for the power flow solutions.

3.4.2 Slack Bus Real Power Generation Limit

Real power generation limit of the slack bus usually expressed by

maximum and minimum limits as

maxmin

sss PPP (3.6)

3.4.3 Voltage Constraints

Too high or too low voltage magnitudes could cause problems to

the end user power apparatus or instability in the power system. The voltage

magnitude constraint is expressed as:

Biii NiVVV :maxmin (3.7)

3.4.4 Generator Reactive Power Capability Limit

The reactive power of a generator is important measures of voltage

magnitude quality, e.g. a low voltage indicate a local shortage of reactive

power. The upper and lower reactive power limits are specified as,

giii NiQgQgQg :maxmin

(3.8)

3.4.5 Reactive Power Generation Limit of Capacitor Banks

The reactive power generation of capacitor bank has a maximum

generating capacity, above which it is not feasible to generate due to technical

or economical reasons. Reactive power generation limits are usually

expressed as maximum and minimum reactive power outputs as:

CCiCiC NiQQQi

:maxmin

(3.9)

32

3.4.6 Transformer Tap Setting Limit

The flow of real power along the transmission line is determined by

the angle of difference of the terminal voltages and the flow of reactive power

is determined mainly by the magnitude difference of terminal voltages. The

value is modified in the search procedure among the existing tap positions and

expressed as:

rkkk Nkttt ;maxmin (3.10)

3.4.7 Transmission Line Flow Limit

The maximum MVA values for transmission lines and transformers

are given due to limitations of the branch material. Excessive power

would damage the transmission elements. This is stated as an inequality

constraint as:

lll NlSS ;max

(3.11)

The equality constraints given by Equations (3.4) and (3.5) are

satisfied by running the Newton Raphson Power flow algorithm. Generator

bus terminal voltages (Vgi), transformer tap settings (tk) and the reactive

power generation of capacitor bank (Qci) are the optimization variables and

are self-restricted between the minimum and maximum value by the

optimization algorithm. The limits on active power generation at the slack bus

(Ps), load bus voltages (Vload) and reactive power generation (Qgi), line flow

(Sl) are state variables which are satisfied by adding a penalty function to the

objective function and minimizing the combined function.

33

3.5 CLASSIFICATION OF RPP VARIABLES

The variables associated with RPP problem can be separated into

categories namely control variable (u) and state variables (x). The control

variables correspond to quantities that can be arbitrarily manipulated within

their limits, in order to minimize the objective function. These include

generator bus terminal voltages (Vgi), reactive power generation of capacitor

banks (QCi) and transformer tap ratio (tk). The state variables correspond to

quantities that are set as a result of the control variables but must be

monitored. The state variables include load bus voltage magnitude (Vload) and

reactive power generations (Qgi) of the generator outputs.

3.6 SUMMARY OF THE RPP PROBLEM

In summary, the optimal reactive power planning problem can be

stated as the minimization of the objective function given by equation (3.1)

subject to the constraints (3.4) to (3.11). In terms of the control and state

variable representations, the reactive power planning problem can be written

in compact form as follows:

Minimize f(x,u) (3.12)

Subject to g(x,u) = 0, (3.13)

h(x,u) 0 (3.14)

where f(x,u) is the objective function. Equation (3.13) represents the equality

constraints of the system and Equation (3.14) represents the inequality

constraints of the system, while imposing maximum or minimum operating

limits on variables associated with the component parts of the system. The

former are usually treated as hard and the later may be regarded as soft.

34

Reactive Power Planning problem is a non linear optimization

problem with continuous and discrete variables, non linear objective function

and constraints. Particle swarm optimization technique is proposed in this

thesis to solve this complex optimization problem. PSO is a population- based

approach for solving the complex optimization problems. PSO simulates the

behavior of bird flocking. The background of PSO is given in appendix 1. The

important advantage of this approach is that it uses only the objective function

information and hence is not restricted by the nature of the search space such

as smoothness, convexity, uni-modality etc.

3.7 PARTICLE SWARM OPTIMIZATION

Particle swarm optimization simulates the behavior of bird

flocking. Suppose, a group of birds are randomly searching food in an area.

There is only one piece of food in the area being searched. All the birds do

not know where that food is. But they know how far the food is in each

iteration. So what’s the best strategy to find the food? The effective one is to

follow the bird, which is nearest to the food.

3.7.1 Algorithm

In PSO, each single solution is a “bird” in the search space. Here it

is called as “particle”. All of particles have fitness values, which are evaluated

by the fitness function to be optimized, and have velocities, which direct the

flying of the particles. The particles are “flown” through the problem space by

following the current optimum particles. PSO is initialized with a group of

random particles (solutions) and then searches for optima by updating

generations. In every iteration, each particle is updated by following two

“best” values. The first one is the best solution (fitness) it has achieved so far.

35

(The fitness value is also stored.) This value is called pbest. Another “best”

value that is tracked by the particle swarm optimizer is the best value,

obtained so far by any particle in the population. This best value is a global

best and called gbest. When a particle takes part of the population as its

topological neighbors, the best value is a local best and is called pbest.

After finding the two best values, the particle updates its velocity

and positions with following equation (3.15) and (3.16)

)()(2)()(1 idgdidididid XPrandcXPrandcVV (3.15)

ididid VXX (3.16)

Figure 3.1 concept of modification of searching point by PSO

SK : Current searching point

SK+1 : Modified searching point

VK : Current velocity

VK+1 : Modified velocity

Vpbest : Velocity based on pbest

Vgbest: Velocity based on gbest

VK

Xy

VK+1

Y

SK

SK+1

Vgbest

Vpbest

36

Vid is the particle velocity; Xid is the current particle (solution). Pid

and Pgd are pbest and gbest. rand ( ) is a random number between (0,1). cl, c2

are learning factors. Usually c1=c2=2.

Particle velocities on each dimension are clamped to a maximum

velocity Vmax, if the sum of accelerations would cause the velocity on that

dimension to exceed Vmax - which is a parameter specified by the user. Then

the velocity on that dimension is limited to Vmax.

3.7.2 Features of the velocity update equation

Refer to equation (3.15), the right side of which consists of three

parts; the first part is the previous velocity of the particle; the second and third

parts are the ones contributing to the change of the velocity of a particle.

Without these two parts, the particles will keep on “flying” at the current

speed in the same direction until they hit the boundary. PSO will not find a

acceptable solution unless there are acceptable solutions on their “flying”

trajectories. But that is a rare case. On the other hand, refer to equation (3.15)

without the first part. Then the “flying” particle velocities are only determined

by their current positions and their best positions in history. The velocity itself

is memory less. Assume at the beginning, the particle I has the best global

position, then the particle I will be “flying” at the velocity 0, that is, it will

keep still until another particle takes over the global best position. At the same

time, each other particle will be “flying” toward its weighted centroid of its

own best position and the global best position of the population.

The recommended choice for constant c1 and c2 is 2. Under this

condition, the particles statistically contrast swarm to the current global best

position until another particle takes over from which time all the particles

statistically contract to the new global best position. Therefore, it can be

imagined that the search process for PSO without the first part is a process

where the search space statistically shrinks through the generations. It

37

resembles a local search algorithm. Displaying the “flying” process on a

screen can illuminate this more clearly. From the screen, it can be easily seen

that without the first part of equation (3.15), all the particles will tend to move

toward the same position, that is, the search area is contracting through the

generations. Only when the global optimum is within the initial search space,

then there is a chance for PSO to find the solution. The final solution is

heavily dependent on the initial seeds (population). So it is more likely to

exhibit local search ability without the first part.

On the other hand, by adding the first part, the particles have a

tendency to expand the search space, that is, they have the ability to explore

the new area. So they more likely have global search ability by adding the

first part. Both the local search and global search will benefit solving some

kinds of problems.

3.7.3 PSO Control Parameters

There are a few parameters that need to be tuned in PSO. The list of

the parameters and their typical values are discussed in the following

subsections.

Number of particles

The typical range is 20-40. Actually for most of the problems 30

particles is large enough to get good results. For some difficult or special

problems, one can try 100 or 200 particles as well.

Vmax

It determines the maximum change one particle can take during an

iteration. Usually the range of the particle is taken as the Vmax. For example,

the particle (x1, x2, x3) x1 belongs [-10, 10], then Vmax=20.

38

Learning factors

c1 and c2 usually equal to 2. However, other settings were also

used in different works. But usually c1 equals to c2 and ranges from [0,4].

Stop condition

The maximum number of iterations the PSO execute and the

minimum error requirement. This stop condition depends on the problem to

be optimized.

3.7.4 Inertia weight

In PSO, there is a tradeoff between the global and local search. For

different problems, there should be different balances between the local

search ability and global search ability. Considering this, an inertia weight w

is brought into the equation (3.15) as shown in equation (3.17). This w plays

the role of balancing the global search and local search. It can be positive

constant or even a positive linear or nonlinear function of time.

Equation (3.17) and (3.18) describe the velocity and position update

equations with an inertia weight included.

)()(2)()(1 idgdidididid XPrandcXPrandcVWV (3.17)

ididid VXX (3.18)

The use of the inertia weight w has provided improved performance

in a number of applications. As originally developed, inertia weight (w) often

is decreased linearly from about 0.9 to 0.4 during a run. Suitable selection of

the inertia weight provides a balance between global and local exploration

39

and exploitation, and results in lesser iterations on average to find an optimal

solution.

3.8 DISCRETE VARIABLE HANDLING

Although the PSO solves optimization problems over continuous

spaces, minor modification to the algorithm allow PSO to solve mixed integer

optimization problems. This is achieved with the use of an operator that

rounds the variable to the nearest integer value, when the value lies between

two integer values. This operator is included after the initialization and before

the fitness calculation.

X1….D= [Y1… k, round (Zk+1…D)] (3.19)

where, X is the D dimensional parameter vector,

Y is the k dimensional vector of continuous parameters, and

Z is the vector of (D-K) discrete parameters.

3.9 PSO IMPLEMENTATION FOR RPP

3.9.1 Problem Representation

Each particle in the PSO population represents a candidate solution

for the given problem. The elements of that solution consist of all the

optimization variables of the problem. For the reactive power planning

problem under consideration, generator terminal voltages ( giV ), the

transformer tap positions (tk) and the Capacitor settings (QCi) are the

optimization variables. Generator bus voltage is represented as floating point

numbers, whereas the transformer tap position and reactive power generation

of capacitor are represented as integers.

40

With this representation, a typical particle of the RPP problem will

look like the following:

0.981 0.970 … 1.05 0.95 0.925 … 1.025 3 2 …. 5

V1 V2 Vn t1 t2 tn Qc1 Qc2 Qcn

3.9.2 Evaluation Function

Particle Swarm Optimization searches for the optimal solution by

maximizing a given fitness function, and therefore an evaluation function

which provides a measure of the quality of the problem solution must be

provided. In the reactive power optimization problem under consideration, the

objective is to minimize the total cost and maximize the voltage profile while

satisfying the constraints (3.4) to (3.11). The equality constraints are satisfied

by running the Newton Raphson power flow algorithm. The inequality

constraints on the control variables are taken into account in the problem

representation itself, and the constraints on the state variables are taken into

consideration by adding a quadratic penalty function to the objective function.

With the inclusion of penalty function the new objective function becomes,

Min f =F+PQN

1j

jVPSPgN

1j

jQP +lN

1j

jLP (3.20)

Here, SP,VPj ,QPj and LPj are the penalty terms for the reference bus

generator active power limit violation, load bus voltage limit violation;

reactive power generation limit violation and line flow limit violation

respectively. These quantities are defined by the following equations:

SP =

otherwise

PPifPPK

PPifPPK

sssss

sssss

0

min2min

max2max

(3.21)

41

VPj =

otherwise

VVifVVK

VVifVVK

jjjjv

jjjjv

0

)(

)(

min2min

max2max

(3.22)

QPj =

otherwise

QQifQQK

QQifQQK

jjjjq

jjjjq

0

)(

)(

min2min

max2max

(3.23)

LPj =otherwise

LLifLLK jjjjl

0

)( max2max

(3.24)

Where, Ks, Kv, Kq and Kl are the penalty factors. The success of the penalty

function approach lies in the proper choice of these penalty factors. The

penalty factors are selected by trial and error approach. Since PSO maximizes

the fitness function, the minimization objective function f is transformed to a

fitness function to be maximized as,

Fitness =f

k (3.25)

where k is a large constant.

3.10 SIMULATION RESULTS

The proposed PSO-based approach for solving the reactive power

planning was applied to IEEE 30-bus, IEEE 57-bus test system, IEEE 118-bus

test system and a practical 76-bus Indian system. The generator active power

generation was kept fixed except for the slack bus. The base power and

parameters of cost are given in Table 3.1. The program was written in

MATLAB and executed on a PC with 2.4 GHZ Intel Pentium IV processor.

The results of the simulation are presented below.

42

Table 3.1 Base Power and cost parameter

SB ei Cci dl

(MVA) ($/p.u.wh) ($) ($./p.u.VAR) Case1 Case 2 Case 3 Case 4

100 6000 1000 3000,000 8760 8760 8760 8760

Table 3.2 Variable limits (in p.u) of IEEE 30-bus test system

Bus 1 2 5 8 11 13

Qgmax 1.5 0.6 0.48734 0.6245 0.4 0.45

Qgmin -0.2 -0.2 -0.15 -0.15 -0.1 -0.15

Vmax Vmin Tmax Tmin Qcmax Qc

min

1.10 0.90 1.1 0.9 5.0 0.0

Case 1A: Reactive Power Planning in IEEE 30-bus system

The one-line diagram of the IEEE 30-bus test system is shown in

Figure 3.2.

Figure 3.2 IEEE-30 bus test system

43

The IEEE 30 bus system has 6 generators, 24 load buses, 41

transmission lines, 4 transformer taps and 2 shunt elements. The transmission

line parameters and the system base load are taken from (Alsac et al 1974).

The variable limits are given in Table 3.2. The real power settings of the

generator are taken from (Alsac et al 1974). The possible locations for

capacitor installation are buses 10,12,15,17,20,21,23,24 and 29. The proposed

algorithm was run with minimization of total cost as the objective function.

The total cost consisting of fixed installation cost, purchase cost and operating

cost are calculated and minimized in the base case. The PSO based algorithm

was tested with different parameter settings and best results are obtained with

the following setting:

No. of Generations : 100

Population Size : 30

c1 : 1.7

c2 : 1.5

Wmax : 0.9

Wmin : 0.4

Trial and error approach was followed to select the suitable values

of the penalty factors Ks, Kv, Kq and Kl. The values of Ks, Kv, Kq and Kl

selected in this case are 3,5,2,4 respectively.

The PSO algorithm reaches a minimum cost of 2,591,830$ in this

case. The algorithm took 50 sec to reach the optimal solution. The optimal

control variables obtained by the proposed approach and other approaches are

given in Table 3.3. Corresponding to these control variables, it was found that

there was no limit violation. The convergence characteristics of PSO

algorithm is shown in Figure 3.3. The minimum cost obtained by the

proposed algorithm is compared with evolutionary programming (Lai et al

44

1997) approach and the results are presented in Table 3.4. The minimum cost

obtained by this method is less then the value reported in Lai et al (1997) and

Durairaj et al (2006). This shows the effectiveness of the proposed approach

in solving the RPP problem.

Table 3.3 Optimal Control Variables of Case 1A

Control Variable SettingControl

VariablesBroyden

methodGA EP PSO

V1, V2, V5,

V8,V11, V13

(p.u)

1.006, 0.998,

0.982, 0.995,

1.001, 0.989

1.0246, 1.0167,

0.9976, 0.9857,

0.9929, 0.9714

1.074, 1.065,

1.043, 1.042,

1.069, 1.058

1.09, 1.08,

1.05, 1.06,

1.04, 1.05

t6-9, t6-10, t4-12,

t28-27(p.u)

1.009, 1.010,

1.013, 1.004

1.0000, 1.0750,

1.0250, 0.9000

0.981, 1.042,

1.029, 1.037

1.05, 0.95,

1.025, 1.00

Capacitor

setting(p.u)

0.086, 0.047,

0.094, 0.105

(locations 6,

17,18, 27)

0,0,0,0 0,0,0,0 1,5, 3,5, 5, 4,

5, 5,1

(locations 10,

12, 15,17, 20,

21, 23, 24, 29)

Ploss (MW) 5.736 5.0355 4.963 4.72

Cost ($) 4, 013, 280 2, 646, 700 2, 608, 500 2, 591, 830

Table 3.4 Comparison of results of total cost

Case 1AMethod

Total Cost($) Ploss(MW)

Broyden method (Lai et al 1997) 4,013,280 5.736

GA (Durairaj et al 2006) 2,646,700 5.0355

EP (Lai et al 1997) 2,608,500 4.963

Proposed method 2,591,830 4.72

45

Figure 3.3 Convergence Characteristics of PSO method

Case 1B. RPP including contingency state Voltage Deviation

Contingency analysis was carried out in IEEE 30-bus system and

voltage deviation defined as the sum of difference between the load bus

voltage and nominal voltage (1 p.u) (for each contingency) was evaluated.

From the Voltage deviation evaluated, the line outages 9-11, 9-10, 12-13, 12-

15, 15-23 and 28-27 are identified as severe contingencies. Singular values

are evaluated for the above severe contingencies and base case and tabulated

in Table 3.5. From the singular values, the buses with low value of singular

values are identified as weak buses and tabulated in Table 3.6. The common

weak buses for different contingency cases are 24, 25, 26, 27, 29 and 30.

These buses are chosen as the candidate buses for placing capacitor. The PSO

algorithm reaches a minimum cost of 2,681,720$ in this case. The algorithm

took 50 sec to reach the optimal solution. The optimal values of control

variables are given in Table 3.7. Corresponding to these control variables, it is

found that there was no limit violation. Table 3.8 gives the deviation in

voltage and the minimum value of bus voltage in the system under the six

46

contingency states before and after the application of the optimization

algorithm. From the results presented in the table, it is observed that the

voltage deviation has reduced and the minimum voltage has increased after

the application of the optimization algorithm in all the six contingency cases.

Table 3.5 Minimum Singular values for IEEE 30 Bus System

Singular Values

BusNumber

Base

case

Lineoutage

9-11

Lineoutage

9-10

Line

outage

12-13

Lineoutage

12-15

Line

outage

15-23

Line

outage

28-27

3

4

6

7

9

10

12

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

102.78

79.48

60.40

51.30

30.00

29.32

21.70

18.56

17.53

17.00

14.87

13.36

11.30

10.58

9.75

6.61

5.87

4.91

4.22

3.30

2.90

1.32

0.92

0.46

102.11

77.22

60.05

49.51

29.53

28.79

21.52

17.55

16.88

15.50

13.75

12.52

11.13

10.35

9.19

6.43

5.78

4.85

4.12

3.14

2.74

1.29

0.83

0.43

102.61

76.06

60.25

40.09

29.38

28.43

21.59

17.45

16.89

14.62

13.27

11.62

10.57

10.14

8.64

6.41

5.77

4.87

4.10

3.08

2.66

1.29

0.72

0.37

101.90

77.18

59.72

49.52

28.50

23.39

21.53

18.18

16.94

15.82

13.56

13.06

10.91

10.10

9.47

6.20

5.70

4.81

3.86

2.86

2.65

1.28

0.72

0.39

102.72

78.72

60.36

50.91

29.07

21.94

21.41

18.30

17.01

15.05

14.73

13.15

10.94

9.97

9.17

6.55

5.67

4.85

3.93

2.95

2.71

1.30

0.76

0.41

102.76

79.06

60.40

51.16

29.81

28.91

21.67

18.46

17.05

15.21

14.24

13.26

11.14

9.71

9.10

6.51

5.44

4.79

3.44

3.24

1.60

1.26

0.91

0.41

102.66

77.80

60.24

50.61

29.66

28.99

19.40

18.41

17.29

16.59

14.62

11.72

10.40

9.90

9.19

6.48

4.84

4.17

3.70

3.25

2.80

1.14

0.84

0.16

47

Table 3.6 Weak Buses

Line outage Weak Buses

Base Case 24,25,26,27,29,30

9-11 24,25,26,27,29,30

9-10 24,25,26,27,29,30

12-13 24,25,26,27,29,30

12-15 23 ,25,26,27,29,30

15-23 23 ,25,26,27,29,30

28-27 24,25,26,27,29,30

Table 3.7 Optimal Control Variables of Case 1B

Control Variable Control Variable setting

V1,V2,V5,V8,V11,V13 1.05,1.06,1.04,1.08,1.05,1.09

t 6-9,t 6-10,t 4-12,t 28-27 1.00, 1.025,0.95,1.00

C24,C25,C26,C27,C29,C30 5,5,4,3,5,5

Ploss (MW)

Cost ($)

Minimum Voltage

4.81

2,681,720

0.91

Table 3.8 Voltage Deviation and Minimum Voltage before and after

Optimization

Voltage Deviations Minimum VoltageOutaged

line BeforeOptimization

AfterOptimization

% ofreduction

BeforeOptimization

AfterOptimization

% ofrise

9-11 1.3134 0.6559 50.06 0.8894 0.9404 5.70

9-10 1.3797 0.7641 44.61 0.8940 0.9085 1.60

12-13 1.5890 0.4872 69.33 0.8853 0.9405 6.23

12-15 1.1601 0.4844 58.24 0.8970 0.9447 5.31

15-23 1.0116 0.5414 46.48 0.8945 0.9230 3.18

28-27 1.5453 0.8274 46.45 0.7772 0.9081 16.80

48

Case 1C: Reactive power planning in IEEE 30-bus system for a 24 hour

load cycle

Next, the proposed PSO-based approach was applied for solving

the reactive power planning for a 24 hour load cycle. The active and reactive

loads during different periods of time are given in Table 3.9. The load curve

with time in X- axis and active power and reactive power load in Y – axis is

given in Figure 3.4.

Table 3.9 Active and Reactive Load with duration

Time Duration % load P load Q load

12 mid night to 6 A.M 6 Hrs 50% 1.417 0.631

6 AM to 10 A.M 4 Hrs 100% 2.834 1.262

10 AM to 6 PM 8 Hrs 60% 1.700 0.757

6 PM to 10 PM 4 Hrs 120% 3.684 1.640

10 PM to 12 PM 2 Hrs 80% 2.267 1.000

Figure 3.4 Load Curve

The same 9 locations were considered for capacitor installation.

The proposed algorithm was run with minimization of total cost as the

objective function. The optimal values of control variables obtained for the

49

five load conditions in the load duration are given in Table 3.10. The real

power loss, minimum cost and minimum voltage obtained for the different

load condition are also tabulated. The algorithm took 50 sec to reach the

optimal solution.

Table 3.10 Optimal Control Variables of Case1C

Control variable settingsControl

Variables50% of

Base Load(6 Hrs)

100% ofBase Load

(4 Hrs)

60% ofBase Load

(8 Hrs)

120% ofBase Load

(4 Hrs)

80 % ofBase Load

(2 Hrs)V1

V2

V5

V8

V11

V13

t6-9

t6-10

t4-12

t28-27

C10

C12

C15

C17

C20

C21

C23

C24

C29

1.02

1.02

1.05

1.01

0.95

1.01

1.10

1.10

0.925

0.95

3

1

4

5

1

2

4

4

5

1.09

1.08

1.05

1.06

1.04

1.05

1.05

0.95

1.025

1.00

1

5

3

5

5

4

5

5

1

1.02

1.01

0.98

0.99

1.07

1.04

1.025

1.000

1.000

1.000

3

3

1

3

5

3

5

3

3

1.04

1.03

1.01

1.00

1.08

1.06

1.00

0.925

0.975

0.950

5

3

3

5

4

5

4

2

4

1.04

1.03

1.02

1.01

0.98

1.04

1.025

1.00

1.025

1.025

3

3

4

5

4

2

3

5

4

Ploss (MW)

Cost ($)

Minimum

Voltage

2.45

93,882

1.07

4.72

1,12,133

0.99

3.52

97,690

0.98

5.52

1,15,325

0.96

3.66

1,08,439

0.97

Among the five loading condition, the real power loss obtained in

the 50% load condition is less (2.45 MW). The highest loss obtained is 5.52

MW for 120% loading condition. From the above, it was observed that, when

50

the loading is high, the real power loss is high and vice versa. The cost

obtained in the 50% loading condition for a time duration of 6 hours is less

(93,882$). The cost obtained in the 120% loading condition for a time

duration of 4 hours is high (1, 15,325$). From the above it was observed that,

the cost is proportional to the real power loss and time duration. The

minimum load bus voltage for 120% load condition is low (0.96 p.u) and high

(1.07 p.u) in the 50% loading condition. From the above, it was observed that,

when the load is high, the minimum voltage is low and when the load is low,

the minimum voltage is high.

Case 2: Contingency Constrained Reactive Power Planning for IEEE 57-

bus system

The IEEE 57 bus system has 7 generators, 50 load buses, 80

transmission lines and 17 transformer taps. Two different cases are considered

in this system. In case 2A, the proposed PSO algorithm is applied to minimize

the total cost in base case without including the contingency constraint. In

case 2B, the algorithm is applied to minimize the total cost in base case after

including contingency constraint. The possible locations of capacitor

installation are buses 25, 30, 32, 34, 35 and 53 to supply reactive power. The

variable limits are given in Table 3.11.

Table 3.11 Variable limits (p.u) of IEEE 57-bus test system

Bus 1 2 3 6 8 9 12

Qgmax 2.0 0.50 0.60 0.25 2.0 0.9 1.55

Qgmin -1.4 -0.17 -0.1 -0.08 -1.4 -0.03 -1.5

Vg max Vg

min T max T min Qc max Qc

min min

loadV max

loadV

1.10 0.90 1.1 0.9 5.0 0.0 0.90 1.05

51

The PSO based algorithm was tested with different parameter

settings and best results are obtained with following setting:

No. of Generations : 100

Population Size : 30

C1 : 1.7

C2 : 1.5

Wmax : 0.9

Wmin : 0.4

The optimal values of the control variables, loss, cost, and

minimum voltage for case 2A are given in second column of Table 3.12. The

minimum cost obtained by this algorithm is compared with genetic algorithm

approach (Subamalini et al 2006) and the results are presented in Table 3.13.

The cost obtained by this method is less than the value reported in

(Subamalini et al 2006).

Next, the single line contingency analysis is performed in IEEE 57-

bus system. From the contingency analysis, the line outage 25-30 is identified

as the severe contingency based on large voltage deviation with a value of

1.545 and a minimum voltage of 0.60. The voltage limit violated buses in

contingency state are 23,24,25,26,27,28,29 and 33. The voltage deviation

occurred during this contingency is added as an additional constraint in case

2B. The optimal values of the control variables, loss, cost, time and minimum

voltage in this case are given in third column of Table 3.12. The voltage

violations which were present earlier have been completely alleviated. The

voltage magnitudes of the above buses before optimization and after

optimization are displayed in Figure 3.5. From this figure, it is inferred that

the voltage magnitudes of the severe buses were improved after the

52

optimization. Also, it is inferred that the minimum voltage is raised from 0.60

to 0.98 after the application of the proposed PSO algorithm. Also, the voltage

deviation is reduced to 0.45 after optimization.

Further, before the application of the algorithm voltage violations

were present in the buses. But, they are corrected after the optimization. Table

3.14 gives the voltage magnitude for a selected list of buses for contingency

25-30. Improvement in voltage profile at the load buses is evident from the

results. The algorithm took 52 sec to reach the optimal solution. This shows

the effectiveness of the proposed algorithm in solving the contingency

constrained reactive power planning problem.

Table 3.12 Optimal values of Control variables of Case 2

Control Variables Setting

Control Variable Case 2A Case 2B

V1,V2,V3,V6,V8,V9,V12 1.08,1.1,1.08,1.06,1.05

1.05,1.05

1.07,1.06,1.06,1.07,1.07

1.08,1.07

t 4-18,t 4-18,,t 21-20,t 24-25,

t 24-25,t 24-26,t 7-29,t 34-32

t 11-41,t 15-45,t 14-46,t 10-51

t 13-49,t 11-43,t 40-56,t 39-57

t 9-55

1.025,1.0,1.025,1.1

1.1,1.0,1.0,0.925,1.0,

1.025,1.025,0.925,0.95,

1.025,1.025,1.0,1.0

1.075,1.0,1.025,1.025

1.025,1.025,0.95

0.975,1.05,1.0,0.95

1.025,0.975,1.0,1.05

1.1,1.05

C25,C30,C34,C32,C35,C53 5,2,5,5,3,5 3,3,3,5,5,4

Loss ( M W)

Cost ($)

Minimum Voltage

Time

25.12

13,284,070

0.99

52 sec

25.83

13,651,240

0.98

53 sec

53

Table 3.13 Comparison of cost in base case for IEEE 57-bus system

Method Total cost ($) Ploss in MW

GA (Subamalini et al)

Proposed Method

14,561,000

13,284,070

25.9654

25.12

Table 3.14 Improvement of voltage profile for IEEE 57 bus system

Voltage Magnitude

S.NoBus

NoBefore

Optimization

After

Optimization

1. 23 0.60 0.98

2. 24 0.62 1.00

3. 25 0.75 1.02

4. 26 0.76 1.01

5. 27 0.90 1.00

6. 28 0.92 0.99

7. 29 0.94 1.02

8. 33 0.91 1.01

Figure 3.5 Voltage magnitudes of severe buses in contingency condition

54

Case 3: Reactive Power Planning of Practical 76-Bus Indian System

The proposed PSO approach was applied to solve the RPP problem

in a practical Indian power system. The system under consideration is a

regional grid of Indian power system, consisting of 13 generator buses, 63

load buses, 116 transmission lines, 18 tap changing transformers and

switchable VAR compensators are located at 12 places. The total load on the

system is 3668 MW and 2591 MVAR. The variable limits are given in

Table 3.15. The base power and parameter of costs are given in Table 3.1.

Table 3.15 Variable limits (p.u) of Practical 76-bus Indian system

Bus 1 2 3 4 5 6 7

Qgmax 1.0 2.0 1.0 3.0 4.0 2.2 2.2

Qgmin -0.6 -1.0 -0.5 -1.5 -2.0 -1.0 -1.0

Bus 8 9 10 11 12 13

Qgmax 2.2 0.8 0.35 0.4 1.0 1.5

Qgmin -1.0 -0.4 -0.2 -0.5 -1.5 -1.0

V max V min T max T min Qc max Qc

min

1.10 0.90 1.1 0.9 5.0 0.0

To obtain the optimal values of control variables the PSO based

algorithm was run with different parameter settings.

The best parameter settings are:

No of generation : 100

Population size : 30

c1 : 1.4

c2 : 1.6

Wmax : 0.8

Wmin : 0.4

55

The algorithm reaches a minimum cost of 120, 46, 18, 870 .̀ The

algorithm took 55 sec to reach the optimal solution. The optimal control

variable settings obtained in this case are given in Table 3.16. The loss

obtained in this case is 50.28 MW which is less than the loss obtained by

conventional linear programming method (Durairaj et al 2005). From the

comparison, it is found that the proposed method is more effective in solving

the RPP problem than the other methods.

Table 3.16 Optimal control variables for practical 76 bus Indian system

Vvar Tvar Cvar

0.99

1.02

1.03

1.05

1.00

1.05

1.06

1.01

0.98

1.02

1.01

1.05

1.04

0.975

1.00

1.025

1.025

1.00

1.025

0.975

0.975

1.00

0.975

1.025

1.10

1.00

0.975

1.00

0.925

0.975

1.05

3

5

3

3

3

5

3

3

3

3

2

2

Cost = 120,46,18,870 `

PLoss = 50.28 MW

Vmin =0.95

56

Case 4: Reactive power planning for IEEE 118 – bus system

The IEEE 118 bus system has 54 generator buses, 64 load buses, 9

tap changing transformers and 14 capacitor installed buses. The one – line

diagram, bus data, generator data and transmission line data are given in

appendix 6. The variable limits are given in Table 3.17.

Table 3.17 Variable limits (p.u) of IEEE 118-bus test system

Vmax Vmin Tmax Tmin Qcmax Qcmin

1.10 0.90 1.1 0.9 5.0 0.0

The possible locations for capacitor installation are buses 5, 34, 37,

44, 46, 48, 74, 79, 82, 83, 105, 107 and 110. The proposed PSO based

algorithm was run with minimization of total cost as the objective function.

The total cost consisting of fixed installation cost, purchase cost and operating

cost are calculated and minimized in this case. The PSO algorithm reaches a

minimum cost of 802, 792 $ with a minimum loss of 1.32 p.u. The algorithm

took 60 sec to reach the optimal solution. The optimal values of control

variables are given in second column of Table 3.18. Corresponding to these

control variables, it was found that there was no limit violation.

Table 3.18 Optimal Control Variables of case 4 (IEEE 118 Bus System)

Control Variable Control Variable Setting

V1, V4, V6, V8, V10, V12, V15, V18, V19,

V24, V25, V26, V27, V31, V32, V34, V36,

V40, V42, V46, V49, V54, V55, V56, V59,

V61, V62, V65, V66, V69, V70, V72, V73,

V74, V76, V77, V80, V85, V87, V89,

V90, V91, V92, V99, V100, V103, V104, V105,

V107, V110, V111, V112, V113, V116

0.99, 1.02, 0.97, 0.96, 0.94,0.95, 1.03,

1.00, 1.01, 1.02, 0.98, 0.95, 1.05, 1.02,

0.98, 1.00, 0.99, 0.97, 1.01, 0.99, 1.05,

1.03, 1.02, 1.01, 1.00, 0.97, 0.95, 0.97,

1.00, 0.99, 0.97, 0.97, 1.02, 0.98, 1.02,

1.01, 1.02, 1.04, 1.05, 0.94, 0.96, 0.97,

0.96, 0.97, 0.95, 1.02, 1.03, 1.01, 0.99,

1.02,0.97, 1.01, 0.96, 1.02

57

Table 3.18 (Continued)

Control Variable Control Variable Setting

t8-5, t26-25, t30-7, t38-37, t63-59,

t64-61, t65-66, t68-69, t81-80

1.10, 1.10, 1.10, 1.075, 0.975, 1.00, 1.10,

0.925, 1.00

C5, C34, C37, C44, C45, C46, C48, C74, C79,

C82, C83, C105, C107, C110

1, 1, 2, 4, 4, 4,2, 4, 4, 1, 3, 1, 3, 2

Ploss

Cost ($)

Minimum Voltage

1.32

802,792

0.95

3.11 CONCLUSION

This chapter has presented a particle swarm optimization approach

for solving the reactive power planning problem. The algorithm minimizes

the operation cost and allocation cost of reactive power sources and improves

the voltage profile by adjusting the control variables namely generator voltage

magnitude, tap setting transformer and Capacitor bank. To handle the mixed

variables a flexible representation scheme has been proposed. Simulation

results on IEEE 30-bus test system, IEEE 57-bus system, practical 76-bus

Indian system and IEEE 118- bus test system demonstrate the effectiveness of

the proposed approach in minimizing the cost and improving the voltage

profile of the systems in base case and contingency conditions.