^{2}Deputy Executive Engineer, JNTU, Hyderabad, India.
^{3}Professor, Department of Electrical and Electronics Engineering, JNTU, Hyderabad, India.
Keywords: Particle Swarm Optimization; Particle Distribution; Optimal Reactive Power; Transmission Loss;
ΔP = Incremental change in bus real power.
ΔQ = Incremental change in bus reactive Power injection
Δθ = incremental change in bus voltage angle.
ΔV = Incremental change in bus voltage Magnitude
Jpθ , JPV , JQθ , JQV jacobian matrix are the submatrixes of the System voltage stability is affected by both P and Q.
To reduce (1), let ΔP = 0 , then.
$$\Delta \text{V}={\text{J}}^{1}\Delta \text{Q}$$
$${\text{J}}_{\text{R}}=\left({\text{J}}_{\text{QV}}{\text{J}}_{\text{Q\theta}}{\text{J}}_{{\text{P\theta}}^{1}}\text{JPV}\right)$$
${\text{J}}_{\text{R}}\text{}$ is called the reduced Jacobian matrix of the system.
Let $${\text{J}}_{\text{R}}=\text{\xi}\Lambda \text{\eta}$$ Where,
ξ = right eigenvector matrix of JR
η = left eigenvector matrix of JR
^ = diagonal eigenvalue matrix of JR and $${\text{J}}_{{\text{R}}^{1}}=\text{\xi}{\Lambda}^{1}\text{\eta}$$ From (5) and (8), we have $$\Delta \text{V}=\text{\xi}{\Lambda}^{1}\text{\eta}\Delta \text{Q}$$ or $$\Delta \text{V}={\displaystyle \sum}_{\text{I}}\frac{{\text{\xi}}_{\text{i}}{\text{\eta}}_{\text{i}}}{{\text{\lambda}}_{\text{i}}}\Delta \text{Q}$$ Where ξi is the ith column right eigenvector and η the ith row left eigenvector of JR.
λi is the ith Eigen value of JR.
The ith modal reactive power variation is, $$\Delta {\text{Q}}_{\text{mi}}={\text{K}}_{\text{i}}{\text{\xi}}_{\text{i}}$$ where, $${\text{K}}_{\text{i}}={\displaystyle \sum}_{\text{j}}{\text{\xi}}_{{\text{ij}}^{2}}1$$ Where
ξji is the jth element of ξi
The corresponding ith modal voltage variation is $$\Delta {\text{V}}_{\text{mi}}=\left[1/{\text{\lambda}}_{\text{i}}\right]\Delta {\text{Q}}_{\text{mi}}$$ If  λi  =0 then the ith modal voltage will collapse
In (10), let ΔQ = ek where ek has all its elements zero except the kth one being 1. Then, $$\Delta \text{V}=\text{}{\displaystyle \sum}_{\text{i}}\frac{{\eta}_{1{\text{k\xi}}_{1}\text{}}}{{\text{\lambda}}_{1}}$$ ${\eta}_{1\text{k}}$ k th element of ${\eta}_{1\text{}}$
V –Q sensitivity at bus k $$\frac{\partial {\text{V}}_{\text{K}}}{\partial {\text{Q}}_{\text{K}}}={\displaystyle \sum}_{\text{i}}\frac{{\eta}_{1{\text{k\xi}}_{1}\text{}}}{{\text{\lambda}}_{1}}\text{}={\displaystyle \sum}_{\text{i}}\frac{{\text{P}}_{\text{ki}}}{{\text{\lambda}}_{1}}$$
Minimize VD = $\sum}_{\text{k}=1}^{\text{nl}}\left{\text{V}}_{\text{k}}1.0\right$
Where nl is the number of load busses and Vk is the voltage magnitude at bus k.
Load flow equality constraints: $${\text{P}}_{\text{Gi}}\u2013{\text{P}}_{\text{Di}}{\text{V}}_{\text{i}{\displaystyle \sum}_{\text{j}=1}^{\text{nb}}{\text{V}}_{\text{j}}}\left[\begin{array}{cc}{\text{G}}_{\text{ij}}& \mathrm{cos}{\text{\theta}}_{\text{ij}}\\ +{\text{B}}_{\text{ij}}& \mathrm{sin}{\text{\theta}}_{\text{ij}}\end{array}\right]=0,\text{i}=1,2\dots .,\text{nb}$$
$${\text{Q}}_{\text{Gi}}{\text{Q}}_{\text{Di}}{\text{V}}_{\text{i}{\displaystyle \sum}_{\text{j}=1}^{\text{nb}}{\text{V}}_{\text{j}}}\left[\begin{array}{cc}{\text{G}}_{\text{ij}}& \mathrm{sin}{\text{\theta}}_{\text{ij}}\\ +{\text{B}}_{\text{ij}}& \mathrm{cos}{\text{\theta}}_{\text{ij}}\end{array}\right]=0,\text{i}=1,2\dots .,\text{nb}$$
Generator bus voltage (VGi) inequality constraint:
$${\text{V}}_{\text{Gi}}^{\text{min}}\le {\text{V}}_{\text{Gi}}\le {\text{V}}_{\text{Gi}}^{\text{max}},\text{i}\in \text{ng}$$ Load bus voltage (VLi) inequality constraint:
$${\text{V}}_{\text{Li}}^{\text{min}}\le {\text{V}}_{\text{Li}}\le {\text{V}}_{\text{Li}}^{\text{max}},\text{i}\in \text{nl}$$ Switchable reactive power compensations (QCi) inequality constraint:
$${\text{Q}}_{\text{Ci}}^{\text{min}}\le {\text{Q}}_{\text{Ci}}\le {\text{Q}}_{\text{Ci}}^{\text{max}},\text{i}\in \text{nc}$$ Reactive power generation (QGi) inequality constraint:
$${\text{Q}}_{\text{Gi}}^{\text{min}}\le {\text{Q}}_{\text{Gi}}\le {\text{Q}}_{\text{Gi}}^{\text{max}},\text{i}\in \text{ng}$$ Transformers tap setting (Ti) inequality constraint:
$${\text{T}}_{\text{i}}^{\text{min}}\le {\text{T}}_{\text{i}}\le {\text{T}}_{\text{i}}^{\text{max}},\text{i}\in \text{nt}$$ Transmission line flow (SLi) inequality constraint:
$${\text{S}}_{\text{Li}}^{\text{min}}\le {\text{S}}_{\text{Li}}^{\text{max}},\text{i}\in \text{nl}$$ Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers.
Given a particle field with personal best point P_{i} and for which the global best point is P_{g} the position of the candidate solution point, c is determined according to: $${\overrightarrow{p}}_{m}=\frac{{\overrightarrow{p}}_{i}+{\overrightarrow{p}}_{g}}{2}$$ $$\overrightarrow{{\text{\sigma}}^{2}}=\left\overrightarrow{{\text{p}}_{\text{i}}}\overrightarrow{{\text{p}}_{\text{g}}}\right$$ $$\overrightarrow{c}=\overrightarrow{N}\left({\overrightarrow{p}}_{m}{\overrightarrow{\sigma}}^{2}\right)$$
Initialization range lbound, ubound
Particle field population size n_{Pop}
Candidate solution point pool size n_{pool}
Weighting function ù ( )
Generate candidate solution point pool C with size ${\text{n}}_{\text{pool}}$
For each particle field $\text{i}\in \text{p}$ do $${\overrightarrow{p}}_{i}=\overrightarrow{U}\left[1bound,\text{ubound}\right]$$ if $f({\overrightarrow{p}}_{i})<f({\overrightarrow{p}}_{j})$ then $${\overrightarrow{p}}_{i}\Leftarrow {\overrightarrow{p}}_{j}$$ End if
End for
While termination criteria not met do
For each candidate solution point $\overrightarrow{\text{c}}\in \text{C}$ do Select particle field $i\in p$ with probability $\frac{\text{\omega}\left(\text{i}\right)}{{{\displaystyle \sum}}^{\text{}}\text{p}\in {\text{P}}^{\text{\omega}\left(\text{p}\right)}}$
$$\overrightarrow{c}\Leftarrow \overrightarrow{N}\left(\frac{{\overrightarrow{p}}_{i}+{\overrightarrow{p}}_{g}}{2},\left[{\overrightarrow{p}}_{i}{\overrightarrow{p}}_{g}\right]\right)$$
$${\overrightarrow{s}}_{i}\Leftarrow {s}_{i}\cup \left\{\overrightarrow{c}\right\}$$ End for
For each particle field $i\in p$ do
Choose point ${\overrightarrow{c}}_{\mathrm{min}}$ from ${S}_{i}$ which minimizes $f({\overrightarrow{c}}_{\mathrm{min}})$
if $f({\overrightarrow{c}}_{\mathrm{min}}<f(p))$ then
${\overrightarrow{p}}_{i}\Leftarrow {\overrightarrow{c}}_{\mathrm{min}}$
If $f({\overrightarrow{p}}_{i})<({\overrightarrow{p}}_{j})$
Then ${\overrightarrow{p}}_{i}\Leftarrow {\overrightarrow{p}}_{j}$
End if
End if
${S}_{i}\Leftarrow \varphi $
End for
End
Control Variables 
Variable Setting 
V1 V2 V5 V8 V11 V13 T11 T12 T15 T36 Qc10 Qc12 Qc15 Qc17 Qc20 Qc23 Qc24 Qc29 Real power loss SVSM 
1.047 1.046 1.041 1.030 1.004 1.031 1.00 1.00 1.01 1.01 3 3 2 0 2 3 3 2 4.2869 0.2479 
Control Variables 
Variable Setting 
V1 V2 V5 V8 V11 V13 T11 T12 T15 T36 Qc10 Qc12 Qc15 Qc17 Qc20 Qc23 Qc24 Qc29 Real power loss SVSM 
1.049 1.047 1.044 1.032 1.005 1.034 0.090 0.090 0.090 0.090 3 3 2 3 0 2 2 3 4.9896 0.2484 
Sl.No 
Contingency 
ORPD Setting 
VSCRPD Setting 
1 
2827 
0.1419 
0.1434 
2 
412 
0.1642 
0.1650 
3 
13 
0.1761 
0.1772 
4 
24 
0.2022 
0.2043 
State variables 
limits 
ORPD 
VSCRPD 

Lower 
upper 

Q1 
20 
152 
1.3422 
1.3269 
Q2 
20 
61 
8.9900 
9.8232 
Q5 
15 
49.92 
25.920 
26.001 
Q8 
10 
63.52 
38.8200 
40.802 
Q11 
15 
42 
2.9300 
5.002 
Q13 
15 
48 
8.1025 
6.033 
V3 
0.95 
1.05 
1.0372 
1.0392 
V4 
0.95 
1.05 
1.0307 
1.0328 
V6 
0.95 
1.05 
1.0282 
1.0298 
V7 
0.95 
1.05 
1.0101 
1.0152 
V9 
0.95 
1.05 
1.0462 
1.0412 
V10 
0.95 
1.05 
1.0482 
1.0498 
V12 
0.95 
1.05 
1.0400 
1.0466 
V14 
0.95 
1.05 
1.0474 
1.0443 
V15 
0.95 
1.05 
1.0457 
1.0413 
V16 
0.95 
1.05 
1.0426 
1.0405 
V17 
0.95 
1.05 
1.0382 
1.0396 
V18 
0.95 
1.05 
1.0392 
1.0400 
V19 
0.95 
1.05 
1.0381 
1.0394 
V20 
0.95 
1.05 
1.0112 
1.0194 
V21 
0.95 
1.05 
1.0435 
1.0243 
V22 
0.95 
1.05 
1.0448 
1.0396 
V23 
0.95 
1.05 
1.0472 
1.0372 
V24 
0.95 
1.05 
1.0484 
1.0372 
V25 
0.95 
1.05 
1.0142 
1.0192 
V26 
0.95 
1.05 
1.0494 
1.0422 
V27 
0.95 
1.05 
1.0472 
1.0452 
V28 
0.95 
1.05 
1.0243 
1.0283 
V29 
0.95 
1.05 
1.0439 
1.0419 
V30 
0.95 
1.05 
1.0418 
1.0397 
Method 
Minimum loss 
Evolutionary programming [41] 
5.0159 
Genetic algorithm [42] 
4.665 
Real coded GA with Lindex as SVSM [43] 
4.568 
Real coded genetic algorithm [44] 
4.5015 
Proposed PDA method 
4.2869 
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