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This paper projects Improved Canis Rufus Floridanus (ICRF) Optimization Algorithm for solving optimal reactive power dispatch problem. Projected ICRF algorithm combines the Canis Rufus Floridanus algorithm with particle swarm optimization (PSO) algorithm. When the PSO algorithm has been intermingled with Canis Rufus Floridanus (ICRF) Optimization algorithm, at first exploration will be done and gradually it will be moved to phase of exploitation. Also in this approach social interaction within the swarm also considered with communication diversity. So due the hybridization both Exploration & Exploitation capability of the projected Improved Canis Rufus Floridanus (ICRF) Optimization Algorithm has been improved. Projected algorithm is evaluated in standard IEEE 30 bus test system. Results indicate that proposed algorithm perform well in solving the optimal reactive power dispatch problem. Real power losses are reduced by the proposed algorithm when compared to other standard algorithms & voltage stability index has increased from 0.2462 to 0.2485, which is an improvement in the system voltage stability. To determine the voltage security of the system, contingency analysis was conducted using the control variable setting obtained.
optimal reactive power, transmission loss, canis rufus floridanus, particle swarm optimization
The main objective of optimal reactive power problem is to reduce the actual power loss. Various techniques problem (Lee et al., 1984; Deeb and Shahidehpour, 1988; Bjelogrlic et al., 1990; Granville, 1994; Grudinin, 1998; Yan et al., 2006) have been utilized but have the complexity in handling constraints. Different types of evolutionary algorithms (Mukherjee and Mukherjee, 2015; Hu et al., 2010; Morgan et al., 2015; Sulaiman et al., 2015; Pandiarajan and Babulal, 2016; Morgan et al., 2016; Mei et al., 2016) have been utilized in various stages to solve the problem. Many algorithms may good in Exploration & but very poor in Exploitation, some algorithms will good in Exploitation but lack in Exploration. This paper projects Improved Canis Rufus Floridanus (ICRF) Optimization Algorithm for solving optimal reactive power dispatch problem. Projected ICRF algorithm combines the Canis Rufus Floridanus algorithm with particle swarm optimization (PSO) algorithm. When the PSO algorithm has been intermingled with Canis Rufus Floridanus (ICRF) Optimization algorithm, at first exploration will be done and gradually it will be moved to phase of exploitation. So due the hybridization both Exploration & Exploitation capability of the projected Improved Canis Rufus Floridanus (ICRF) Optimization Algorithm has been improved. Projected algorithm is evaluated in standard IEEE 30 bus test system. Results indicate that proposed algorithm perform well in solving the optimal reactive power dispatch problem. Real power losses are reduced by the proposed algorithm when compared to other standard algorithms & voltage stability index has increased from 0.2462 to 0.2485, which is an improvement in the system voltage stability. To determine the voltage security of the system, contingency analysis was conducted using the control variable setting obtained.
2.1. Modal analysis for voltage stability evaluation
Modal analysis is one among best methods for voltage stability enhancement in power systems. The steady state system power flow equations are given by.
$\left[ \begin{array} { l } { \Delta P } \\ { \Delta Q } \end{array} \right] = \left[ \begin{array} { l l } { J _ { p \theta } } & { J _ { p v } } \\ { J _ { q \theta } } & { J _ { Q V } } \end{array} \right] \left[ \begin{array} { l } { \Delta \theta } \\ { \Delta V } \end{array} \right]$ (1)
Where
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 sub-matrixes of the System voltage stability is affected by both P and Q.
To reduce (1), let ΔP=0, then.
$\Delta Q = \left[ J _ { Q V } - J _ { Q \theta } J _ { P \theta ^ { - 1 } } J _ { P V } \right] \Delta V = J _ { R } \Delta V$ (2)
$\Delta V = J ^ { - 1 } - \Delta Q$ (3)
Where
$J _ { R } = \left( J _ { Q V } - J _ { Q \theta } J _ { P \theta ^ { - 1 } } J P V \right)$ (4)
$J_R$ is called the reduced Jacobian matrix of the system.
2.2. Modes of voltage instability
Voltage Stability characteristics of the system have been identified by computing the Eigen values and Eigen vectors.
Let
$J _ { R } = \xi \wedge \eta$ (5)
Where,
ξ=right eigenvector matrix of JR
η=left eigenvector matrix of JR
∧=diagonal eigenvalue matrix of JR and
$J _ { R ^ { - 1 } } = \xi _ { \wedge } ^ { - 1 } \eta$ (6)
From (5) and (8), we have
$\Delta V = \xi \wedge ^ { - 1 } \eta \Delta Q$ (7)
or
$\Delta V = \sum _ { I } \frac { \xi _ { i } \eta _ { i } } { \lambda _ { i } } \Delta Q$ (8)
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,
$\Delta Q _ { m i } = K _ { i } \xi _ { i }$ (9)
Where
$K _ { i } = \sum _ { j } \xi _ {{ i j } ^2} - 1$ (10)
Where
ξji is the jth element of ξi
The corresponding ith modal voltage variation is
$\Delta V _ { m i } = \left[ 1 / \lambda _ { i } \right] \Delta Q _ { m i }$ (11)
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,
$\Delta V = \sum _ { i } \frac { \eta _ { 1 k } \xi _ { 1 } } { \lambda _ { 1 } }$ (12)
$\mathrm{n}_{1 \mathrm{k}}$ kth element of
V–Q sensitivity at bus k
$\frac { \partial V _ { K } } { \partial Q _ { K } } = \sum _ { i } \frac { \eta _ { 1 k } \xi _ { 1 } } { \lambda _ { 1 } } = \sum _ { i } \frac { P _ { k i } } { \lambda _ { 1 } }$ (13)
To minimize the system real power loss,
$\left.\mathrm { P } _ { \text {loss } } = \sum _ { \mathrm { k } = ( \mathrm { i } , \mathrm { j } ) } ^ { \mathrm { n } } \mathrm { g } _ { \mathrm { k } \left( \mathrm { V } _ { \mathrm { i } } ^ { 2 } + \mathrm { V } _ { \mathrm { j } } ^ { 2 } - 2 \mathrm { V } _ { \mathrm { i } } \right.} \mathrm { v } _ { \mathrm { j } } \cos \theta _ { \mathrm { ij } } \right)$ (14)
Voltage deviation magnitudes (VD) is stated as Minimize
$\mathrm { VD } = \sum _ { \mathrm { k } = 1 } ^ { \mathrm { nl } } \left| \mathrm { V } _ { \mathrm { k } } - 1.0 \right|$ (15)
Load flow equality constraints are:
$\mathrm { P } _ { \mathrm { Gi } } - \mathrm { P } _ { \mathrm { Di } } - \mathrm { V } _ { \mathrm { i } } \sum _ { \mathrm { j } = 1 } ^ { \mathrm { nb } } \mathrm { v } _ { \mathrm { j } } \left[ \begin{array} { c c } { \mathrm { G } _ { \mathrm { ij } } } & { \cos \theta _ { \mathrm { ij } } } \\ { + \mathrm { B } _ { \mathrm { ij } } } & { \sin \theta _ { \mathrm { ij } } } \end{array} \right] = 0 , \mathrm { i } = 1,2 \ldots , \mathrm { nb }$ (16)
$\mathrm { Q } _ { \mathrm { Gi } } - \mathrm { Q } _ { \mathrm { Di } } - \mathrm { V } _ { \mathrm { i } } \sum _ { \mathrm { j } = 1 } ^ { \mathrm { nb } } \mathrm { v } _ { \mathrm { j } } \left[ \begin{array} { c c } { \mathrm { G } _ { \mathrm { ij } } } & { \sin \theta _ { \mathrm { ij } } } \\ { + \mathrm { B } _ { \mathrm { ij } } } & { \cos \theta _ { \mathrm { ij } } } \end{array} \right] = 0 , \mathrm { i } = 1,2 \ldots , \mathrm { nb }$ (17)
Inequality constraints are:
$V _ { \mathrm { Gi } } ^ { \min } \leq V _ { \mathrm { Gi } } \leq V _ { \mathrm { Gi } } ^ { \max } , \mathrm { i } \in \mathrm { ng }$ (18)
$\mathrm { V } _ { \mathrm { Li } } ^ { \min } \leq \mathrm { V } _ { \mathrm { Li } } \leq \mathrm { V } _ { \mathrm { Li } i } ^ { \max } , \mathrm { i } \in \mathrm { nl }$ (19)
$\mathrm { Q } _ { \mathrm { Ci } } ^ { \min } \leq \mathrm { Q } _ { \mathrm { Ci } } \leq \mathrm { Q } _ { \mathrm { Ci } } ^ { \max } , \mathrm { i } \in \mathrm { nc }$ (20)
$\mathrm { Q } _ { \mathrm { Gi } } ^ { \min } \leq \mathrm { Q } _ { \mathrm { Gi } } \leq \mathrm { Q } _ { \mathrm { Gi } } ^ { \max } , \mathrm { i } \in \mathrm { ng }$ (21)
$\mathrm { T } _ { \mathrm { i } } ^ { \mathrm { min } } \leq \mathrm { T } _ { \mathrm { i } } \leq \mathrm { T } _ { \mathrm { i } } ^ { \mathrm { max } } , \mathrm { i } \in \mathrm { nt }$ (22)
$S _ { \mathrm { Li } } ^ { \min } \leq S _ { \mathrm { Li } } ^ { \max } , i \in \mathrm { nl }$ (23)
Canis Rufus Floridanus optimization algorithm imitates the collective organization and other activities of Canis Rufus Floridanus. α, β and γ are the three fittest candidate solutions has been assumed in the regions of exploration space. Other Canis Rufus Floridanus is denoted as 'φ' and it will enhance α, β and γ to encircle, hunt, attack prey; in the formulated algorithm searching towards improved solutions. Actions of Canis Rufus Floridanus are mathematically written as:
$\vec { Z } = | \vec { M } \cdot \overrightarrow { X _ { P } } ( t ) - \vec { X } ( t ) |$ (24)
$\vec { X } ( t + 1 ) = \overrightarrow { X _ { P } } ( t ) - \vec { N } \cdot \vec { Z }$ (25)
Where t indicates the current iteration, $\vec { N } = 2 \vec { b } \cdot \overrightarrow { r _ { 1 } } - \vec { b } , \vec { M } = 2 \cdot \overrightarrow { r _ { 2 } } , \widehat { X _ { P } }$ the position vector of the prey, $\overrightarrow { \mathrm { X } }$ is the position vector of a Canis Rufus Floridanus, $\overrightarrow { \mathrm { b } }$ is linearly decreased from 1.99 to 0, and $\overrightarrow { \mathrm { r } _ { 1 } }$ and $\overrightarrow { \mathrm { r } _ { 2 } }$ are random vectors in [0, 1].
Hunting behavior of Canis Rufus Floridanus are formulated as,
$\overrightarrow { Z _ { \alpha } } = | \overrightarrow { M _ { 1 } } , \overrightarrow { X _ { \alpha } } - \vec { X } |$
$\overrightarrow { Z _ { \beta } } = | \overrightarrow { M _ { 2 } } , \overrightarrow { x _ { \beta } } - \vec { X } |$
$\overrightarrow { Z _ { \gamma } } = | \overrightarrow { M _ { 3 } } , \overrightarrow { X _ { \gamma } } - \vec { X } |$ (26)
$\overrightarrow { X _ { 1 } } = \overrightarrow { X _ { \alpha } } - \overrightarrow { N _ { 1 } } \cdot \overrightarrow { Z _ { \alpha } }$
$\overrightarrow { X _ { 2 } } = \overrightarrow { X _ { \beta } } - \overrightarrow { N _ { 2 } } \cdot \overrightarrow { Z _ { \beta } }$
$\overrightarrow { X _ { 3 } } = \overrightarrow { X _ { Y } } - \overrightarrow { N _ { 3 } } \cdot \overrightarrow { Z _ { Y } }$ (27)
$\vec { X } ( t + 1 ) = \frac { \overrightarrow { x _ { 1 } } + \overrightarrow { x _ { 2 } } + \overrightarrow { x _ { 3 } } } { 3 }$ (28)
Position of Canis Rufus Floridanus was updated by equation (28) and to discrete the position the following equation formulated,
$f l a g _ { i , j } = \left\{ \begin{array} { c c } { 1 } & { X _ { i , j } > 0.498 } \\ { 0 } & { \text { otherwise } } \end{array} \right.$ (29)
Where i, indicates the jth position of the ith Canis Rufus Floridanus, $flag_{i,j}$ indicates about the total features of Canis Rufus Floridanus.
In this formulation particle swarm optimization is utilized to enrich the exploration & latter exploitation. Position & velocity of the particles are defined by,
$v _ { t + 1 } ^ { i } = \omega _ { t } \cdot v _ { t } ^ { i } + c g _ { 1 } \cdot R m _ { 1 } \cdot \left( m _ { t } ^ { i } - y _ { t } ^ { i } \right) + c g _ { 2 } \cdot \operatorname { Rm } _ { 2 } \cdot \left( m _ { t } ^ { g } - y _ { t } ^ { i } \right)$ (30)
$y _ { t + 1 } ^ { i } = y _ { t } ^ { i } + v _ { t + 1 } ^ { i }$ (31)
The current position of particle is $\mathrm { y } _ { \mathrm { t } } ^ { \mathrm { i } }$ & search velocity is $\mathrm { v} _ { \mathrm { t } } ^ { \mathrm { i } }$. Global best-found position is. $\mathrm { m } _ { \mathrm { t } } ^ { \mathrm { g } }$. In uniformly distributed interval (0, 1) Rm1 & Rm2 are arbitrary numbers. Where cg1 and cg2 are scaling parameters. is the particle inertia. The variable $\omega _ { t } $ is modernized as
$\omega _ { t } = \left( \omega _ { \max } - \omega _ { \min } \right) \cdot \frac { \left( t _ { \max } - t \right) } { t _ { \max } } + \omega _ { \min }$ (32)
Maximum and minimum of $\omega _ { t } $ is represented by $\omega _ { max } $ and $\omega _ { min } $; maximum number of iterations is given by $t _ { max } $. Until termination conditions are met this process will be repeated.
To examine the social interactions within the swarm, when a particle i updates its position based on the position of a particle j (the best neighbor of particle i is the particle j) at a given iteration t social interaction happens in the PSO. Weight of an edge (i, j) is equal to the number of times the particle i was the best neighbor of the particle j or vice-versa .Additionally, they used a time window to control the recency of the analysis, so at iteration t with window tw is defined as follows,
$I _ { i j } ^ { t w } = \sum _ { t ^ { \prime } = t - t _ { w } + 1 } ^ { t } \left[ \delta _ { i , n j \left( t ^ { \prime } \right) } + \delta _ { j , n i \left( t ^ { \prime } \right) } \right] ,$ with $t > t _ { w } \geq 1$ (33)
$A _ { t _ { w } }$ measures the diversity in the information flow for a given time window. The communication diversity CD is defined as following,
$C D ( t ) = 1 - \frac { 1 } { | T | | S | } \sum _ { t _ { w } ^ { \prime } \in t } A _ { t _ { w } } = t _ { w } ^ { \prime } ( t )$ (34)
Where |S| is the number of particles in the swarm and T is a set of time windows. Thus, swarms exhibiting high CD (low values for $A _ { t _ { w } }$) have the ability to have diverse information flows, while low values for CD imply in swarms with only few information flows (high value for $A _ { t _ { w } }$). An ideal set T would be one taking into account all time windows (i.e., interactions from tw=1 until tw=t).
Canis Rufus Floridanus; α, β and γ determine the position of the prey. $\overrightarrow { \mathrm { N } } = 2 \overrightarrow { \mathrm { b } } \cdot \overrightarrow { \mathrm { r } _ { 1 } } - \overrightarrow { \mathrm { b } }$ directs the exploration & exploitation process by reducing the value from 1.99 to 0.When $| \vec { N } | < 1$ it converged towards the prey & If $| \vec { N } | > 1$ diverged away. The first best Minimum loss and variables are accumulated as "α" position, score & as like second best, third best accumulated as "β" and "γ" position & score.
The efficiency of the proposed Improved Canis Rufus Floridanus (ICRF) optimization algorithm is demonstrated by testing it on standard IEEE-30 bus system. The IEEE-30 bus system has 6 generator buses, 24 load buses and 41 transmission lines of which four branches are (6-9), (6-10), (4-12) and (28-27) - are with the tap setting transformers. The lower voltage magnitude limits at all buses are 0.95 p.u. and the upper limits are 1.1 for all the PV buses and 1.05 p.u. for all the PQ buses and the reference bus. The simulation results have been presented in Tables 1, 2, 3 &4. The optimal values of the control variables along with the minimum loss obtained are given in Table 1. Corresponding to this control variable setting, it was found that there are no limit violations in any of the state variables.
Table 1. Results of ICRF–ORPD optimal control variables
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 |
1.03100 1.03200 1.03900 1.03100 1.00000 1.03000 1.0000 1.0000 1.0000 1.0100 3 2 2 0 3 2 3 2 4.2406 0.2462 |
Optimal Reactive Power Dispatch (ORPD) problem together with voltage stability constraint problem was handled in this case as a multi-objective optimization problem where both power loss and maximum voltage stability margin of the system were optimized simultaneously.
Table 2 indicates the optimal values of these control variables. Also it is found that there are no limit violations of the state variables. It indicates that voltage stability index has increased from 0.2462 to 0.2485, which is an improvement in the system voltage stability.
To determine the voltage security of the system, contingency analysis was conducted using the control variable setting obtained in case 1 and case 2. The Eigen values equivalents to the four critical contingencies are given in Table 3. From this result it is observed that the Eigen value has been improved considerably for all contingencies in the second case.
Table 2. Results of ICRF-Voltage stability control reactive power dispatch (VSCRPD) optimal control variables
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 |
1.04500 1.04100 1.04000 1.02900 1.00000 1.03000 0.09000 0.09000 0.09000 0.09000 2 2 2 3 0 2 2 3 4.9886 0.2485 |
Sl.No |
Contingency |
ORPD Setting |
VSCRPD Setting |
1 |
28-27 |
0.1419 |
0.1434 |
2 |
4-12 |
0.1642 |
0.1650 |
3 |
1-3 |
0.1761 |
0.1772 |
4 |
2-4 |
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 |
38.820 |
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 |
In the Table 5 shows the proposed algorithm powerfully reduces the real power losses when compared to other given standard algorithms.
Table 5. Comparison of real power loss
Method |
Minimum loss |
Method; Evolutionary programming (Wu and Ma, 1995) |
5.01590 |
Method; Genetic algorithm (Durairaj et al., 2006) |
4.6650 |
Method; Real coded GA with Lindex as SVSM (Devaraj, 2007) |
4.5680 |
Method; Real coded genetic algorithm (Jeyanthy and Devaraj, 2010) |
4.50150 |
Proposed ICRF method |
4.24060 |
In this paper, the Improved Canis Rufus Floridanus (ICRF) Optimization Algorithm has been successfully solved Optimal Reactive Power Dispatch problem. Efficiency of the projected Improved Canis Rufus Floridanus (ICRF) Optimization Algorithm has been evaluated in standard IEEE 30 bus test system. Real power losses are reduced by the proposed algorithm when compared to other standard algorithms & voltage stability index has increased from 0.2462 to 0.2485, which is an improvement in the system voltage stability. To determine the voltage security of the system, contingency analysis was conducted using the control variable setting obtained.
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