Real power loss diminution by camelopard optimization algorithm

Modal analysis for voltage stability evaluation

Power flow equations of the steady state system is given by,

$\left[ \begin{array} { c } { \Delta \mathrm { P } } \\ { \Delta \mathrm { Q } } \end{array} \right] = \left[ \begin{array} { l } { \mathrm { J } _ { \mathrm { p } \theta } \mathrm { J } _ { \mathrm { pv } } } \\ { \mathrm { J } _ { \mathrm { q } \theta } \mathrm { J } _ { \mathrm { QV } } } \end{array} \right] \left[ \begin{array} { l } { \Delta \theta } \\ { \Delta V } \end{array} \right]$   (1)

Where

ΔP = bus real powerchange incrementally.

ΔQ = bus reactive Power injectionchange incrementally.

Δθ = bus voltage angle change incrementally.

ΔV = bus voltage Magnitudechange incrementally.

Jpθ, JPV, JQθ, JQV are sub-matrixes of the System voltage stability in jacobian matrix and both P and Q get affected by this.

Presume ΔP = 0, then equation (1) can be written as:

$\left.\Delta \mathrm { Q } = \left[ \mathrm { J } _ { \mathrm { QV } } - \mathrm { J } _ { \mathrm { Q } \theta } \mathrm { J } _ { \mathrm { P } \mathrm { \theta } ^ { - 1 } } \right] _ { \mathrm { PV } } \right] \Delta \mathrm { V } = \mathrm { J } _ { \mathrm { R } } \Delta \mathrm { V }$        (2)

$\Delta \mathrm { V } = \mathrm { J } ^ { - 1 } - \Delta \mathrm { Q }$   (3)

Where

$\mathrm { J } _ { \mathrm { R } } = \left( \mathrm { J } _ { \mathrm { QV } } - \mathrm { J } _ { \mathrm { Q } \theta } \mathrm { J } _ { \mathrm { P } \theta ^ { - 1 } } \mathrm { JPV } \right)$      (4)

J_R denote the reduced Jacobian matrix of the system.

2.1. Modes of voltage instability

Voltage Stability characteristics of the system have been identified through computation of the Eigen values and Eigen vectors.

$\mathrm { J } _ { \mathrm { R } } = \xi \wedge \mathrm { \eta }$   (5)

Where,

ξ denote the  right eigenvector matrix of JR, ηdenote the  left eigenvector matrix of JR, ∧ denote the  diagonal eigenvalue matrix of JR.

$\mathrm { J } _ { \mathrm { R } ^ { - 1 } } = \xi ^ { - 1 } \mathrm { \eta }$   (6)

From the equations (5) and (6),

$\Delta \mathrm { V } = \xi \wedge ^ { - 1 } \eta \Delta \mathrm { Q }$  (7)

or

$\Delta \mathrm { V } = \sum _ { \mathrm { I } } \frac { \mathfrak { \xi} _ { 1 } \eta _ { \mathrm { i } } } { \lambda _ { \mathrm { i } } } \Delta \mathrm { Q }$    (8)

ξi denote the ith column right eigenvector & η is the ith row left eigenvector of JR.

λi indicate the ith Eigen value of JR.

reactive power variation ofthe ith modalis given by,

$\Delta \mathrm { Q } _ { \mathrm { mi } } = \mathrm { K } _ { \mathrm { i } } \xi _ { \mathrm { i } }$  (9)

where,

$\mathrm { K } _ { \mathrm { i } } = \Sigma _ { \mathrm { j } } \xi _ { \mathrm { ij } ^ { 2 } } - 1$    (10)

Whereξji is the jth element of ξi

ith modal voltage variation is mathematically given by,

$\Delta \mathrm { V } _ { \mathrm { mi } } = \left[ 1 / \lambda _ { \mathrm { i } } \right] \Delta \mathrm { Q } _ { \mathrm { mi } }$    (11)

When the value of |λi| =0 then the ith modal voltage will get collapsed.

In equation (8), when ΔQ = ek  is assumed ,then ek has all its elements zero except the kth one being 1. Then

$\Delta \mathrm { V } = \sum _ { \mathrm { i } } \frac { \mathrm { n } _ { 1 \mathrm { k } } \xi _ { 1 } } { \lambda _ { 1 } }$     (12)

$\eta_{1k}$ is k th element of $\eta_1$

At bus k V –Q sensitivity is given by,

$\frac { \partial \mathrm { v } _ { \mathrm { K } } } { \partial \mathrm { Q } _ { \mathrm { K } } } = \sum _ { \mathrm { i } } \frac { \eta _ { 1 \mathrm { k } } \xi _ { 1 } } { \lambda _ { 1 } } = \sum _ { \mathrm { i } } \frac { \mathrm { P } _ { \mathrm { ki } } } { \lambda _ { 1 } }$     (13)

Minimization of actual power loss and augmentation of static voltage stability margin index (SVSM) is main key to solve optimal reactive power dispatch problem. Voltage stability evaluation has been done through modal analysis method.

2.2. Minimization of real power loss

Real power loss (Ploss) minimization is given as,

$\mathrm { P } _ { \text {loss } } = \sum _ { \mathrm { k } = ( \mathrm { i } , \mathrm { j } ) } ^ { \mathrm { n } } \mathrm { g } _ { \mathrm { k } \left( \mathrm { V } _ { \mathrm { i } } ^ { 2 } + \mathrm { V } _ { \mathrm { j } } ^ { 2 } - 2 \mathrm { V } _ { \mathrm { i } } \mathrm { V } _ { \mathrm { j } } \cos \theta _ { \mathrm { ij } } \right) }$   (14)

Where n is the number of transmission lines, gk is the conductance of branch k, Vi and Vj are voltage magnitude at bus i and bus j, and θij is the voltage angle difference between bus i and bus j.

2.3. Minimization of voltage deviation

Formula for reducing the voltage deviation magnitudes (VD) is derived as follows,

Minimize $\mathrm { VD } = \sum _ { \mathrm { k } = 1 } ^ { \mathrm { nl } } \left| \mathrm { V } _ { \mathrm { k } } - 1.0 \right|$    (15)

Where nl is the number of load busses and Vk is the voltage magnitude at bus k.

2.4. System constraints

Load flow equality constraints:

$\mathrm { P } _ { \mathrm { Gi } } - \mathrm { P } _ { \mathrm { Di } } - \mathrm { V } _ { \mathrm { i } } \sum _ { \mathrm { j } = 1 } ^ { \mathrm { nb } } \mathrm { v } _ { \mathrm { j } } \left[ \begin{array} { c c } { \mathrm { G } _ { \mathrm { ij } } } & { \cos \theta _ { \mathrm { ij } } } \\ { + \mathrm { B } _ { \mathrm { ij } } } & { \sin \theta _ { \mathrm { ij } } } \end{array} \right] = 0 , \mathrm { i } = 1,2 \ldots , \mathrm { nb }$     (16)

$\mathrm { P } _ { \mathrm { Gi } } - \mathrm { P } _ { \mathrm { Di } } - \mathrm { V } _ { \mathrm { i } } \sum _ { \mathrm { j } = 1 } ^ { \mathrm { nb } } \mathrm { v } _ { \mathrm { j } } \left[ \begin{array} { c c } { \mathrm { G } _ { \mathrm { ij } } } & { \sin \theta _ { \mathrm { ij } } } \\ { + \mathrm { B } _ { \mathrm { ij } } } & { \cos \theta _ { \mathrm { ij } } } \end{array} \right] = 0 , \mathrm { i } = 1,2 \ldots , \mathrm { nb }$    (17)

where, nb is the number of buses, P_G and Q_G are the real and reactive power of the generator, P_D and Q_D are the real and reactive load of the generator, and G_ij and B_ij are the mutual conductance and susceptance between bus i and bus j.

$\mathrm { V } _ { \mathrm { Gi } } ^ { \min } \leq \mathrm { V } _ { \mathrm { Gi } } \leq \mathrm { V } _ { \mathrm { Gi } } ^ { \max } , \mathrm { i } \in \mathrm { ng }$   (18)

$V _ { \mathrm { Li } } ^ { \min } \leq V _ { \mathrm { Li } } \leq V _ { \mathrm { Li } } ^ { \max } , \mathrm { i } \in \mathrm { nl }$    (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)

$S _ { \mathrm { Li } } ^ { \min } \leq S _ { \mathrm { Li } } ^ { \max } , \mathrm { i } \in \mathrm { nl }$   (23)

Camelopard optimization Algorithm (COA) is tested in standard IEEE 30-bus system. In Table 1control variables are given.

Table 1. Limits

Power limits of the generators are listed in table 2.

Table 2. Generators power limits

Control variables obtained after optimization given in table 3.COA performance presented in table 4. Comparison of active power loss is given in table 5. Fig 1 gives comparison of real power loss

Table 3. Values of control variable after optimization

Table 4. COA performance

Table 5. Evaluation of outcome

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Figure 1. Comparison of real power loss

Table 6. Generator data

Table 7. Comparison of losses

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Figure 2. Comparison of loss

Secondly IEEE 57 bus system is used as test system to validate the performance of the proposed algorithm. Total active and reactive power demands in the system are 1247.89 MW and 338.04 MVAR, respectively. Generator data the system is given in Table 6. The optimum loss comparison is presented in Table 7. Fig 2. Gives the comparaison of losses.

Table 8. Reactive power sources limits

Table 9. Evaluation of results

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Figure 3. Comparison of actual loss

Table 9. shows the comparaison of results.

Then IEEE 118 bus system is used as test system to validate the performance of the proposed algorithm. Table 8 shows limit values.

Finally IEEE 300 bus system is used as test system and Table10 shows the comparaison of real power loss.

With Considering Voltage Stability Evaluation Criterion in IEEE 30 bus system projected algorithm has been verified. Table 11 shows the optimal control variables.

Table 10. Comparison of real power loss

Table 11. COA-ORPD based control variables

Static voltage stability index rises from 0.2382 to 0.2396.

In table 12 optimal (control variables) are given.

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Figure 4. Comparison of  active power loss

Table 12. Value of COA -voltage stability control reactive power dispatch optimal control variables

In Table 13 Eigen values are given.

Table 13. Values of settings

In table 14 values for limit violation checking has been given with upper & lower limits.

Table 14. Limits of violation

In table 15 over all comparison of real power loss has been given. It indicates that proposed algorithm efficiently reduced power loss. Fig 5. Gives Comparison of real power loss

Table 15. Comparison of losses

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Figure 5. Comparison of real power loss