Multi distributed generation placement using ant-lion optimization

This paper the mainly concentrate on improvement of bus voltages and minimization of active power loss in distribution network utilizing one DG, multi DG system. By using BW/FW sweep analysis initially, calculate system reactive and active power loss and voltages at every bus. Then the reduction of power loss indices also with per unit bus voltages, which are considered as, inputs for an advanced algorithm, which determine the best DG place and size.

2.1. Objective functions

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Figure 1. Two bus single line diagram

The above Figure 1 represents a two-bus radial network. This paper deals the issue of active power loss minimization and improvement of voltage at all the buses in the test system after and before integrating DGs in best sites with proper capacity. The main objective function is to enhance the voltage profile and the active power loss reducing as follows:

$F=\left\{ \begin{align} & \min .active\text{ }power\text{ }loss\text{ }({{f}_{1}}) \\  & \min .Voltage\text{ }deviation\text{ }at\text{ }every\text{ }bus\text{ }({{f}_{2}}) \\  & \max .Voltage\text{ }Stability\text{ }Index\text{ }(VSI)\text{ }({{f}_{3}}) \\ \end{align} \right.$    (1)

The mathematical formula for calculating power loss in the network is (Elgerd and Happ, 1971),

$\min active\text{ }power\text{ }loss\text{ }\left( {{f}_{1}} \right)=\min \left( {{P}_{L}} \right)$      (2)

Where, ${{P}_{L}}=\sum\limits_{i=1}^{N}{\sum\limits_{j=1}^{n}{\left[ {{\alpha }_{ij}}\left( {{P}_{i}}{{P}_{j}}+{{Q}_{i}}{{Q}_{j}} \right)+{{\beta }_{ij}}\left( {{Q}_{i}}{{P}_{j}}-{{P}_{i}}{{Q}_{j}} \right) \right]}}$, ${{\alpha }_{ij}}=\frac{{{r}_{ij}}}{{{V}_{i}}{{V}_{j}}}\cos \left( {{\delta }_{i}}-{{\delta }_{j}} \right)$, ${{\beta }_{ij}}=\frac{{{r}_{ij}}}{{{V}_{i}}{{V}_{j}}}\sin \left( {{\delta }_{i}}-{{\delta }_{j}} \right)$ and ${{Z}_{ij}}={{r}_{ij}}+j{{x}_{ij}}$ is the ij^th element of [Z_bus] matrix, N–Total bus numbers.

Another objective function is to improve the system bus voltages, mathematically represented as:

$\min Voltage\text{ }deviation\left( {{f}_{2}} \right)=\sum\limits_{ni=1}^{{{N}_{n}}}{{{\left( {{V}_{ni}}-{{V}_{rated}} \right)}^{2}}}$   (3)

Where, N_n is the number of system buses, V_ni specifies the i^th bus voltage and V_rated specifies the rated bus voltage of the network that is equal to1.0 p.u.

The final objective function in this paper is Voltage Stability Index (VSI) is represented as follows:

$\max (VSI(ni))={{\left| {{V}_{mi}} \right|}^{4}}-4\left[ {{P}_{ni}}(ni){{R}_{ni}}+{{Q}_{ni}}(ni){{X}_{ni}} \right]{{\left| {{V}_{mi}} \right|}^{2}}- 4{{\left[ {{P}_{ni}}(ni){{R}_{ni}}+{{Q}_{ni}}(ni){{X}_{ni}} \right]}^{2}}$    (4)

Where, P_ni(ni) is Total active power fed through bus ni, Q_ni(ni) is Total reactive power through bus ni, R_ni is i-th branch resistance, X_ni is i-th branch reactance and SI(ni) is VSI (Voltage stability Index) of ni node.

${{P}_{i}}={{P}_{Gi}}-{{P}_{Di}}$    (5)

${{Q}_{i}}={{Q}_{Gi}}-{{Q}_{Di}}$     (6)

Where, P_Giand Q_Gispecifies real and reactive generated power at bus i respectively; P_Diand Q_Dispecifies real and reactive demand of load at bus i respectively.

2.2. Constraints

2.2.1. Voltage

The deviation of voltage is specified in the limits of 0.9 p.u. and 1.05 p.u.

${{V}_{i,\min }}\le {{V}_{i}}\le {{V}_{i,\max }}$        (7)

Where, V_min is minimum voltage at bus i, V_i is the voltage at bus i and V_max is the maximum voltage at bus i.

2.2.2. Power

$P{{L}_{afterDG}}\le P{{L}_{beforeDG}}$    (8)

Where, PL_afterDG is the Power Loss after DG installation and PL_beforeDG is the Power Loss before DG installation.

2.2.3. Distributed generation

$\sum\limits_{i=1}^{Nbus}{0\le {{P}_{DG}}\le {{P}_{Di}}}$    (9)

$\sum\limits_{i=1}^{Nbus}{0\le {{Q}_{DG}}\le {{Q}_{Di}}}$     (10)

Where, P_DGand Q_DGare the active and reactive injected by DG at the same bus i, P_Diand Q_Diare the active and reactive load demand at the same bus i.

In 2015 Mirjalili was presented an Ant-Lion Optimizer (ALO) which was inspired by nature (Raju et al., 2016). ALO emulates hunting behaviour of ant-lion’s from nature. The important two phases in ALO life cycle are Larvae phase and adult phases. Larvae phase of Ant lions for hunt and reproduction in adult phase. Larvae phase is the motivation for ALO algorithm. ALO algorithm has mainly five operations those are ants’ movement randomly, trap building, ants’ trapping in traps, catching of food/prays and re-building of traps. Ants’ positions randomly are stored in the matrix M_ant is given as equation (11):

${{M}_{ant}}=\left( \begin{matrix}   An{{t}_{1,1}} & An{{t}_{1,2}} & \ldots  & An{{t}_{1,n}}  \\   An{{t}_{2,1}} & An{{t}_{2,2}} & \ldots  & An{{t}_{2,d}}  \\   \vdots  & \vdots  & \vdots  & \vdots   \\   An{{t}_{n,1}} & An{{t}_{n,2}} & \ldots  & An{{t}_{n,d}}  \\\end{matrix} \right)$       (11)

Here A_m,n describes the m^th value of variable of n^th ant, n denotes the ants number (population size). Ant’s fitness should be kept within the M_OA matrix in terms of the objective function f is given as equation (12):

${{M}_{OA}}=\left( \begin{matrix}   f\left[ An{{t}_{1,1}},An{{t}_{1,2}},\ldots ,An{{t}_{1,n}} \right]  \\   f\left[ An{{t}_{2,1}},An{{t}_{2,2}},\ldots ,An{{t}_{2,d}} \right]  \\   \vdots   \\   f\left[ An{{t}_{n,1}},An{{t}_{n,2}},\ldots ,An{{t}_{n,d}} \right]  \\\end{matrix} \right)$   (12)

The location and fitness of ant-lion are shown by the matrices M_antlion and M_OAL, which is given as equation (13):

${{\text{M}}_{\text{antlion}}}\text{=}{{\text{M}}_{\text{OAL}}}\text{=}\left( \begin{matrix}   \text{f}\left( \left[ \text{A}{{\text{L}}_{\text{1,1}}}\text{,A}{{\text{L}}_{\text{1,2}}}\text{,,A}{{\text{L}}_{\text{1,n}}} \right] \right)  \\   \text{f}\left( \left[ \text{A}{{\text{L}}_{\text{2,1}}}\text{,A}{{\text{L}}_{\text{2,2}}}\text{,,A}{{\text{L}}_{\text{2,n}}} \right] \right)  \\   \text{M}  \\   \text{f}\left( \left[ \text{A}{{\text{L}}_{\text{n,1}}}\text{,A}{{\text{L}}_{\text{n,2}}}\text{,,A}{{\text{L}}_{\text{n,n}}} \right] \right)  \\\end{matrix} \right)$   (13)

4.1. Ants movement randomly

ALO approach mimics the interaction among ants and ant-lions in the trap. To describe this interaction, ants should move along the search area and ant-lions were permitted to hunt ants and become best fitter utilizing traps. All ants move randomly for food search. Ants’ random movement is shown by equation (14):

$X(t)=\left[ 0,cummsum(2r({{t}_{1}})-1),cummsum(2r({{t}_{2}})-1),...,cummsum(2r({{t}_{n}})-1) \right]$  (14)

Here cummsum is cummulative sum, n signifies as number of ants, t described as iteration and r(t) denotes the random number generation, rand signifies a random number created with constant distribution in [0,1], which given as equation (15):

$r(t)=\left\{ \begin{matrix}   01\,\,\,when\text{ }rand>0.50  \\   00\,\,\,when\,rand<0.50  \\\end{matrix} \right.$       (15)

Normalized form is utilized to confine the arbitrary movement within the search area. Location of ants will be changed by equation (16):

$X_{m}^{t}=\frac{(X_{m}^{t}-{{a}_{m}})({{b}_{m}}-c_{m}^{t})}{dd_{m}^{t}-{{a}_{m}}}+{{c}_{t}}$      (16)

Here a_m and b_m are ants random walk minimum and maximum, c^t_m and dd^t_m are m^th variable at t^th iteration minimum and maximum.

4.2. Trapping of ants

Ant’s random walk is affected by traps constructed by ant-lion’s traps. Ants trapping expression into the ant-lion’s pits is mathematically represented by equations (17) and (18):

$c_{m}^{t}=Ant-lion_{n}^{t}+{{c}^{t}}$         (17)

$dd_{m}^{t}=Ant-lion_{n}^{t}+d{{d}^{t}}$       (18)

4.3. Trap construction

By using Roulette wheel technique the fittest ant-lion is selected during iterations used in ALO. This mechanism gives high opportunities for ants catching.

4.4. Ants sliding into pits

Relative to fitness of ant lions they can prepare traps and ants were assumed to randomly move. Ant lions shoot sands away from the pit centre when an ant-lion recognises that there is an ant inside trap. This behaviour slips down the trapped ant when is trying to escape. For describe this behaviour, the area of random walk of ant's hyper sphere is adaptively condensed. Ants slipping into pits are represented by equations (19) and (20):

${{c}^{t}}=\frac{{{c}^{t}}}{I}$      (19)

$d{{d}^{t}}=\frac{d{{d}^{t}}}{I}$     (20)

Here I=10.0^w t/s, t specifies present iteration, S specifies the maximum iterations and w specifies a constant. The w measurement is described by equation (21):

$w=\left\{ \begin{matrix}   02\,\,\,\,when\text{ }\,t>0.10S  \\   03\,\,\,\,when\text{ }\,t>0.50S  \\   04\,\,\,\,when\text{ }\,t>0.750S  \\   05\,\,\,\,when\text{ }\,t>0.90S  \\   06\,\,\,\,when\text{ }\,t>0.950S  \\\end{matrix} \right.$    (21)

4.5. Pray/food catching and pit reconstruction

Than the chosen ant lion, if the ant having a perfect objective functions later it modifies their place to the new location of the hunted ant to advance its catching chance of new ant. Later, in order to catch new prey ant-lion should update the location (Rajan et al., 2017). This is given by equation (22):

$Ant\,lion_{n}^{t}=Ant_{m}^{t}\,\,if\,f(Ant_{m}^{t})>f(Ant\,lion_{j}^{t})$    (22)

4.6. Elitism

Elitism was utilized to have the finest results in each phase. Ant lion can be considered as elite when it is globally best. For this by Roulette wheel all ants were assumed to coordinate with an ant-lion and the elite is described by equation (23):

$An{{t}^{t}}=\frac{R_{A}^{t}+R_{B}^{t}}{2}$       (23)

Here R_A and R_B is the ant walk randomly about an ant-lion particularly and elite at iteration t^th simultaneously. This represents that ant walk randomly is highly constrained around the elite ant lions and as a consequently, algorithm experiences poor exploration. The concept of meta-heuristic algorithms is extremely requires appropriate balance among exploration and exploitation (Moradi and Abedini, 2012).

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Figure 3. ALO algorithm analysis flowchart

Table 1. Total summary of results obtained by multi DG placement using various algorithms

The percentage of power loss reduction of connected multi DGs are given in the Table 1. With 1-DG, the percentage of minimization in active power loss by using ALO is 53.86%, with 2-DGs, it is 57.50%, and with 3-DGs, it is 64.16%. It clearly understand from the outcomes that with increase in DGs number, the percentage minimization in real power loss is much effective.

Table 2. Comparison analysis of VSI

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Figure 5. Voltage comparisons using GSA, BAT and ALO methods individually

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Figure 6. Voltage comparisons using GSA, BAT and ALO methods

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Figure 7. Power loss comparisons using GSA, BAT and ALO methods individually

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Figure 8. Power loss comparisons using GSA, BAT and ALO methods