Novel Extreme Learning Machine and Chaotic in-Built Opposition Based – Quantum Ruddy Turnstone Optimization Algorithms for Real Power Loss Dwindling and Voltage Consistency Enhancement

Kanagasabai, Lenin

doi:10.1007/s40866-022-00149-8

Novel Extreme Learning Machine and Chaotic in-Built Opposition Based – Quantum Ruddy Turnstone Optimization Algorithms for Real Power Loss Dwindling and Voltage Consistency Enhancement

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Published: 29 June 2022

Volume 7, article number 25, (2022)
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Novel Extreme Learning Machine and Chaotic in-Built Opposition Based – Quantum Ruddy Turnstone Optimization Algorithms for Real Power Loss Dwindling and Voltage Consistency Enhancement

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Lenin Kanagasabai ORCID: orcid.org/0000-0002-4367-5783¹

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Abstract

This paper proposes Ruddy turnstone optimization (RTO) algorithm, Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Chaotic based Ruddy turnstone optimization (CRTO) algorithm, Quantum based Ruddy turnstone Optimization (QRTO) Algorithm, Opposition based Ruddy turnstone optimization (ORTO) Algorithm and chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm for genuine loss lessening. Important goals of the paper are Power fidelity extension, power eccentricity minimization and genuine loss lessening. The leading stimulus is in the sculpting of Repositioning –Peripatetic and argumentative actions of Ruddy turnstone. Ruddy turnstone will guzzle wiretaps, young insect and it subsists in bundling style. The constellation of Ruddy turnstone, which mobile from one place to alternate in the sequence of repositioning and the fresh investigation agent position is to evade the smash amongst their contiguous Ruddy turnstone. In the proposed Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Ruddy turnstone Optimization Algorithm approach enhances Extreme Learning Machine features to determine an optimal skeleton of Extreme Learning Machine for enhanced canons. In Chaotic based Ruddy turnstone optimization (CRTO) algorithm Exploration and Exploitation are augmented. In Quantum based Ruddy turnstone Optimization (QRTO) Algorithm, features emulate the analogous performance with the certain stage as they route in a credible powdered of median. Opposition based Ruddy turnstone optimization (ORTO) Algorithm employs Laplace distribution to enhance the exploration skill. Then examining the prospect to widen the exploration, a new method endorses stimulating capricious statistics used in formation stage regulator factor in Ruddy turnstone Optimization Algorithm. In the projected chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm, the transaction of erratic figures is completed with the irrational digits enthused by Laplace distribution to amplify the support of the probability of formation level inside the exploration zone. Proposed Ruddy turnstone optimization (RTO) algorithm, Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Chaotic based Ruddy turnstone optimization (CRTO) algorithm, Quantum based Ruddy turnstone Optimization (QRTO) Algorithm, Opposition based Ruddy turnstone optimization (ORTO) Algorithm and chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm are corroborated in Garver’s 6-bus test system, IEEE 30, 57, 118, 300, 354 bus test systems and Practical system - WDN 220 KV (Unified Egyptian Transmission Network (UETN)). Loss lessening, voltage divergence curtailing, and voltage constancy index augmentation has been attained.

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Introduction

Loss dripping problem is envisioned as one of the noteworthy circumstances for safe and fiscal operation of system. It is consummate by appropriate organization of the edifice apparatus used to cope up the power flow with the goal of diminishing the true power losses and progress the voltage outline of the structure. Many preceding studies solved the power dispatch problem as the optimization of one or two objective functions calculated distinctly. Loss lessening, power variance curtailing, and power reliability expansion has been attained.

Related Study

Zhu et al. [1] solved the problem by modified interior point method. Quintana et al. [2] solved by successive quadratic programming. Jan et al. [3] used Newton-Raphson. Terra et al. [4] did Security-constrained mode. Grudinin [5] used successive quadratic programming. Mohamed Ebeed et al. [6] used Marine Predators Algorithm. Zahir Sahli et al. [7] used Hybrid Algorithm. Davoodi et al. [8] used semidefinite method. Bingane et al. [9] applied Tight-and-cheap conic relaxation approach. Sahli et al. [10] applied Hybridized PSO-Tabu. Mouassa et al. [11] applied Ant lion algorithm for solving the problem. Mandal et al. [12] solved by using quasi-oppositional. Khazali et al. [13] solved the problem by harmony search procedure. Tran et al. [14] solved by fractal search procedure. Polprasert et al. [15] solved the problem by using enhanced pseudo-gradient method. Thanh et al. [16] solved the problem by an Operative Metaheuristic Procedure. Raghuwanshi et al. [17] utilized Bagging based ELM. Yu X et al. [18] applied Dual-Weighted Kernel ELM. Han et al. [19] used kernel ELM. From Illinois Center [20] for a Smarter Electric Grid (ICSEG) IEEE 30 bus system data obtained. Dai et al. [21] used Seeker optimization procedure for solving the problem. Subbaraj et al. [22] used self-adaptive real coded Genetic procedure to solve the problem. Pandya et al. [23] applied Particle swarm optimization to solve the problem. Ali Nasser Hussain et al. [24] applied Amended Particle Swarm Optimization to solve the problem. Vishnu et al. [25] applied an Enhanced Particle Swarm Optimization to solve the problem. Vodchits Angelina et al. [26] did Development of a Design Algorithm for the Logistics. Vodchits Angelina et al. [27] did the work on organization of logistic systems of scientific productions. Vodchits Angelina et al. [28] solved the Problems and organizational and technical solutions. Khunkitti et al. [29] used Slime Mould Algorithm. Diab et al. [30] used Optimization Techniques. Reddy [31, 32] solved the problem by faster and cuckoo search algorithms. Sridhar et al. [33] used ALO method. Suja [34] used moth flame optimization procedure. Darvish [35] applied grasshopper optimization algorithm. Sharma, et al. [36] used hybrid ABC-PSO. Saravanan et al. [37] used dragonfly algorithm. Bentouati, Bet al. [38] applied improved moth-swarm algorithm. Menon, et al. [39] applied OS–DPLL. Saxena, et al. [40] used STATCOM. Kazmi, et al. [41] Worked on Loop Configuration. Kola Sampangi, et al. [42] worked in EDN. Zaidan, Majeed et al. [43] worked in var. comp.optimal Location. Lakshmi Priya et al. [44] used GWO-BSA. Ahmadnia Sajjad et al. [45] worked in ESR. Azimi, Mohammad et al. [46] worked in Thyristor-controlled Phase Shifting. Juneja Kapil [47] used a Fuzzy-Controlled Differential Evolution to solve the problem. Kien et al. [48] used Discrete Values of Capacitors and Tap Changers. Souhil et al. [49] applied ant lion optimization algorithm. Tudose et al. [50] applied Improved Salp Swarm Algorithm. Nagarajan et al. [51] applied Levy Interior Search Algorithm. Mei et al. [52] applied moth-flame optimization technique. Nuaekaew et al. [53] applied grey wolf optimizer. Khazali et al. [13] applied harmony search algorithm. Gonggui et al. [54] applied enhanced PSO algorithm. From ee. Washington [55] IEEE 57-Bus Test System data obtained. From Power Systems Test Case Archive, University of Washington [56] data obtained. From ee. Washington [57] IEEE 118-Bus Test System data obtained. Lin, et al. [58] did work in chaotic Lévy flight bat algorithm. Hakli et al. [59] did work in particle swarm optimization algorithm Nagarajan, et al. [51] did work in Levy Interior Search Algorithm. Davidchack et al., [60] did work in chaotic systems. Inoue, et al., [61] did work in Application of chaos. Dinkar, et al. [62] did in Opposition Based Laplacian. Gai-gewang et al. [63] did work in Cauchy mutation. Tizhoosh [64] did the work in Opposition-based learning. Verma et al. [65] done in Modified Artificial Bee. Romero et al. [66] worked in AC Model. Mahmoudabadi, et al., [67] worked in Deregulated Environment. Asadamongkol, et al. [68] worked in Generalized Benders Decomposition. Mohamed eta l [69] worked in binary bat algorithm. Abou et al. [70] worked in System Expansion Planning. Emad M. Ahmed et al. [77] did work in modern distribution networks. Almalaq et al. [78] did work towards Increasing Hosting Capacity of Modern Power Systems. Ismael et al. [79] applied Hybrid PSOGSA Optimization Algorithm. Mahmoud, et al. [80] worked in Real Egyptian Distribution System. Muhyaddin Rawa et al. [81] applied improved grey wolf algorithm.

Proposed Methodology

This paper proposes Ruddy turnstone optimization (RTO) algorithm, Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Chaotic based Ruddy turnstone optimization (CRTO) algorithm, Quantum based Ruddy turnstone Optimization (QRTO) Algorithm, Opposition based Ruddy turnstone optimization (ORTO) Algorithm and chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm for solving the loss dripping problem. The leading stimulus is in the sculpting of Repositioning –Peripatetic and argumentative actions of Ruddy turnstone. Ruddy turnstone will guzzle wiretaps, young insect and it subsists in bundling style. The constellation of Ruddy turnstone, which mobile from one place to alternate in the sequence of repositioning and the fresh investigation agent position is to evade the smash amongst their contiguous Ruddy turnstone.

In the proposed Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Ruddy turnstone Optimization approach enhances Extreme Learning Machine features to determine an optimal carcass of Extreme Learning Machine for enhanced standards. In principally all elements don’t own any info about the explication area. In preliminary phases of iteration, the Ruddy turnstone contestants are multifarious in milieu and exponential spare generates boundless unpremeditated amounts which contribute the rudiments to lodging the entire explication zone. Disparately, all over end stage of iterations, rudiments are surrounded by Ruddy turnstone contestants and all an optimal situation with similar pattern.

Chaotic sequences are combined into Ruddy turnstone Optimization and it termed as - Chaotic based Ruddy turnstone optimization (CRTO) algorithm. This integration will augment the Exploration and Exploitation. Tinkerbell chaotic map engendering standards are implemented.

Quantum mechanics has been combined with Ruddy turnstone Optimization Algorithm and it titled as Quantum based Ruddy turnstone Optimization (QRTO) Algorithm. In quantum method, features emulate the analogous performance with the certain stage as they route in a credible powdered of median.

Ruddy turnstone Optimization Algorithm, even though the initiative of contestant explications slants to touch an optimal solution, yet several are get entombed and not adept of emotive in the route of the dominant solution. It significances to snare in local optima and it accordingly enforces into primary and slow convergence. Subsequently Opposition based Ruddy turnstone optimization (ORTO) Algorithm employs Laplace distribution to enhance the exploration skill. Then examining the prospect to widen the exploration, a new method endorses stimulating capricious statistics used in formation stage regulator factor in Ruddy turnstone Optimization Algorithm. In the proposed procedure, the exchanging of capricious statistics is done with the illogical numbers stimulated by Laplace distribution to enlarge the assistance of the probability of formation stage in the exploration zone.

In the projected chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm, the transaction of erratic figures is completed with the irrational digits enthused by Laplace distribution to amplify the support of the probability of formation level inside the exploration zone. Chaotic sequences will augment the Exploration and Exploitation. Quantum features emulate the analogous performance with the certain stage as they route in a credible powdered of median.

Important aspects of the Ruddy turnstone optimization (RTO) algorithm, Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Chaotic based Ruddy turnstone optimization (CRTO) algorithm, Quantum based Ruddy turnstone Optimization (QRTO) Algorithm, Opposition based Ruddy turnstone optimization (ORTO) Algorithm and chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm

In Ruddy turnstone optimization algorithm the constellation of Ruddy turnstone, which mobile from one place to alternate in the sequence of repositioning and the fresh investigation agent position is to evade the smash amongst their contiguous Ruddy turnstone.
In the proposed Extreme Learning Machine based Ruddy turnstone Optimization Algorithm, Ruddy turnstone Optimization approach enhances Extreme Learning Machine features to determine an optimal carcass of Extreme Learning Machine for enhanced standards.
In Chaotic based Ruddy turnstone optimization algorithm Exploration and Exploitation are augmented. In Quantum based Ruddy turnstone Optimization Algorithm, features emulate the analogous performance with the certain stage as they route in a credible powdered of median.
Opposition based Ruddy turnstone optimization Algorithm employs Laplace distribution to enhance the exploration skill. Then examining the prospect to widen the exploration, a new method endorses stimulating capricious statistics used in formation stage regulator factor in Ruddy turnstone Optimization Algorithm.
In the projected chaotic in-built Opposition based – Quantum Ruddy turnstone optimization algorithm, the transaction of erratic figures is completed with the irrational digits enthused by Laplace distribution to amplify the support of the probability of formation level inside the exploration zone.

Critical Outcome of the Work

Significant goals of the paper are Voltage constancy augmentation, voltage deviance minimization and Actual power loss lessening. Proposed Ruddy turnstone optimization (RTO) algorithm, Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Chaotic based Ruddy turnstone optimization (CRTO) algorithm, Quantum based Ruddy turnstone Optimization (QRTO) Algorithm, Opposition based Ruddy turnstone optimization (ORTO) Algorithm and chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm is corroborated in Garver’s 6-bus test system, IEEE 30, 57, 118, 300, 354 bus test systems and Practical system - WDN 220 KV (Unified Egyptian Transmission Network (UETN)). Loss lessening, power divergence curtailing, and power fidelity augmentation has been attained.

Problem Formulation

Loss minimization [13, 52,53,54] is demarcated by

$$\mathit{\operatorname{Min}}\ \overset{\sim }{F}\left(\overline{g},\overline{h}\right)$$

(1)

$$M\left(\overline{g},\overline{h}\right)=0$$

(2)

$$N\left(\overline{g},\overline{h}\right)=0$$

(3)

$$g=\left[{VLG}_1,..,{VLG}_{Ng};{QC}_1,..,{QC}_{Nc};{T}_1,..,{T}_{N_T}\right]$$

(4)

$$h=\left[{PG}_{slack};{VL}_1,..,{VL}_{N_{Load}};{QG}_1,..,{QG}_{Ng};{SL}_1,..,{SL}_{N_T}\right]$$

(5)

$${F}_1={P}_{Min}=\mathit{\operatorname{Min}}\left[{\sum}_m^{NTL}{G}_m\left[{V}_i^2+{V}_j^2-2\ast {V}_i{V}_j\mathit{\cos}{\varnothing}_{ij}\right]\right]$$

(6)

$${F}_2=\mathit{\operatorname{Min}}\left[\sum\nolimits_{i=1}^{N_{LB}}{\left|{V}_{Lk}-{V}_{Lk}^{desired}\right|}^2+\sum\nolimits_{i=1}^{Ng}{\left|{Q}_{GK}-{Q}_{KG}^{Lim}\right|}^2\right]$$

(7)

$${F}_3=\mathit{\operatorname{Minimize}}\ {L}_{MaxImum}$$

(8)

$${L}_{Max}=\mathit{\operatorname{Max}}\left[{L}_j\right];j=1;{N}_{LB}$$

(9)

And

$$\left\{\begin{array}{c}{L}_j=1-\sum_{i=1}^{NPV}{F}_{ji}\frac{V_i}{V_j}\\ {}{F}_{ji}=-{\left[{Y}_1\right]}^1\left[{Y}_2\right]\end{array}\right.$$

(10)

$${L}_{Max}=\mathit{\operatorname{Max}}\left[1-{\left[{Y}_1\right]}^{-1}\left[{Y}_2\right]\times \frac{V_i}{V_j}\right]$$

(11)

Parity constraints

$$0=PG_i-PD_i-V_i{\textstyle\sum_{j\in N_B}}V_j\left[G_{ijcos}\left[\varnothing_i-\varnothing_j\right]+B_{ijsin}\left[\varnothing_i-\varnothing_j\right]\right]$$

(12)

$$0=QG_i-QD_i-V_i{\textstyle\sum_{j\in N_B}}V_j\left[G_{ijsin}\left[\varnothing_i-\varnothing_j\right]+B_{ijcos}\left[\varnothing_i-\varnothing_j\right]\right]$$

(13)

Disparity constraints

$${\mathrm{P}}_{\mathrm{gsl}}^{\mathrm{min}}\le {\mathrm{P}}_{\mathrm{gsl}}\le {\mathrm{P}}_{\mathrm{gsl}}^{\mathrm{max}}$$

(14)

$${\mathrm{Q}}_{\mathrm{g}\mathrm{i}}^{\mathrm{min}}\le {\mathrm{Q}}_{\mathrm{g}\mathrm{i}}\le {\mathrm{Q}}_{\mathrm{g}\mathrm{i}}^{\mathrm{max}},\mathrm{i}\in {\mathrm{N}}_{\mathrm{g}}$$

(15)

$${\mathrm{VL}}_{\mathrm{i}}^{\mathrm{min}}\le {\mathrm{VL}}_{\mathrm{i}}\le {\mathrm{VL}}_{\mathrm{i}}^{\mathrm{max}},\mathrm{i}\in \mathrm{NL}$$

(16)

$${\mathrm{T}}_{\mathrm{i}}^{\mathrm{min}}\le {\mathrm{T}}_{\mathrm{i}}\le {\mathrm{T}}_{\mathrm{i}}^{\mathrm{max}},\mathrm{i}\in {\mathrm{N}}_{\mathrm{T}}$$

(17)

$${\mathrm{Q}}_{\mathrm{c}}^{\mathrm{min}}\le {\mathrm{Q}}_{\mathrm{c}}\le {\mathrm{Q}}_{\mathrm{C}}^{\mathrm{max}},\mathrm{i}\in {\mathrm{N}}_{\mathrm{C}}$$

(18)

$$\left|{SL}_i\right|\le {S}_{L_i}^{max},\mathrm{i}\in {\mathrm{N}}_{\mathrm{TL}}$$

(19)

$${\mathrm{VG}}_{\mathrm{i}}^{\mathrm{min}}\le {\mathrm{VG}}_{\mathrm{i}}\le {\mathrm{VG}}_{\mathrm{i}}^{\mathrm{max}},\mathrm{i}\in {\mathrm{N}}_{\mathrm{g}}$$

(20)

$$Multi\ objective\ fitness\ (MOF)={F}_1+{r}_i{F}_2+u{F}_3={F}_1+\left[\sum\nolimits_{i=1}^{NL}{x}_v{\left[{VL}_i-{VL}_i^{min}\right]}^2+{\sum}_{i=1}^{NG}{r}_g{\left[{QG}_i-{QG}_i^{min}\right]}^2\right]+{r}_f{F}_3$$

(21)

$${VL}_i^{min imum}=\left\{\begin{array}{c}{VL}_i^{max},{VL}_i>{VL}_i^{max}\\ {}{VL}_i^{min},{VL}_i<{VL}_i^{min}\end{array}\right.$$

(22)

$${QG}_i^{min imum}=\left\{\begin{array}{c}{QG}_i^{max},{QG}_i>{QG}_i^{max}\\ {}{QG}_i^{min},{QG}_i<{QG}_i^{min}\end{array}\right.$$

(23)

Ruddy Turnstone Optimization Algorithm

In this paper Ruddy turnstone Optimization (RTO) Algorithm is applied to solve the loss lessening problem. The leading stimulus is in the sculpting of Repositioning –Peripatetic and argumentative actions of Ruddy turnstone. Ruddy turnstone will guzzle wiretaps, young insect and it subsists in bundling style. The constellation of Ruddy turnstone, which mobile from one place to alternate in the sequence of repositioning and the fresh investigation agent position is to evade the smash amongst their contiguous Ruddy turnstone.

$$\overrightarrow{Rt_S}={Rt}_M+\overrightarrow{Q_P}(t)$$

(24)

$${\displaystyle \begin{array}{c}\begin{array}{c} where\overrightarrow{Rt_S}\ is\ position\ without\ any\ \mathrm{Hindrance}\ \mathrm{between}\ \mathrm{each}\ \mathrm{other}\\ {}{Rt}_M\ is\ \mathrm{Mobility}\ \mathrm{of}\ \mathrm{the}\ \mathrm{Ruddy}\ \mathrm{turnstone}\ \end{array}\\ {}\overrightarrow{Q_P}(t) is\ present\ location\ and\ t\ indiate\ the\ present\ iteraion\ \\ {}{Rt}_M={Rt}_F-\left(t\times \frac{Rt_F}{\mathit{\max}. iter}\right)\ \end{array}}$$

(25)

$${\displaystyle \begin{array}{c} where\ t=0,1,2,3,..,\mathit{\max}. iter\\ {}{Rt}_F\ is\ the\ regulating\ frequency\\ {} when\ {Rt}_M=2\ then\ {Rt}_F\kern0.5em gradually\ decrease\ 2\ to\ 0\kern0.75em \end{array}}$$

Examination mediator’s passage in the direction of the dominant representative,

$$\overrightarrow{L_S}={Rt}_R\times \left(\overrightarrow{Q_{Best}}(t)-\overrightarrow{Q_P}(t)\right)$$

(26)

$${Rt}_R=0.5\times R$$

(27)

$${\displaystyle \begin{array}{c} where\ {Rt}_R\ is\ random\ parameter\ for\ controlling\ the\ exploration\ \\ {}\overrightarrow{Q_{Best}}(t)\ specify\ the\ best\ position\ \\ {}\overrightarrow{L_S}\ is\ location\ of\ the\ examination\ agent\end{array}}$$

Rendering to premium position, Ruddy turnstone will appraise its location,

$$\overrightarrow{M_S}=\overrightarrow{Rt_S}+\overrightarrow{L_S}$$

(28)

$$where\ \overrightarrow{M_S}\ indicate\ the\ space\ between\ examination\ and\ excellent\ agent$$

In the course of drive the Ruddy turnstone will execute Whorl action and the cantankerous actions of Ruddy turnstone is scientifically demarcated as,

$${U}^{\prime }=B\times \sin (t)$$

(29)

$${V}^{\prime }=B\times \cos (t)$$

(30)

$${W}^{\prime }=B\times (t)$$

(31)

$${\displaystyle \begin{array}{c} where\ b\ specify\ the\ radius\ of\ Whorl\ action\\ {}B=x\times {e}^{ny}\end{array}}$$

(32)

$${\displaystyle \begin{array}{c} where\ x\ and\ y\ are\ the\ constants\ to\ describe\ the\ \mathrm{Whorl}\ \mathrm{action}\\ {}n\in \left[0\le n\le 2\pi \right]\\ {}x\ and\ y=1\end{array}}$$

Modernizing the Location of the Exploration agents is accomplished by,

$$\overrightarrow{Q_P}(t)=\left(\overrightarrow{Rt_S}\times \left({U}^{\prime }+{V}^{\prime }+{W}^{\prime}\right)\times \overrightarrow{Q_{Best}}(t)\right)$$

(34)

Figure 1 shows the schematic diagram of Ruddy turnstone optimization (RTO) algorithm.

Extreme Learning Machine Based Ruddy Turnstone Optimization Algorithm

In the proposed Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Ruddy turnstone Optimization Algorithm approach enhances Extreme Learning Machine features to determine an optimal skeleton of Extreme Learning Machine for enhanced canons. ELM is applied and learning speed of feed-forward neural networks is composed of input, hidden and output layer [17,18,19].

The correlating neurons weight matrix of input to hidden layer is defined as,

$$Weight\ (Wht)=\left[\begin{array}{c}{wht}_1^T\\ {}{wht}_2^t\\ {};\\ {}{wht}_l^T\end{array}\right]=\left[\begin{array}{ccc}{wht}_{11}& \cdots & {wht}_{1n}\\ {}\vdots & \ddots & \vdots \\ {}{wht}_{L1}& \cdots & {wht}_{Ln}\end{array}\right]$$

(35)

$$\left( nwm.\beta \right)=\left[\begin{array}{c} nwm.{\beta}_1^T\\ {} nwm.{\beta}_2^t\\ {};\\ {} nwm.{\beta}_l^T\end{array}\right]=\left[\begin{array}{ccc} nwm.{\beta}_{11}& \cdots & nwm.{\beta}_{1n}\\ {}\vdots & \ddots & \vdots \\ {} nwm.{\beta}_{L1}& \cdots & nwm.{\beta}_{Ln}\end{array}\right]$$

(36)

$$\mathrm{Neurons}\ \mathrm{hidden}\ \mathrm{layer}\ \mathrm{bias}\ \mathrm{vector}\ \left(\mathrm{bsv}\right)={\left[\begin{array}{c}{bsv}_1\\ {}{bsv}_2\\ {}:\\ {}{bsv}_L\end{array}\right]}_{L\times 1}$$

(37)

For N impulsive e (B_i, F_i); F_i = [F_i1, F_i2, .., F_idn]^E ∈ MN^dn, C_i = [C_i1, C_i2, .., C_idn]^E ∈ MN^dn,

$$(C)=\left[\begin{array}{c}{C}_1^T\\ {}{C}_2^t\\ {};\\ {}{C}_l^T\end{array}\right]=\left[\begin{array}{ccc}{C}_{11}& \cdots & {C}_{1n}\\ {}\vdots & \ddots & \vdots \\ {}{C}_{L1}& \cdots & {C}_{Ln}\end{array}\right]$$

(38)

$$\sum\nolimits_{i=1}^N nwm.{\beta}_i\cdot k\left({\omega}_i{F}_j+{a}_i\right)={C}_j,j=1,2,3,..,N$$

(39)

$$(O)\cdot \left( nwm.\beta \right)=C$$

(40)

$$O\left({F}_1,..{F}_L;{\omega}_1,..,{\omega}_L;{a}_1,..,{a}_l\right)=\left[\begin{array}{ccc}k\left({\omega}_1{F}_1+{a}_1\right)& \cdots & k\left({\omega}_L{F}_1+{a}_L\right)\\ {}\vdots & \ddots & \vdots \\ {}k\left({\omega}_1{F}_N+{a}_1\right)& \cdots & k\left({\omega}_L{F}_N+{a}_L\right)\end{array}\right]$$

(41)

$$nwm.\beta ={O}^{-1}\cdot C$$

(42)

In the proposed Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm, Ruddy turnstone Optimization approach enhances Extreme Learning Machine features to determine an optimal carcass of Extreme Learning Machine for enhanced standards. In principally all elements don’t own any info about the explication area. In preliminary phases of iteration, the Ruddy turnstone contestants are multifarious in milieu and exponential spare generates boundless unpremeditated amounts which contribute the rudiments to lodging the entire explication zone. Disparately, all over end stage of iterations, rudiments are surrounded by Ruddy turnstone contestants and all an optimal situation with similar pattern. Figure 2 shows the schematic diagram of Extreme Learning Machine based Ruddy turnstone Optimization (ELMRTO) Algorithm.

Chaotic Based Ruddy Turnstone Optimization Algorithm

Chaotic sequences are combined into Ruddy turnstone Optimization and it termed as - Chaotic based Ruddy turnstone optimization (CRTO) algorithm. This integration will augment the Exploration and Exploitation. Tinkerbell chaotic map [60, 61] engendering standards are implemented. Figure 3 shows the schematic diagram of Chaotic based Ruddy turnstone optimization (CRTO) algorithm.

$${u}_{t+1}={u}_t^2-{v}_t^2+a\cdot {u}_t+b\cdot {v}_t$$

(43)

$${v}_{t+1}=2{u}_t{v}_t+c\cdot {u}_t+d\cdot {v}_t$$

(44)

$${\displaystyle \begin{array}{c}\begin{array}{c}\begin{array}{c} where\ a,b,c\ and\ d\ are\ non- zero\ parameters\ \\ {}a=0.900\\ {}b=-0.600\end{array}\\ {}c=2.000\\ {}d=0.500\end{array}\\ {} At\ primary\ stage\kern0.5em {u}_o and\ {v}_o=0.10\end{array}}$$

The functional value by linear scaling in Tinkerbell chaotic map [60, 61] is demarcated as,

$${u}_{t+1}^{\ast }={u}_{t+1}- minimum(u)/ maximum(u)- minimum(u)$$

(45)

Quantum Based Ruddy Turnstone Optimization Algorithm

Quantum mechanics [71,72,73,74,75,76] has been combined with Ruddy turnstone Optimization Algorithm and it titled as Quantum based Ruddy turnstone Optimization (QRTO) Algorithm. In quantum method, features emulate the analogous performance with the certain stage as they route in a credible powdered of median. The wave utility in the Quantum mechanics [71,72,73,74,75,76] is demarcated as,

$${\left|\varPsi \right|}^2\cdot dx\cdot dy\cdot dz= Quantum\cdot dx\cdot dy\cdot dz$$

(46)

$$where\ \varPsi\ indicate\ probability\ density\ functional\ value$$

The time contingent Schrodinger equation [71,72,73,74,75,76] is smeared to evaluate the wave utility is demarcated as,

(47)

$$where\ Hor\ specify\ the\ \mathrm{Hamiltonian}\ \mathrm{operator}$$

$$Hor=-{h}^2/2m\cdot {\Delta }^2+V(x)$$

(48)

In the quantum pattern ∆Fit consecutively achieve as particle and it successively passages in delta potential in the direction of center. Figure 4 shows the schematic diagram of Quantum based Ruddy turnstone Optimization (QRTO) Algorithm.

$$\mathrm{Schrodinger}\left(\mathrm{Time}-\mathrm{independent}\right) is\ \frac{d^2\varPsi }{d{z}^2}+\frac{2m}{h^2}\left[G+\gamma \delta (z)\right]\varPsi =0$$

(49)

$$\varPsi (z)=\frac{1}{\sqrt{L}}{e}^{-\frac{\left|z\right|}{L}}$$

(50)

$$Quantum(z)={\left|\varPsi (z)\right|}^2=\frac{1}{\sqrt{L}}{e}^{-\frac{\left|z\right|}{L}}$$

(51)

$$z=\pm \frac{L}{2} In\left(1/g\right)$$

(52)

Opposition Based Ruddy Turnstone Optimization Algorithm

Ruddy turnstone Optimization Algorithm, even though the initiative of contestant explications slants to touch an optimal solution, yet several are get entombed and not adept of emotive in the route of the dominant solution. It significances to snare in local optima and it accordingly enforces into primary and slow convergence. Subsequently Opposition based Ruddy turnstone optimization (ORTO) Algorithm employs Laplace distribution to enhance the exploration skill. Then examining the prospect to widen the exploration, a new method endorses stimulating capricious statistics used in formation stage regulator factor in Ruddy turnstone Optimization Algorithm. In the proposed procedure, the exchanging of capricious statistics is done with the illogical numbers stimulated by Laplace distribution [62,63,64] to enlarge the assistance of the probability of formation stage in the exploration zone.

$$function(v)=\left\{\begin{array}{c}\frac{1}{2}\ \mathit{\exp}\left(-\left|v-c\right|/d\right),b\le c\\ {}1-\frac{1}{2}\ \mathit{\exp}\left(-\left|v-c\right|/d\right),b>c\end{array}\right.$$

(53)

The probability propagation function of Laplace dispersal is,

$$function\left(v;c,d\right)=1/2v\ \mathit{\exp}\left(-\left|v-c\right|/d\right),-\infty <c<\infty$$

(54)

$$where\ c\in \left(-\infty, \infty \right)$$

Opposition based learning (OBL) is one of the influential approaches to improve the convergence quickness of procedures [62,63,64]. The flourishing use of the Opposition based learning includes evaluation of opposite populace and dominant populace in the analogous generation to regulate the superior contestant explication. The perception of opposite number requirements is to be delineated to explicate Opposition based learning. Figure 5 shows the schematic diagram of Opposition based Ruddy turnstone optimization (ORTO) Algorithm.

Let O (Z ∈ [c, d]) be a palpable figure and the O^o (opposite figure) can be delineated as,

$${O}^o=c+d-U$$

(55)

In the exploration area it has been protracted as,

$${O}_i^o={c}_i+{d}_i-{U}_i$$

(56)

$${\displaystyle \begin{array}{c}\mathrm{Where}\ \left({O}_1,{O}_2,..{O}_d\right)\ indicate\ dimensional\ exploration\ zone\\ {}{O}_i\in \left[{c}_i,{d}_i\right],i\to \left\{1,2,3,..d\right\}\end{array}}$$

The perception of Opposition based learning is employed in the initialization procedure and in iterations by means of the cohort vaulting level.

a.
Min f
b.
if f (O^∗) ≤ f (O); then O = O^∗
c.
Or else
d.
Sustain with O in successive generations

An opposite component is assimilated after streamlining and produced the distinguished component

$${Rt}_i(iter)=\left({LB}_i+{UB}_i-{Rt}_e(iter)\right)$$

(57)

$$where\ LB, UB\ are\ lower\ and\ upper\ bound$$

At that moment the Flexible speeding up factor (F_s) balance the exploration and exploitation and scientifically demarcated as,

$${F}_s={F}_{max}-{iter}_p\cdot {F}_{max}-{F}_{min}/{iter}_{max}$$

(58)

The Opposition based learning method engaged round the distinguished component and it demarcated as,

$${Rt}_i(iter)={F}_s\ast \left({LB}_i+{UB}_i-{Rt}_e(iter)\right)$$

(59)

Chaotic in-Built Opposition Based – Quantum Ruddy Turnstone Optimization Algorithm

In the projected chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm, the transaction of erratic figures is completed with the irrational digits enthused by Laplace distribution to amplify the support of the probability of formation level inside the exploration zone. Chaotic sequences will augment the Exploration and Exploitation. Quantum features emulate the analogous performance with the certain stage as they route in a credible powdered of median. Figure 6 shows the schematic diagram of chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm.

Computational Complexity

The off line incorrectness is computed as,

$$off- line\ \mathrm{erroneousness}=\frac{1}{\mathit{\max}\ iter}\sum_{t=1}^{\max iter} Present\ {\mathrm{erroneousness}}_t$$

Categories of computation complication of O(nlogn) and O(n²) in the finest and least case correspondingly. The generalized total calculation of complexity is apportioned as follows:

$${\displaystyle \begin{array}{c}\begin{array}{c}O\left(\mathrm{Q}\ \right)\\ {}O(nlogn)=O\left(T\left(O(s)+O\left(p\ \right)\right)\right)\end{array}\\ {}O\left({n}^2\right)=O\left(t\left({n}^2+n\times d\right)\right)=O\left({tn}^2+ tnd\right)\\ {}O\left(\mathit{\operatorname{Max}}\ iter\ast {N}^2\ast D\right)\ast O\left( obj. fun\right)+O\left(\mathit{\operatorname{Max}}\ iter\ast t\ast N\right)+O\left(\mathit{\operatorname{Max}}\ iter\ast N\ast D\right)\end{array}}$$

Simulation Results

Proposed Extreme Learning Machine based Ruddy turnstone optimization (RTO) algorithm, Ruddy turnstone Optimization (ELMRTO) Algorithm, Chaotic based Ruddy turnstone optimization (CRTO) algorithm, Quantum based Ruddy turnstone Optimization (QRTO) Algorithm, Opposition based Ruddy turnstone optimization (ORTO) Algorithm and chaotic in-built Opposition based – Quantum Ruddy turnstone optimization (COQRTO) algorithm are corroborated in Garver’s 6-bus test system, IEEE 30, 57, 118, 300, 354 bus test systems and Practical system - WDN 220 KV (Unified Egyptian Transmission Network (UETN)). At first proposed algorithm is reviewed in Garver’s 6-bus test system [65]. It has six buses and lines, three generators and five loads [65] at buses. Table 1 shows the loss appraisal and Table 2 shows the voltage aberration evaluation. Figures 7 and 8 give the graphical appraisal between the methods.

Table 1 Loss appraisal

Novel Extreme Learning Machine and Chaotic in-Built Opposition Based – Quantum Ruddy Turnstone Optimization Algorithms for Real Power Loss Dwindling and Voltage Consistency Enhancement

Abstract

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Introduction

Related Study

Proposed Methodology

Critical Outcome of the Work

Problem Formulation

Ruddy Turnstone Optimization Algorithm

Extreme Learning Machine Based Ruddy Turnstone Optimization Algorithm

Chaotic Based Ruddy Turnstone Optimization Algorithm

Quantum Based Ruddy Turnstone Optimization Algorithm

Opposition Based Ruddy Turnstone Optimization Algorithm

Chaotic in-Built Opposition Based – Quantum Ruddy Turnstone Optimization Algorithm

Computational Complexity

Simulation Results

Conclusion

Scope of Future Work

Data Availability

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