1 Introduction

In electrical power supply systems, frequency is a significant, as well as, fundamental parameter that specifies stability between power generation and power consumption [1]. Thus, the frequency component can be considered as a functional gauge to identify anomalous operating conditions [2]. In order to ensure regulated power supply to the clients and the utilities, it is mandatory requirement to scrutinize power quality of electrical grid through parameter estimation of power signal i.e., amplitudes, phases and frequencies [3, 4]. Research community has shown considerable interest in parameter estimation of power signal in power planning and distribution systems, for instance, Xu and Ding [5] presented the stochastic gradient (SG) and least squares (LS) procedures, Xu et al. [6] developed the hierarchical parameter estimation method, Li et al. [7] applied the reconstructing time sample techniques, Cao and Liu [8] gave the concept of the hierarchical identification approach, Phan et al. [3] described dedicated state space method, Chen et al. [9, 10] provided fast Fourier transform methods and Chaudhary et al. [11,12,13] provided the scheme of fractional adaptive filtering. The parameters of modeling power signals are identified by variety of the procedures introduced recently [14,15,16,17,18,19,20]. These are all deterministic procedures with their own perks, benefits and limitations while stochastic computing paradigm based on bioinspired heuristic looks promising to be explored exhaustively in the domain of power signal modeling.

The backtracking search optimization algorithm (BSA)-based evolutionary computing solvers have been explored to solving many problems arising in engineering and technological domains. A few latest applications of these schemes are in fluid dynamics [21], nanotechnology [22], wireless networks [23], system identification [24], power electronics [25], control [26], electric machines [27], economic load dispatch problems [28], nonlinear electric circuits [29], signal processing [30], biomedical [31], bioinformatics [32], finance [33]. All these contributions motivate authors to explore in meta-heuristic paradigm of BSA for accurate, reliable and robust system identification problems arising in power signal models. The aim of this research study is to exploit the well-known strength of BSA for parameter estimation of power signal systems. The prominent features of the proposed scheme are:

  • A novel application of evolutionary computational heuristics through BSA is presented for effective, viable and reliable estimation of parameters in power signal modeling problems.

  • Approximation theory is exploited for formulation of fitness function in terms of mean squared errors of actual and approximated parameters as well as responses for power signal models.

  • Variants of BSA are implemented for parameter estimation of power signals with different degrees of freedom based on amplitude, frequency and phase for number of noise variances.

  • Performance verification is ascertained through statistical results in terms of mean weight deviation, root mean square error and Thiel inequality coefficient-based evaluations metrics. Further, significance of the model is evaluated on the basis of ANOVA test.

Rest of the paper is organized as follows: Sect. 2 presents the necessary details of power signal modeling problem, designed methodology for parameter estimation is provided in Sect. 3, simulation of experimentation with interpretations is given in Sect. 4, while the conclusions and future recommendations are listed in Sect. 5.

2 System model: power signals

A distorted electric signal s(t) from an AC power system can be expressed in the form of Fourier series [6, 34]:

$$s(t) = \sum\limits_{k = 1}^{K} {\left( {c_{k} \cos \,k\omega t + d_{k} sin\,k\omega t} \right)} + v(t)$$
(1)

here, c and d are the Fourier coefficients, K is the harmonics index and ω is the fundamental frequency of the AC system.

The generic description of alternating current electrical signal can be derived from Eq. (1) and given below in terms of amplitudes, frequencies and phases as [8]:

$$s(t) = \sum\limits_{i = 1}^{k} {r_{i} \sin \left( {x_{i} (t)} \right)} + v(t),$$
(2)

for xi(t) = ωit + φi.

The amplitudes, the frequencies and phases (in radians) are represented as \(\varvec{r} = [r_{1} ,r_{2} , \ldots ,r_{k} ]\), \(\varvec{\omega}= [\omega_{1} ,\omega_{2} , \ldots ,\omega_{k} ],\) and \(\varvec{\varphi } = [\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{k} ],\), respectively. For all non-zero magnitudes of \(\varphi\), the whole signal waveform appears to be shifted in time scale by \(\varphi /\omega\) seconds. A positive magnitude of \(\varphi\) indicates an advance while a negative value is for a delay in the electric signal. Eqs. (1)–(2) have been reported from many studies of electrical and electronic engineering [35,36,37,38] and reference therein.

In this experimental procedure, \(t_{n} = nh\) is the sampling whose sampling period is h. The observed data are \(\left\{ {t_{k} ,s(t_{k} )} \right\}\). Let \(r_{n} = s(t_{n} )\) for inference, then, the discretized electrical signal on the basis of sinusoidal function is formalized as:

$$s_{n} = \sum\limits_{k = 1}^{K} {\left( {c_{k} \cos \,k\omega t_{n} + d_{k} sin\,k\omega t_{n} } \right)} + \nu_{n} \,\;\;\,{\text{for}}\,\,\,n = 1,2, \ldots N.$$
(3)

The expression for the power signal (2) in discrete form as:

$$s_{n} = \sum\limits_{i = 1}^{k} {r_{i} \sin (\omega_{i} t_{n} + \varphi_{i} )} + \nu_{n} \,\,\;\;{\text{for}}\,\,\,n = 1,2, \ldots N.$$
(4)

Amplitudes, frequencies and phases are components of the electrical signals described in Eqs. (1)–(4). In addition to individual any arbitrary combination of these parameters based on can be formulized for the parameter estimation problems. The unknown fundamental components of power signal to be estimated are given as:

$$\varvec{\vartheta}_{cd} = \varvec{cd} = \left[ {c_{1} ,c_{2} , \ldots ,c_{k} ,d_{1} ,d_{2} , \ldots ,d_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{k}$$
(5)
$$\varvec{\vartheta}_{r} = \varvec{r} = \left[ {r_{1} ,r_{2} , \ldots ,r_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{k}$$
(6)
$$\varvec{\vartheta}_{\omega } = \omega = \left[ {\omega_{1} ,\omega_{2} , \ldots ,\omega_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{k}$$
(7)
$$\varvec{\vartheta}_{\varphi } = \varphi = \left[ {\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{k}$$
(8)
$$\varvec{\vartheta}_{r,\,\omega } = [\varvec{r},\varvec{\omega}] = \left[ {r_{1} ,r_{2} , \ldots ,r_{k} ,\;\omega_{1} ,\omega_{2} , \ldots ,\omega_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{2k}$$
(9)
$$\varvec{\vartheta}_{r,\,\varphi } = [\varvec{r},\varvec{\varphi }] = \left[ {r_{1} ,r_{2} , \ldots ,r_{k} ,\;\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{2k}$$
(10)
$$\varvec{\theta}_{\omega ,\,\varphi } = [\varvec{\omega},\varvec{\varphi }] = \left[ {\omega_{1} ,\omega_{2} , \ldots ,\omega_{k} ,\;\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{2k}$$
(11)
$$\begin{aligned}\varvec{\vartheta}_{r,\,\omega ,\,\varphi } & = [\varvec{r},\varvec{\omega},\varvec{\varphi }] \\ & = \left[ {r_{1} ,r_{2} , \ldots ,r_{k} ,\;\omega_{1} ,\omega_{2} , \ldots ,\omega_{k} ,\;\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{k} } \right]^{\text{T}} \in {\mathbb{R}}^{3k} . \\ \end{aligned}$$
(12)

The individual parameter estimation problems are given in Eqs. (6)–(8), while parametric Eqs. (9)–(12) indicate the integrated parameter estimation systems.

3 Proposed methodology

In this section, proposed methodology for parameter estimation of signal modeling problem is presented in two steps; fitness function formulation and learning procedure by exploitation of meta-heuristics of BSA. The framework of the proposed methodology is presented graphically in Fig. 1.

Fig. 1
figure 1

Graphical flow diagram of parameter estimation of signal modeling problem

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3.1 Construction of fitness function

In the first step, the fitness/merit function e is constructed in mean square sense as:

$$e = e_{1} + e_{2} ,$$
(13)

where \(e_{1}\) is the difference actual \(s\) and estimated \(\hat{s}\) response and is expressed as:

$$e_{1} = \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {s_{n} - \hat{s}_{n} } \right)^{2} } ,$$
(14)

while \(e_{2}\) is an error term associated with parameter vector,

$$e_{2} = \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\vartheta_{n} - \hat{\vartheta }_{n} } \right)^{2} } .$$
(15)

For example, the fitness function formulation for power signal model represented in Eq. (3) with decision variables as given in Eq. (5) in case of e1 is given as:

$$e_{1} = \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\begin{array}{*{20}l} {\sum\limits_{k = 1}^{K} {\left( {c_{k} \cos \,k\omega t_{n} + d_{k} sin\,k\omega t_{n} } \right)} + \nu_{n} } \hfill \\ {\quad - \sum\limits_{k = 1}^{K} {\left( {\hat{c}_{k} \cos \,k\omega t_{n} + \hat{d}_{k} sin\,k\omega t_{n} } \right)} } \hfill \\ \end{array} } \right)^{2} } .$$
(16)

Accordingly, \(e_{2}\) is error function associated with the parameter vector in case of Eq. (5) is given as:

$$e_{2} = \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\vartheta_{{cd_{n} }} - \hat{\vartheta }_{{cd_{n} }} } \right)^{2} } ,$$
(17)

using Eq. (5), we have

$$e_{2} = \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\begin{array}{*{20}l} {\left( {c_{1} ,c_{2} , \ldots ,c_{k} ,d_{1} ,d_{2} , \ldots ,d_{k} } \right)_{n} } \hfill \\ {\quad - \left( {\hat{c}_{1} ,\hat{c}_{2} , \ldots ,\hat{c}_{k} ,\hat{d}_{1} ,\hat{d}_{2} , \ldots ,\hat{d}_{k} } \right)_{n} } \hfill \\ \end{array} } \right)^{2} } .$$
(18)

Now, the fitness function e as given in Eq. (13) is written as:

$$e = \left[ {\begin{array}{*{20}l} {\frac{1}{N}\sum\limits_{k = 1}^{K} {\left( {\begin{array}{*{20}l} {\sum\limits_{k = 1}^{K} {\left( {c_{k} \cos \,k\omega t_{n} + d_{k} sin\,k\omega t_{n} } \right)} } \hfill \\ {\quad + \nu_{n} - \sum\limits_{k = 1}^{K} {\left( {\hat{c}_{k} \cos \,k\omega t_{n} + \hat{d}_{k} sin\,k\omega t_{n} } \right)} } \hfill \\ \end{array} } \right)^{2} } } \hfill \\ {\quad + \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\begin{array}{*{20}l} {\left( {c_{1} ,c_{2} , \ldots ,c_{k} ,d_{1} ,d_{2} , \ldots ,d_{k} } \right)_{n} } \hfill \\ {\quad - \left( {\hat{c}_{1} ,\hat{c}_{2} , \ldots ,\hat{c}_{k} ,\hat{d}_{1} ,\hat{d}_{2} , \ldots ,\hat{d}_{k} } \right)_{n} } \hfill \\ \end{array} } \right)^{2} } } \hfill \\ \end{array} } \right].$$
(19)

Similarly, the fitness function formulation for power signal model represented in Eq. (4) with decision variables as given in Eq. (11) is given as:

$$e = \left[ {\begin{array}{*{20}l} {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\begin{array}{*{20}l} {\sum\limits_{i = 1}^{k} {r_{i} \sin (\omega_{i} t_{n} + \varphi_{i} )} + \nu_{n} } \hfill \\ {\quad - \sum\limits_{i = 1}^{k} {r_{i} \sin (\hat{\omega }_{i} t_{n} + \hat{\varphi }_{i} )} } \hfill \\ \end{array} } \right)^{2} } } \hfill \\ {\quad + \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\begin{array}{*{20}l} {\left( {\omega_{1} ,\omega_{2} , \ldots ,\omega_{k} ,\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{k} } \right)_{n} } \hfill \\ {\quad - \left( {\hat{\omega }_{1} ,\hat{\omega }_{2} , \ldots ,\hat{\omega }_{k} ,\hat{\varphi }_{1} ,\hat{\varphi }_{2} , \ldots ,\hat{\varphi }_{k} } \right)_{n} } \hfill \\ \end{array} } \right)^{2} } } \hfill \\ \end{array} } \right].$$
(20)

Accordingly, the fitness function formulation for power signal model represented in Eq. (4) with decision variables as given in Eq. (11) is given as:

$$e = \left[ {\begin{array}{*{20}l} {\frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\begin{array}{*{20}l} {\sum\limits_{i = 1}^{k} {r_{i} \sin (\omega_{i} t_{n} + \varphi_{i} )} + \nu_{n} } \hfill \\ {\quad - \sum\limits_{i = 1}^{k} {\hat{r}_{i} \sin (\hat{\omega }_{i} t_{n} + \hat{\varphi }_{i} )} } \hfill \\ \end{array} } \right)^{2} } } \hfill \\ {\quad + \frac{1}{N}\sum\limits_{n = 1}^{N} {\left( {\begin{array}{*{20}l} {\left( {r_{1} ,r_{2} , \ldots ,r_{k} ,\omega_{1} ,\omega_{2} , \ldots ,\omega_{k} ,\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{k} } \right)_{n} } \hfill \\ {\quad - \left( {\hat{r}_{1} ,\hat{r}_{2} , \ldots ,\hat{r}_{k} ,\hat{\omega }_{1} ,\hat{\omega }_{2} , \ldots ,\hat{\omega }_{k} ,\hat{\varphi }_{1} ,\hat{\varphi }_{2} , \ldots ,\hat{\varphi }_{k} } \right)_{n} } \hfill \\ \end{array} } \right)^{2} } } \hfill \\ \end{array} } \right].$$
(21)

On a similar pattern, the rest of the fitness functions are constructed for different power signal models.

Now, objective is optimization of the error functions given in Eqs. (19)–(21) in such a way that as e approaches zero, the adaptive parameter vectors of decision variable \(\hat{\varvec{\vartheta }}_{cd}\), \(\hat{\varvec{\vartheta }}_{\omega ,\varphi }\) and \(\hat{\varvec{\vartheta }}_{r,\omega ,\varphi }\) matches the desired variables \(\varvec{\vartheta}_{cd}\), \(\varvec{\vartheta}_{\omega ,\varphi }\) and \(\varvec{\vartheta}_{r,\omega ,\varphi }\) of the power signal models, respectively.

3.2 Learning method: variants of BSA

The second phase of designed methodology, we introduced BSA for optimization of decision variable of power signal model.

BSA is an initial population-based stochastic algorithm introduced by Civicioglu [39] in 2012 for the solution of constraint and unconstraint optimization problem. In BSA, trial population is generated using three basic recombination operators including selection, mutation and crossover. BSA employs random mutation process and non-uniform crossover strategy which is relatively complex than traditional ones. BSA belongs to the class of global optimization technique designed to solve high dimensional multimodal optimization problems having simple structure, single control parameter and additional benefit of possessing a memory. Few recent potential applications of BSA include beach realignment [40], solving constrained engineering problem [41], parameter estimation [42], economic load dispatch problem [43], wireless communication [44, 45], photovoltaic models [46] and nonlinear system identification [47]. Graphical flow chart describing the procedural steps of BSA is shown in Fig. 1, while necessary further detailed of these steps is provide in the pseudocode presented in Table 1.

Table 1 Pseudocode for BSA algorithm for optimization of periodic signal modeling
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The performance analysis for parameter estimation of signal modeling problem has also been carried out based on fitness function, normalizing error calculation, root mean squared error (RMSE) and Thiel’s inequality coefficient (TIC). The mathematical definitions of these performance indices can be seen in [47] for interested readers.

4 Numerical experimentation

Simulations are performed for three different examples of power signals parameter estimation problems through the evolutionary computing heuristics of BSA under varying noise scenarios. The variants of BSA are designed by means of different population size (PS) and generations (GENS) as tabulated in Table 2.

Table 2 Description of six BSA variants
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Example 1

In this case study, the power signal estimation problem with known amplitude while, unknown frequency and phase parameters is taken. The mathematical expressions for Example 1 are written as [8]:

$$\begin{aligned} s(t) & = r_{1} \sin (\omega_{1} t + \varphi_{1} ) + r_{2} \sin (\omega_{2} t + \varphi_{2} ) + r_{3} \sin (\omega_{3} t + \varphi_{3} ) \\\varvec{\vartheta}& = \left[ {\omega_{1} ,\omega_{2} ,\omega_{3} ,\varphi_{1} ,\varphi_{2} ,\varphi_{3} } \right]^{\text{T}} \\\varvec{\vartheta}& = [0.07, 0.1, 0.2 , 0.95, 0.8, 0.76]^{\text{T}} . \\ \end{aligned}$$
(22)

Example 2

The power signal modeling problem with unknown amplitude, frequency and phase in the parameter vector is taken. The mathematical expressions for Example 2 are given as [8]:

$$\begin{aligned} s(t) & = r_{1} \sin (\omega_{1} t + \varphi_{1} ) + r_{2} \sin (\omega_{2} t + \varphi_{2} ) + r_{3} \sin (\omega_{3} t + \varphi_{3} ) \\\varvec{\vartheta}& = \left[ {r_{1} ,r_{2} ,r_{3} ,\omega_{1} ,\omega_{2} ,\omega_{3} ,\varphi_{1} ,\varphi_{2} ,\varphi_{3} } \right]^{\text{T}} \\\varvec{\vartheta}& = [0.07, 0.1, 0.2 , 0.95, 0.8, 0.76]^{\text{T}} . \\ \end{aligned}$$
(23)

Example 3

The power signal modeling problem with unknown amplitude in the parameter vector is presented mathematically as [34]:

$$\begin{aligned} s(t) & = c_{1} \cos \,\omega t + d_{1} sin\,\omega t + c_{2} \cos \,2\omega t + d_{2} sin\,2\omega t \\\varvec{\vartheta}& = \left[ {c_{1} ,c_{2} ,d_{1} ,d_{2} } \right]^{\text{T}} \\\varvec{\vartheta}& = \left[ {300,\,40,\,20,\,10} \right]^{\text{T}} . \\ \end{aligned}$$
(24)

In the simulations, s(t) is taken as the input signal and v(t) represents noise signal having zero mean and three different noise levels i.e., no noise, 30 db and 70 db. The power signal modeling problem as described in Eqs. (22)–(24) is executed with the proposed scheme and the objective function is implemented for N = 20 snap shots. The meta-heuristic algorithm, BSA together with all its six variants is performed for 100 independent runs. The results of each BSA variant against the values of fitness function are shown graphically in Fig. 2, for all three signal models given in Examples 1–3 and noise levels. It is observed from the results presented in Fig. 2 that all the variants of BSA are convergent for all noise levels, but convergence of the variant-III of BSA is slightly better than all others. Comparison of the actual signals is also made with the approximated signal, and the resultant graphs are presented in Fig. 3 for all three examples. The actual and estimation decision variables are listed in Table 3 for Example 1 in case of all three noise levels, while these results for Examples 2 and 3 are provided in Table 4 for each noise variation. The estimated signals overlap the actual signals consistently, as well as, relatively small difference between actual and estimation parameters which prove the accuracy of the scheme. However, with the rise in the noise level, there is a decrease in the accuracy level is observed for all six variants of BSA.

Fig. 2
figure 2

Convergence graphs of BSA for parameter estimation of power signal modeling problem with noise variation scenarios

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Fig. 3
figure 3

Comparison plots of BSA for parameter estimation of power signal modeling problem with noise variation scenarios

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Table 3 Comparison of true parameters with the proposed results of parameter estimation of power signal models model Example 1 for all noise variances
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Table 4 Comparison of true parameters with the proposed results of parameter estimation of power signal models model Examples 2 and 3 for all noise variances
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In order to access the minute difference, the values of absoluter error (AE), i.e., difference between actual and estimated parameter, is calculated for each case. The values of AE are plotted in Fig. 4 for Example 1 in case of all six variants of BSA, while the results for Examples 2 and 3 are shown in Fig. 5. It is seen that the range of AE values lie around 10−02 to 10−04 for BSA-I while, for BSA-II, range is around 10−05 to 10−06 and similar trend for rest of the variants is observed.

Fig. 4
figure 4

Comparison of the accuracy of BSA variants for parameter estimation of power signal modeling problem with noise variation scenarios in case of Example 1

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Fig. 5
figure 5

Comparison of the accuracy of BSA variants for parameter estimation of power signal modeling problem with noise variation scenarios in case of Examples 2 and 3

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The performance indices magnitudes based on fitness e, normalized error δ, MAE, RMSE and TIC for best independent run of the scheme are listed in Table 5 for each noise variation in case of all six variants of BSA. Near-to-optimal magnitudes of all these metrics achieved for all three examples of power signal estimation problem which evidently demonstrate the accuracy of the proposed scheme. Additionally, the computational complexity measures in terms of mean execution time, iteration consumed and function counts (FC) during the whole optimization procedure are also tabulated in Table 5 for all three example for each scenario. The complexity analysis shows that BSA-III consumed more iterations and time than other variants but is comparatively accurate from rest of the methodologies.

Table 5 Performance comparison of all BSA variants through satistical and complexity operators
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The results of each variant of BSA against the fitness values are graphically presented in Fig. 4g–l for several independent runs, i.e., 100 trials, all three noise levels in case of Example 1, while these illustrations for Examples 2 and 3 are shown in Fig. 5g–l. All these graphs are given in sorted and zoomed plots for better assessment of the results. It is evident from the plots that all the variants of the BSA converge but accuracy degrades as noise increases.

Convergence analysis is performed to evaluate reliability of the scheme for attaining the various accuracy levels on the basis of fitness gauges, i.e., fitness ε ≤ 10−03, 10−04, 10−05 and 10−06. The results of percentage independent runs fulfilling these criterions are listed in Table 6 for each scenario of all three example of power signal estimation problem. It is seen that nearly 100% of the independent runs meet the primary level of the basic fitness measure and also even a few trials attained relatively tough criteria. The results are comparatively more accurate and convergent for BSA-IV and BSA-V for rest of the scheme.

Table 6 Convergence analysis of all examples using BSA variants
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The statistical performances in terms of MAE, RMSE and TIC metrics are also evaluated and results of these indices are shown in Figs. 6 and 7 for each scenario of Examples 1 and 2, respectively. In Fig. 6a–d, MAE magnitudes are plotted for 100 independent trials of all the variants of BSA algorithm on semi log scale. The results show that MAE are near 10−3 to 10−2, 10−4 to 10−2, 10−3 and 10−3 to 10−4 for BSA I, II, III and IV, respectively. Histogram plots are also shown in Fig. 6e–l for all six variants of BSAs. The smaller magnitudes of RMSE verify the accuracy of the designed methodology. In Fig. 6m, TIC magnitudes as stacked bar graph are shown for BSA-III and BSA-IV each noise variance-based scenario of Example 1. These results demonstrate that magnitudes of TIC lie around 10−5 to 10−2. Similarly plots for Examples 2 and 3 are given in Fig. 7a–d, e–l and m in case of MAE, RMSE and TIC values, respectively. The similar trend of results is seen as in case of Example 1.

Fig. 6
figure 6

Comparative study of BSA variants on the basis of different performance indices for parameter estimation of power signal modeling problem

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Fig. 7
figure 7

Comparative study of BSA variants on the basis of different performance indices for parameter estimation of power signal modeling problem

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Further evaluation about the performance of the BSA variants for periodic signals estimation is made using ANOVA test. The results are computed for all three examples and presented in Table 7 for SNR 70 dB. With the consideration of assumption of uniform variances, the null hypothesis of homogeneous variances at the significance level α = 0.05 is accepted, as the respective probability values i.e., p values, attained for absolute errors for Examples 1, 2 and 3 are 0.283, 0.811 and 0.828, respectively. The result of ANOVA established that the expected values show uniformity, and there is no strong evidence against the null hypothesis. Therefore, it is quite evident that all the means are equivalent.

Table 7 ANOVA test results for Examples 1, 2 and 3 of power signal model with 70 dB SNR
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5 Conclusions

Evolutionary computational paradigm through variants of BSA are exploited for effective, viable and reliable solution of parameter estimation of power signal models with various noise level-based scenarios. Comparative study of the designed algorithms for both noise less and noisy environment, i.e., no noise, SNR = 70 db and 30 db, depict the accuracy of all six variants of BSA technique. The study shows that the accuracy level decreases as noise increases for all variants of BSA; despite, all the results are quite precise. Analysis based on performance measures, i.e., normilizing error, AEs, MAD, RMSE and TIC metrics, demonstrated the efficacy of the proposed scheme. While, the complexity analysis in terms of time consumed, iterations executed and functions count show that the results of BSA-III and BSA-IV are relatively higher than the other variants of the BSA optimization technique; however, their relatively better accuracy for the rest of scheme overshadows this aspect. Power signal modeling with increase degrees of freedom slight deteriorates the performance of the proposed BSA in terms of accuracy and complexity.

One may apply the proposed methodology for solving real-world complex power and energy-related optimization problems [48,49,50,51,52]. Recently introduced meta-heuristic techniques such as firefly, particle swarm optimization and cuckoo search algorithm along with their fractional variants can be exploited to improve the accuracy and convergence in power signal estimation problems.