1 Introduction

Electrical power supply systems are designed to ensure standard power provision to the consumers and the utilities. The ideal sinusoidal behavior of an AC power voltage or current signals distorts due to the excessive employment of power electronics devices, power converters, solid-state power switching devices, renewable power grid integration and nonlinear loads in electrical grid stations [1,2,3,4,5]. The distorted signals deteriorate the rated voltage and current, which badly affect the electrical equipment operations [6]. Due to these reasons, it is essential to monitor power quality of electrical networks by estimating the parameters for amplitudes, phases and frequencies to analyze the power signals. During the last few decades, researchers have made great effort toward estimating parameters of the signal models. Chen et al. provided several fast Fourier transform-based methods for parameter estimation [7]. Li studied least squares-based algorithms for power signals modeling [8]. Zhao et al. considered multifrequency identification technique for parameter estimation of various signal models [9]. Cao proposed hierarchical identification approach for estimation of signal frequency [10]. Ding et al. introduced recursive least squares and stochastic gradient-based parameter estimation techniques [11]. Adaptive filtering techniques were employed for the estimation of signal modeling parameters [12, 13]. Besides simple least square technique variations, i.e., bilinear recursive least square algorithm [14], forgetting factor [15] and variable constraint-based least mean square [16] were also investigated for estimation of signal components. These all are deterministic approaches, which are exhaustively applied for parameter identification of electrical power signals with their own perks and limitations, while stochastic solvers proficiency in nonlinear systems has not yet been widely exploited in harmonic estimation problems of various electrical networks.

The stochastic numerical optimization mechanisms based on evolutionary heuristics have been extensively applied for optimization of constrained and unconstrained problems in various domains [17,18,19,20,21]. Few notable applications of evolutionary computational technique include problems arising in computational fluid dynamics [22], electronic circuits [23], combustion theory [24], optimized power dispatch in electrical power system [25], optimal power flow solution [26], system identification [27], bioinformatics [28], controls [29], satellite systems design [30], magnetohydrodynamics [31], electrical machines [32], energy [33], plasma physics [34], astrophysics [35], atomic physics [36], signal processing [37], economic dispatch problem [38] and finance [39]. In the view of the above techniques, authors investigated the performance of new optimization techniques for stable and accurate parameter estimation of power signal modeling. In this research work, the strength of differential evolution (DE) optimization technique is employed for estimating fundamental signal model components by taking various noise variances and different model dynamics. The comparative performance of the proposed algorithm has also been carried out with renowned heuristic techniques based on genetic algorithms (GAs) and pattern search (PS) for different power signal models. The primary objectives of the proposed study are described as:

  • Novel application of bioinspired heuristic computing through differential evolution is proposed for estimating fundamental components of power signal in the presence of noise with different variance levels.

  • The objective function is formulated by exploiting the competency of approximation theory in mean squared error sense, and unknown adjustable variables of the system model are optimized with efficient global search tools based on differential evolution, genetic algorithms and pattern search method to draw constructive deductions.

  • The efficacy, reliability and stability of the scheme are evaluated through comparison of true parameters of the power signal models with different degree of freedom based on amplitudes, frequencies, phases and their arbitrary combinations.

  • Validation of the performance through statistic operators calculated for multiple executions of the algorithm in terms of mean absolute error (MAE), Nash–Sutcliffe efficiency (NSE) and Theil's inequality coefficient (TIC) along with their global versions.

Rest of the paper is organized as follows: The power signal model, objective function mathematical formulation, detailed description of designed methodology and optimization procedures are presented in Sect. 2. Discussion on the simulation results based on single run and multiple runs is provided in Sects. 3 and 4, respectively, in terms of tabular and graphical illustrations. Concluding remarks are listed in Sect. 5.

2 Design methodology

The proposed methodology for parameter estimation problem in power signal modeling consists of two parts; in the first step, system modeling of power signals is overviewed along with the development of the objective function in the mean squared error sense, while in the later stage, the optimization procedures for minimizing the error function are described. The detailed description of the designed scheme in the block processes is illustrated in Fig. 1.

Fig. 1
figure 1

Graphical abstract of the designed scheme

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2.1 Power signal modeling

The general form of electric alternating current signal can be described in terms of frequencies, phases and amplitudes as:

$$\begin{aligned} r(t) & = \sum\limits_{i = 1}^{n} {c_{i} \sin (\omega_{i} t + \varphi_{i} ) = } \,c_{1} \sin (\omega_{1} t + \varphi_{1} ) \\ & \quad + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + \cdots + c_{n} \sin (\omega_{n} t + \varphi_{n} ), \\ \end{aligned}$$
(1)

where \(\varvec{c} = [c_{1} ,c_{2} , \ldots ,c_{n} ]\) are the amplitudes of the peak deviation of the function from zero, the number of oscillations (cycles) that occur each second of time is \(\varvec{\omega}= [\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} ]\), the frequencies. \(\varvec{\varphi } = [\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{n} ]\) are the phases (in radians) at t = 0 during oscillation. For all nonzero values of \(\varphi\), the complete waveform emerges to be shifted in timescale by \(\varphi /\omega\) seconds. A positive value of \(\varphi\) signifies an advance, while a negative value indicates a delay in the signal. Equation (1) appears in many report studies of engineering and applied sciences [40,41,42] and referenced therein.

In the experiment, the sampling time is \(t_{m} = mh\), where h is sampling period. The observed data are shown as \(\left\{ {t_{k} ,r(t_{k} )} \right\}\). Let \(r_{m} = r(t_{m} )\) for implication; then, the discretized power signal based on sinusoidal function is given as:

$$\begin{aligned} r_{m} & = \sum\limits_{i = 1}^{n} {c_{i} \sin (\omega_{i} t_{m} + \varphi_{i} )} = c_{1} \sin \left( {\omega_{1} t_{m} + \varphi_{1} } \right) \\ & \quad + c_{2} \sin \left( {\omega_{2} t_{m} + \varphi } \right)_{2} + \cdots + c_{n} \sin \left( {\omega_{n} t_{m} + \varphi_{n} } \right), \\ {\text{For}}\,\,m = 1,2, \ldots ,M \\ \end{aligned}$$
(2)

Standalone as well as the arbitrary combination of the parameters based on frequencies, phases and amplitudes of the sinusoidal-based signal model can be formulated for the estimation problems. The unknown parameters to be identified are given as:

$$\varvec{\theta}_{c} = \varvec{c} = \left[ {c_{1} ,c_{2} , \ldots ,c_{n} } \right]^{T} \in {\mathbb{R}}^{n}$$
(3)
$$\varvec{\theta}_{\omega } = \omega = \left[ {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right]^{T} \in {\mathbb{R}}^{n}$$
(4)
$$\varvec{\theta}_{\varphi } = \varphi = \left[ {\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{n} } \right]^{T} \in {\mathbb{R}}^{n}$$
(5)
$$\varvec{\theta}_{c,\,\omega } = [\varvec{c},\varvec{\omega}] = \left[ {c_{1} ,c_{2} , \ldots ,c_{n} ,\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} } \right]^{T} \in {\mathbb{R}}^{2n}$$
(6)
$$\varvec{\theta}_{c,\,\varphi } : = [\varvec{c},\varvec{\varphi }] = \left[ {c_{1} ,c_{2} , \ldots ,c_{n} ,\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{n} } \right]^{T} \in {\mathbb{R}}^{2n}$$
(7)
$$\varvec{\theta}_{\omega ,\,\varphi } : = [\varvec{\omega},\varvec{\varphi }] = \left[ {\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} ,\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{n} } \right]^{T} \in {\mathbb{R}}^{2n}$$
(8)
$$\varvec{\theta}_{c,\,\omega ,\,\varphi } : = [\varvec{c},\,\,\varvec{\omega},\,\,\varvec{\varphi }] = \left[ {c_{1} ,c_{2} , \ldots ,c_{n} ,\omega_{1} ,\omega_{2} , \ldots ,\omega_{n} ,\varphi_{1} ,\varphi_{2} , \ldots ,\varphi_{n} } \right]^{T} \in {\mathbb{R}}^{3n}$$
(9)

The standalone parameter identification problems are based on Eqs. (3–5), while parametric Eqs. (6–9) represent the integrated parameter identification systems.

The fitness function for power signal modeling problem is formulated by exploiting the approximation theory in mean squared error sense as:

$$\varepsilon = \frac{1}{M}\sum\limits_{m = 1}^{M} {\left( {r_{m} - \left( {\hat{r}_{m} + v_{m} } \right)} \right)^{2} }$$
(10)
$$\varepsilon = \frac{1}{M}\sum\limits_{m = 1}^{M} {\left( {\sum\limits_{i = 1}^{n} {c_{i} \sin (\omega_{i} t_{m} + \varphi_{i} )} - \sum\limits_{i = 1}^{n} {\hat{c}_{i} \sin (\hat{\omega }_{i} t_{m} + \hat{\varphi }_{i} )} - v_{m} } \right)^{2} } ,$$
(11)

where \(\varvec{v} = [v_{1} ,v_{2} , \ldots ,v_{m} ]\) is the noise signal with zero mean and constant variance, and \(\hat{r}_{m}\) is the approximate signal with parameter vectors for amplitude, frequency and phase as:

$$\hat{\varvec{\theta }}_{c} = \hat{\varvec{c}} = \left[ {\hat{c}_{1} ,\hat{c}_{2} , \ldots ,\hat{c}_{n} } \right]^{T} \in {\mathbb{R}}^{n} ,$$
(12)
$$\hat{\varvec{\theta }}_{\omega } = \hat{\omega } = \left[ {\hat{\omega }_{1} ,\hat{\omega }_{2} , \ldots ,\hat{\omega }_{n} } \right]^{T} \in {\mathbb{R}}^{n} ,$$
(13)
$$\hat{\varvec{\theta }}_{\varphi } = \hat{\varphi } = \left[ {\hat{\varphi }_{1} ,\hat{\varphi }_{2} , \ldots ,\hat{\varphi }_{n} } \right]^{T} \in {\mathbb{R}}^{n} ,$$
(14)

accordingly, the variables \(\hat{\varvec{\theta }}_{c,\,\omega }\), \(\hat{\varvec{\theta }}_{c,\,\varphi }\), \(\hat{\varvec{\theta }}_{\,\omega ,\varphi }\) and \(\hat{\varvec{\theta }}_{c,\,\omega ,\varphi }\) of integrated parameter identification systems can be defined. Now the requirement is to optimize fitness function (10) such that \(\varepsilon\) approaches 0; then, the adaptive parameters \(\hat{\varvec{\theta }}_{c}\), \(\hat{\varvec{\theta }}_{\omega }\), \(\hat{\varvec{\theta }}_{\,\varphi }\), \(\hat{\varvec{\theta }}_{c,\,\omega }\), \(\hat{\varvec{\theta }}_{c,\,\varphi }\), \(\hat{\varvec{\theta }}_{\,\omega ,\varphi }\) and \(\hat{\varvec{\theta }}_{c,\,\omega ,\varphi }\) match with the true variables \(\varvec{\theta}_{c}\), \(\varvec{\theta}_{\omega }\), \(\varvec{\theta}_{\,\varphi }\), \(\varvec{\theta}_{c,\,\omega }\), \(\varvec{\theta}_{c,\,\varphi }\), \(\varvec{\theta}_{\,\omega ,\varphi }\) and \(\varvec{\theta}_{c,\,\omega ,\varphi }\) of the model.

2.2 Optimization procedure

The well-known global stochastic procedures DE, GAs and PS are used for estimation of signal model parameter vector \(\varvec{\theta}\) by optimizing objective function as described in Eq. (10), and brief introduction of these optimization solvers is given in this section.

DE is a stochastic optimization technique belonging to the class of evolutionary algorithms introduced by Storn and Price [43]. It is population-based search technique which mainly uses mutation as a search tool. Later the selection operation directs the search in the proximity of the feasible regions in the search space. Crossover and recombination operators are also used to search for better solution space. The generic flow graph with coherent designed steps of DE is shown in Fig. 2, while recent applications of DE are seen in solving different problems including text classification [44], reactive power management [45], hydrothermal energy scheduling problem [46], load dispatch involving wind power plant incorporating emission [47], nonlinear systems [48] and nanofluidics systems [49], while the renewed application of DE in diverse field can be seen latest review articles [50,51,52,53] and referenced therein. Genetic algorithms (GAs) belong to a family of evolutionary computational heuristics [54, 55]. Inspired by the theory of evolution, an optimal solution of a problem is determined through three basic genetic operators, i.e., mutation, selection and reproduction. The generic flow graph with essential procedural steps of GAs is presented in Fig. 2a. The global search efficacy of well-tuned, coherent structure and smooth implemented GAs can provide near-optimal solution for multiobjective, multiconstraint, convex and nonconvex problems. This method was introduced by Holland [54, 55] and has several applications in science and engineering [56,57,58,59,60,61]. The PS is direct search optimization technique that does not require gradient information to search for minimal value of the cost function [62]. This method is exploited for nondifferentiable, stochastic and continuous functions. The pattern search algorithm computes a set of points known as a mesh around the initial/current point that tends to approach optimal value. When a current point is used to update a pattern, a mesh is created. The generic flow diagram with intermediate major steps of PS algorithm is illustrated in Fig. 2b [62], while the few recent application addresses with PS algorithm as can be seen in [63,64,65,66].

Fig. 2
figure 2

Schematic workflow of PS and GAs

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Efficiency of global optimizers DE, GAs and PS was the reason to utilize these optimization procedures for finding unknown parameters of signal model. The process flow graph of DE for signal model is presented in Fig. 3, while the detailed procedural steps are given in the form of pseudocode demonstrated in Table 1.

Fig. 3
figure 3

Workflow diagram of DE for power signal modeling

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Table 1 Pseudocode for DE algorithm for optimization of power signal modeling system
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2.3 Performance indices

In this research study, performance evaluation of the designed methodology for parameter estimation of the power signal models is carried out using four performance indices, i.e., mean absolute error (MAE), root of mean squared error (RMSE), estimation error function and coefficient of determination (R2). All these performance measures are briefly discussed in this section.

The performance measure \({\text{MAE}}_{c}\) is described as:

$${\text{MAE}}_{c} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {\left| {\varvec{\theta}_{c,i} - \hat{\varvec{\theta }}_{c,i} } \right|} .$$
(15)

where \(\hat{\varvec{\theta }}\) is an estimated weight vector of true vector \(\varvec{\theta}\). Similarly, the performance measure of \({\text{MAE}}_{c,\omega }\), \({\text{MAE}}_{c,\varphi }\), \({\text{MAE}}_{\omega ,\varphi }\), \({\text{MAE}}_{c,\omega ,\varphi }\) are formulated for \(\hat{\varvec{\theta }}_{c,\omega }\), \(\hat{\varvec{\theta }}_{c,\varphi }\), \(\hat{\varvec{\theta }}_{\omega ,\varphi }\) and \(\hat{\varvec{\theta }}_{c,\omega ,\varphi }\), respectively.

The RMSE is mathematically defined as:

$${\text{RMSE}}_{c} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {(\varvec{\theta}_{c,i} - \hat{\varvec{\theta }}_{c,i} )^{2} ,} }$$
(16)

Likewise, the root-mean-square errors for \(\hat{\varvec{\theta }}_{c,\omega }\), \(\hat{\varvec{\theta }}_{c,\varphi }\), \(\hat{\varvec{\theta }}_{\omega ,\varphi }\) and \(\hat{\varvec{\theta }}_{c,\omega ,\varphi }\) are also formulated mathematically.

Estimation error function \(\delta\) is calculated as:

$$\delta = \frac{{\left\| {\hat{\varvec{\theta }} -\varvec{\theta}} \right\|}}{{\left\|\varvec{\theta}\right\|}},$$
(17)

where || || represents the L2 norm.

Coefficient of determination \(R_{c}^{2}\) is determined as below:

$$R_{c}^{2} = \left( {\frac{{\sum\nolimits_{i = 1}^{n} {(\varvec{\theta}_{c,i} - \bar{\varvec{\theta }})(\hat{\varvec{\theta }}_{c,i} - \varvec{\bar{\hat{\theta }}})} }}{{\sum\nolimits_{i = 1}^{n} {\sqrt {(\varvec{\theta}_{c,i} - \bar{\varvec{\theta }})^{2} } } \sum\nolimits_{i = 1}^{n} {\sqrt {(\hat{\varvec{\theta }}_{c,i} - \varvec{\hat{\bar{\theta }}})^{2} } } }}} \right)^{2} ,\quad for\,\,\bar{\varvec{\theta }}= \frac{1}{n}\sum\limits_{i = 1}^{n} {\varvec{\theta}_{c,i} } ,\,\,\varvec{\bar{\hat{\theta }}}= \frac{1}{n}\sum\limits_{i = 1}^{n} {\hat{\varvec{\theta }}_{c,i} } .$$
(18)

The error in \(R_{c}^{2}\)\(({\text{ER}}_{c}^{2} )\) is shown as:

$${\text{ER}}_{c}^{2} = 1 - R_{c}^{2} .$$
(19)

Similarly, the performance indices \({\text{ER}}_{c,\omega }^{2}\), \({\text{ER}}_{c,\varphi }^{2}\), \({\text{ER}}_{\omega ,\varphi }^{2}\) and \({\text{ER}}_{c,\omega ,\varphi }^{2}\) are also defined for \(\hat{\varvec{\theta }}_{c,\omega }\), \(\hat{\varvec{\theta }}_{c,\varphi }\), \(\hat{\varvec{\theta }}_{\omega ,\varphi }\) and \(\hat{\varvec{\theta }}_{c,\omega ,\varphi }\), respectively.

The average fitness \(\bar{\varepsilon }\) calculated from total number of runs R is illustrated as:

$$\bar{\varepsilon } = \left\{ \begin{aligned} \frac{1}{R}\sum\limits_{r = 1}^{R} {\varepsilon_{r} } \hfill \\ \frac{1}{R}\sum\limits_{r = 1}^{R} {\left( {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {\left( {{\mathbf{\varphi }}^{T} (t_{c,i,r} )\varvec{\theta}+ v(t_{c,i,r} )} \right) - \left( {{\mathbf{\varphi }}^{T} (t_{c,i,r} )\hat{\varvec{\theta }}} \right)} \right)^{2} } } \right).} \hfill \\ \end{aligned} \right.$$
(20)

Similarly, the global MAE (\(G_{{{\text{MAE}}c}}\)) is mathematically evaluated as:

$$G_{{{\text{MAE}}c}} = \frac{1}{R}\sum\limits_{r = 1}^{R} {MAEc_{r} } = \frac{1}{R}\sum\limits_{r = 1}^{R} {\left( {\sum\limits_{i = 1}^{n} {\left| {\varvec{\theta}_{c,i} - \hat{\varvec{\theta }}_{c,i} } \right|} } \right)} ,$$
(21)

where R represents the total number of independent executions, and one independent run is an algorithmic process with different random seeds. The global performance measures \(G_{{{\text{MAE}}c,\omega }}\), \(G_{{{\text{MAE}}c,\varphi }}\), \(G_{{{\text{MAE}}\omega ,\varphi }}\) and \(G_{{{\text{MAE}}c,\omega ,\varphi }}\) are also formulized for \(\hat{\varvec{\theta }}_{c,\omega }\), \(\hat{\varvec{\theta }}_{c,\varphi }\), \(\hat{\varvec{\theta }}_{\omega ,\varphi }\) and \(\hat{\varvec{\theta }}_{c,\omega ,\varphi }\), respectively.

Likewise, the global RMSE (\(G_{{{\text{RMSE}}c}}\)) is formulated as,

$$G_{RMSEc} = \frac{1}{R}\sum\limits_{r = 1}^{R} {RMSEc_{r} } = \frac{1}{R}\sum\limits_{r = 1}^{R} {\left( {\sqrt {\frac{1}{n}\sum\limits_{i = 1}^{n} {(\varvec{\theta}_{c,i} - \hat{\varvec{\theta }}_{c,i} )^{2} } } } \right)} .$$
(22)

For \(\hat{\varvec{\theta }}_{c,\omega }\), \(\hat{\varvec{\theta }}_{c,\varphi }\), \(\hat{\varvec{\theta }}_{\omega ,\varphi }\) and \(\hat{\varvec{\theta }}_{c,\omega ,\varphi }\), the global performance operators are \(G_{MAEc,\omega }\), \(G_{MAEc,\varphi }\), \(G_{MAE\omega ,\varphi }\) and \(G_{MAEc,\omega ,\varphi }\), respectively.

Global \({\text{ER}}_{c}^{2}\) (\(G_{Rc}\)) is mathematically formalized as:

$$G_{Rc} = \left\{ \begin{aligned} \frac{1}{R}\sum\limits_{r = 1}^{R} {{\text{ER}}_{c}^{2} } \hfill \\ \frac{1}{R}\sum\limits_{r = 1}^{R} {\left( {\frac{{\sum\nolimits_{i = 1}^{n} {(\varvec{\theta}_{c,i} - \bar{\varvec{\theta }})(\hat{\varvec{\theta }}_{c,i} - \varvec{\hat{\bar{\theta }}})} }}{{\sum\nolimits_{i = 1}^{n} {\sqrt {(\varvec{\theta}_{c,i} -\varvec{\theta})^{2} } } \sum\nolimits_{i = 1}^{n} {\sqrt {(\hat{\varvec{\theta }}_{c,i} - \varvec{\hat{\bar{\theta }}})^{2} } } }}} \right)^{2} .} \hfill \\ \end{aligned} \right.$$
(23)

The global performance operators \(G_{Rc,\omega }\), \(G_{Rc,\varphi }\), \(G_{R\omega ,\varphi }\) and \(G_{Rc,\omega ,\varphi }\) are also formulated for other four parameter vectors. In case of the perfect models, the standard values of these performance indices MAE, RMSE and \({\text{ER}}^{2}\) should be zero.

3 Simulation with discussion

The simulation results of numerical experimentation for five examples are presented for parameter estimation problem of power signal model with variation in optimization parameters as well as different signal-to-noise ratios using various heuristic computing techniques including DE and GAs and PS algorithms.

Example 1

Power Signal with Unknown Amplitude: In this example, the description of parameter estimation problem of power signal with unknown amplitude as parameter vector is considered as below:

$$\begin{aligned} r(t) & = c_{1} \sin (\omega_{1} t + \varphi_{1} ) + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + c_{3} \sin (\omega_{3} t + \varphi_{3} ) + c_{4} \sin (\omega_{4} t + \varphi_{4} ) \\\varvec{\theta}_{c} & = \left[ {c_{1} ,c_{2} ,c_{3} ,c_{4} } \right]^{T} \\\varvec{\theta}_{c} & = [1.8,0.9,4,2.5]^{\text{T}} . \\ \end{aligned}$$
(24)

Example 2

Power Signal with Unknown Amplitude and Frequency: Here, an estimation problem of power signal with unknown amplitude and frequency is evaluated as:

$$\begin{aligned} r(t) & = c_{1} \sin (\omega_{1} t + \varphi_{1} ) + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + c_{3} \sin (\omega_{3} t + \varphi_{3} ) \\\varvec{\theta}_{c,\omega } & = \left[ {c_{1} ,c_{2} ,c_{3} ,\omega_{1} ,\omega_{2} ,\omega_{3} } \right]^{T} \\\varvec{\theta}_{c,\omega } & = [ 0. 8 , { 0} . 9 , { 0} . 4 , { 0} . 7 , { 0} . 5 , { 0} . 2]^{T} . \\ \end{aligned}$$
(25)

Example 3

Power Signal with Unknown Amplitude and Phase: In this case study, parameter estimation of power signals having unknown amplitude and phase in the parameter vector is considered as follows:

$$\begin{aligned} r(t) & = c_{1} \sin (\omega_{1} t + \varphi_{1} ) + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + c_{3} \sin (\omega_{3} t + \varphi_{3} ) \\\varvec{\theta}_{c,\varphi } & = \left[ {c_{1} ,c_{2} ,c_{3} ,\varphi_{1} ,\varphi_{2} ,\varphi_{3} } \right]^{T} \\\varvec{\theta}_{c,\varphi } & = [ 0. 8 , { 0} . 6 , { 0} . 7 , { 0} . 6 , { 0} . 8 , { 0} . 7 6]^{T} . \\ \end{aligned}$$
(26)

Example 4

Power Signal with Unknown Frequency and Phase: In this study, parameter estimation of power signal with unknown frequency and phase in the parameter vector is shown as follows:

$$\begin{aligned} r(t) & = c_{1} \sin (\omega_{1} t + \varphi_{1} ) + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + c_{3} \sin (\omega_{3} t + \varphi_{3} ) \\\varvec{\theta}_{\omega ,\varphi } & = \left[ {\omega_{1} ,\omega_{2} ,\omega_{3} ,\varphi_{1} ,\varphi_{2} ,\varphi_{3} } \right]^{T} \\\varvec{\theta}_{\omega ,\varphi } & = [ 0. 0 7 , { 0} . 1 , { 0} . 2 , { 0} . 9 5 , { 0} . 8 , { 0} . 7 6]^{T} . \\ \end{aligned}$$
(27)

Example 5

Power Signal with Unknown Amplitude, Frequency and Phase: In this example, signal power parameter estimation model having unknown amplitude, frequency and phase in the parameter vector is described as follows:

$$\begin{aligned} r(t) & = c_{1} \sin (\omega_{1} t + \varphi_{1} ) + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + c_{3} \sin (\omega_{3} t + \varphi_{3} ). \\\varvec{\theta}_{c,\omega ,\varphi } & = \left[ {c_{1} ,c_{2} ,c_{3} ,\omega_{1} ,\omega_{2} ,\omega_{3} ,\varphi_{1} ,\varphi_{2} ,\varphi_{3} } \right]^{T} \\\varvec{\theta}_{c,\omega ,\varphi } & = [ 0. 8 , { 0} . 9 , { 0} . 4 , { 0} . 9 5 , { 0} . 8 , { 0} . 7 6 , { 0} . 0 7 , { 0} . 5 , { 0} . 2]^{T} . \\ \end{aligned}$$
(28)

Example 6

Power Signal with Unknown Frequency and Phase: In this example, signal parameter estimation model having unknown frequency and phase in the parameter vector is considered as follows:

$$\begin{aligned} r(t) & = c_{1} \sin (\omega_{1} t + \varphi_{1} ) + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + c_{3} \sin (\omega_{3} t + \varphi_{3} ) \\ & \quad \ldots + c_{4} \sin (\omega_{4} t + \varphi_{4} ) + c_{5} \sin (\omega_{5} t + \varphi_{5} ). \\\varvec{\vartheta}& = \left[ {\omega_{1} ,\omega_{2} ,\omega_{3} ,\omega_{4} ,\omega_{5} ,\varphi_{1} ,\varphi_{2} ,\varphi_{3} ,\varphi_{4} ,\varphi_{5} } \right]^{T} \\\varvec{\vartheta}& = [0.0 7 ,0. 50,0. 20,0. 30,0. 10,0. 9 5 ,0. 80,0. 7 6 ,0. 60,0. 5 6]^{T} \\ \end{aligned}$$
(29)

Example 7

Power Signal with Unknown Amplitude, Frequency and Phase: In this example, power signal parameter estimation model having unknown amplitude, frequency and phase in the parameter vector is given as follows:

$$\begin{aligned} r(t) & = c_{1} \sin (\omega_{1} t + \varphi_{1} ) + c_{2} \sin (\omega_{2} t + \varphi_{2} ) + c_{3} \sin (\omega_{3} t + \varphi_{3} ) + c_{4} \sin (\omega_{4} t + \varphi_{4} ) \\ & \quad \ldots + c_{5} \sin (\omega_{5} t + \varphi_{5} ) + c_{6} \sin (\omega_{6} t + \varphi_{6} ) + c_{7} \sin (\omega_{7} t + \varphi_{7} ). \\\varvec{\vartheta}& = \left[ {c_{1} ,c_{2} ,c_{3} ,c_{4} ,c_{5} ,c_{6} ,c_{7} ,\omega_{1} ,\omega_{2} ,\omega_{3} ,\omega_{4} ,\omega_{5} ,\omega_{6} ,\omega_{7} ,\varphi_{1} ,\varphi_{2} ,\varphi_{3} ,\varphi_{4} ,\varphi_{5} ,\varphi_{6} ,\varphi_{7} } \right]^{T} \\\varvec{\vartheta}& = [0. 8 0 ,0.0 7 ,0. 9 5 ,0. 9 0 ,0. 5 0 ,0. 8 0 ,0. 4 0 ,0. 2 0 ,0. 7 6 ,0. 10,0. 3 0 ,0. 6 0 ,\\ & \quad \ldots 0.00 2 0 ,0. 1 0 ,0. 5 6 ,0. 1 0 ,0. 1 0 ,0. 3 4 ,0. 1 0 ,0. 9 0 ,0. 2 4]^{T} \\ \end{aligned}$$
(30)

In the present research study, five examples of power signal modeling are taken, the input signal r(k) is a random signal sequence with zero mean and unit variance, and v(t) is a noise signal with zero mean for different variance magnitudes. Three different noise levels are added to the signal which produce three noise scenarios, i.e., no noise scenario, 70-db and 30-db SNR levels, respectively, while the values of sampling period h and the data length L are set to 0.2 s and 2000 steps, respectively. The parameter estimation of the power signal model is performed with the designed methodologies based on DE, GAs and PS techniques, as discussed briefly in the previous section, while the parameter settings of all three algorithms for all seven examples are given in Table 2. However, fitness function (10) is formulated for M = 2000 for each case of all examples and is given as:

$$\varepsilon = \left\{ \begin{aligned} \frac{1}{2000}\sum\limits_{m = 1}^{2000} {\left( {r_{m} - \left( {\hat{r}_{m} + v_{m} } \right)} \right)^{2} } \hfill \\ \frac{1}{2000}\sum\limits_{m = 1}^{2000} {\left( {\sum\limits_{i = 1}^{n} {c_{i} \sin (\omega_{i} t_{m} + \varphi_{i} )} - \sum\limits_{i = 1}^{n} {\hat{c}_{i} \sin (\hat{\omega }_{i} t_{m} + \hat{\varphi }_{i} )} - v_{m} } \right)^{2} ,} \hfill \\ \end{aligned} \right.$$
(31)
Table 2 Parameter settings for all examples of power signal modeling
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For all five examples of power signal model, the optimization of objective function (31) is performed for three SNR levels employing optimization procedures based on DE, GAs and PS. The magnitudes of the performance metrics MAE, MWD, RMSE and ER2 as defined in Eqs. (15) to (18) for each optimization technique are computed for 100 independent executions. The best run of the methodologies is revealed on the basis of the smallest magnitudes of all these performance indices which are listed in Table 3 for all five examples of power signal modeling problems. Comparison of actual signal with the approximated signals is made and is plotted in Figs. 4, 5 and 6 for each designed methodology DE, GAs and PS, respectively. Absolute error (AE) values, i.e., deviation from the reference solution of power signals, are evaluated to further elaborate similarity level of the results with true solutions for all three SNR levels of each defined example.

Table 3 Best iteration of DE, GAs and PS algorithms based on different performance operators for each examples of power signal model
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Fig. 4
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Comparison results for DE in case of power signal models with different SNR levels

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Fig. 5
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Comparison results for GAs in case of power signal models with different SNR levels

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Fig. 6
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Comparison results for PS in case of power signal models with different SNR levels

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The absolute error is computed for the run of the algorithm with minimum MAE value. The results of AE for all five examples of power signal model for each variation are shown graphically in Fig. 4a–f, for DE. Generally, the respective AE magnitudes lie around 10−10, 10−06 and 10−03 for example 1, 10−10, 10−06 and 10−04 for example 2, 10−10, 10−06 and 10−04 for example 3, 10−10, 10−05 and 10−03 in case of example 4 and 10−09, 10−05 and 10−03 in case of example 5, for no noise scenario, 70-db SNR and 30-db SNR, respectively. Curves are also plotted in Fig. 4g–l comparing actual signal with the approximated signal. The computed solutions are found in good agreement with the actual numerical solutions. Comparison of MAE values for the first five examples is shown in subfigure, 4(m). Similar plots are also displayed for GAs, and PS is presented in Figs. 5 and 6, respectively.

The values of performance indices in terms of accuracy measures of fitness, MAE, RMSE and \(ER^{2}\) are computed for the best execution of each algorithm, and results are listed in Table 4 for the first five examples, together with complexity measures based on time consumed, generations executed and function evaluated. It is clear that the values of MAE are around 10−10 to 10−11, 10−05 to 10−04 and 10−04 to 10−03 for no noise level, 70-db SNR and 30-db SNR, respectively, for DE in case of example 1, while for examples 2, 3, 4 and 5 the respective values lie 10−01 to 10−04, 10−03 to 10−10, 10−02 and 10−01 to 10−02. Similarly, the values of performance indices along with complexity measures for examples 6 and 7 are listed in Table 5. The MAE magnitudes of DE and GAs are comparable, while PS algorithm depicts significantly low performance. It is observed from the figures that in example 3, among all designed procedures, GAs generate best results, while DE outperforms in example 5 on the basis of accuracy indicators, while in case of complexity measures, DE is found much more efficient than GAs as DE consumes less time, smaller number of generations and function calls than GAs. Very low values of these performance measures verify the stability of DE and GA schemes for estimating parameters of all examples of the power signal models.

Table 4 Performance indices of accuracy and complexity for each algorithm for the first five examples of power signal model
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Table 5 Performance indices of accuracy and complexity for each algorithm for examples 6 and 7 of power signal model
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It is important to note here that due to increased popularity of memetic computing methodologies, hybrid procedures based on combination of global and local search techniques are also suggested for power signal modeling identification problem. Recently, memetic computing-based methodologies are broadly used to find the candidate solution of the constrained/unconstrained and convex/nonconvex optimization problems, such as GAs aided with active-set method (GAs-ASM) [96], GAs integrated with sequential quadratic programming (GAs-SQP) [36, 61] and GAs hybrid with interior-point method (GAs-IPM) [37]. The hybrid schemes usually give better results than optimization techniques independently; therefore, hybridization of DE, GA and PS with local search mechanisms SQP, IPA and ASM for optimization of power signal models was also employed; however, no significant increase in the accuracy is achieved for all examples. Furthermore, local search techniques such as SQP, IPM and ASM were individually employed for estimation of components of power signal model and it was observed that due to nonconvex nature of the examples, these techniques failed to optimize their cost functions. For this reason, in the current article we restricted our analysis to three global search techniques based on DE, GAs and PS for identification of power signal models.

4 Statistics-based comparative analysis

In order to determine the stability and reliability of the proposed schemes, 100 independent runs were executed for each SNR level for all examples of power signal models and optimized results of fitness function values for several runs of each method are illustrated in Fig. 7. The graphs are given in sorted output and zoomed plots for better perception. The results based on performance metrics MAE, RMSE and \({\text{ER}}^{2}\) are presented in Figs. 8, 9, 10, respectively, for all designed approaches DE, GAs and PS. The results of GAs computed for each SNR level for the first five examples for power signal models are computed, and few plots are also presented in Figs. 7, 8 and 9a, for MAE, RMSE and \({\text{ER}}^{2}\), respectively. These plots are given in semi-logarithmic scale in order to decipher the results more clearly. The MAE values for power signal model of example 4 lie around 10−04 to 10−05, 10−03 to 10−04 and 10−02 to 10−03 for no noise scenario, 70-db and 30-db SNRs, and similar trend is observed for performance measures of RMSE and \({\text{ER}}^{2}\), which shows the robustness of the schemes. For the analysis of consistency in precision, histogram studies are carried out and results based on MAE values for GAs are shown in Fig. 8b–e, and results based on PS are presented graphically in Fig. 8j–m. Empirical cumulative distribution function plots for all designed approaches are also plotted and are illustrated in Fig. 8f–i, while stacked bar plots for MAE values of DE for power signal modeling problem with few noise scenarios of different examples are also shown in Fig. 8n. Likewise, plots for RMSE and \({\text{ER}}^{2}\) are given in Figs. 9 and 10, respectively. All these figures and plots verify that these designed approaches are capable of parameter estimation of power signal model, yet the DE and GAs results are comparatively better than PS. The algorithms DE and GAs are found parallel in terms of accuracy, while in example 5, DE outperforms GAs. The algorithm giving superior fitness value has corresponding better MAE, RMSE and \({\text{ER}}^{2}\) indices, which validate the accuracy through different performance measures.

Fig. 7
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Plot of fitness for 100 runs of the algorithms in terms of sorted and zoomed illustrations

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Fig. 8
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Comparative study of DE, GA and PS based on MAEs for the first five examples (Exps) of power signal model

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Fig. 9
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Comparative study of DE, GA and PS based on RMSE magnitude for the first five examples (Exps) of power signal model

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Fig. 10
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Comparative study of DE, GA and PS based on R2 magnitude for the first five examples (Exps) of power signal model

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Accuracy of the proposed methodology is further evaluated to observe the best, mean and the worst estimated parameter vector obtained on minimum error-based fitness, AEs in three noise scenarios for the first five examples of power signal model computed through 100 independent runs of DE, GAs and PS approaches. Outcomes of statistical measures for the three approaches are presented in Tables 6, 7, 8, 9, 10 for three SNR levels of examples 1–5 with their actual parameters, respectively. The parameter estimation performance of DE, GAs and PS algorithms is dependent on the noise variance, such as for larger SNR level, performance of the designed procedures reduces. Furthermore, by increasing the length of the parameter vector of the power signal model, a small decrease in the accuracy of the proposed algorithms is observed which is understandable as with more degrees of freedom, the optimization problem gets stiffer.

Table 6 Comparison of DE, GA and PS results through statistics for power signal model in case of all SNR levels for example 1
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Table 7 Comparison of DE, GA and PS results through statistics for power signal model in case of all SNR levels for example 2
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Table 8 Comparison of DE, GA and PS results through statistics for power signal model in case of all SNR levels for example 3
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Table 9 Comparison of DE, GA and PS results through statistics for power signal model in case of all SNR levels for example 4
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Table 10 Comparison of DE, GA and PS results through statistics for power signal model in case of all SNR levels for example 5
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In order to draw reliable inference on the precision, the global performance metrics \(\bar{E}\), \(G_{\text{MAE}}\), \(G_{\text{RMSE}}\) and \(G_{{ER^{2} }}\) magnitudes are computed based on multiple executions of DE, GAs and PS as described in Eqs. (11)–(14), respectively. Result is tabulated in Table 11 for the first five examples of the power signal systems for each SNR level. On the basis of fitness, the accuracy level for global operators is found in the order of 10−02 to 10−21 for DE, 10−04 to 10−16 for GAs and 10−01 to 10−16 for PS in all examples of power signal models, whereas the range of performance indices of MAE, RMSE and ER2 is approximately 10−01 to 10−10 for DE and GAs and 10−01 to 10−16 for PS methods. The attained global performance metrics have values almost close to their ideals, which verify the correctness of the DE- and GAs-based methodologies for each example of power signal model.

Table 11 Magnitudes of global performance metrics for each variation of the first five examples of power signal model
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The performance of the proposed algorithms is further evaluated using ANOVA test for power signal modeling problem. The results are generated for the first five examples of the power signal models, and results are presented only for complex scenarios, i.e., example 4 with 70-dB SNR and example 5 with 30-dB SNR, in Tables 12 and 13, respectively. While considering the assumption of homogeneity of variances, the null hypothesis of homogeneous variances at the significance level 0.05 cannot be rejected, as the respective probability values attained for AE in case of DE, GAs and PS are 0.895, 0.810 and 0.821 for example 4, while in case of example 5, the respective probability values are 0.897, 0.695 and 0.107. This indicates that the expected values in the three groups do not have any strong evidence of variation in the response variable of uniformity. Thus, it is quite significant that all means are equal. Additionally, it is seen from the values of confidence interval that DE and GAs provide better results than PS, while DE performs comparatively better than GAs. The same trend prevails with similar inferences for rest of the cases on the basis of ANOVA test.

Table 12 Results of ANOVA test for example 4 of power signal model with 70-dB SNR
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Table 13 Results of ANOVA test for example 5 of power signal model with 30-dB SNR
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Computational complexity for all three algorithms, DE, GAs and PS, is analyzed through calculating the time, each algorithm consumed for hundred independent runs, iteration executed and objective functions evaluated, for optimization of the first five examples of power signal model for different noise levels, and results are tabulated in Table 14. Average time expended, generation executed and functions evaluated are around 16 ± 14, 750 ± 550, 51,200 ± 48,100 for DE, 37 ± 5, 500, 120,240 for GAs, 2.5 ± 1.1, 750 ± 550 and 5827 ± 4100 for PS, respectively. It is observed that with the increase in the length of parameter vector of power signal model, the complexity of DE and PS increases significantly, whereas GAs-based extensive global search does not show such behavior. In general, GAs complexity is much more than that of DE and PS, while its accuracy is found parallel with DE. All the simulations are performed in this research work on HP ProDesk 400 G3, Processor core i7, 3.4 GHz, RAM 8 GB, using MATLAB 2016a running in Windows 8.1 professional environment.

Table 14 Comparison of computational complexity comparison for each SNR level of the first five power signal model examples
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5 Concluding remarks

Strength of nature-inspired heuristics based on DE, GAs and PS is efficiently exploited in power signal modeling through estimating the necessary components of the power signal. Comparative studies reveal that all the three proposed algorithms are effective; however, estimation ability reduced for the model with more degrees of freedom for each approach, but the estimation of DE outperforms its counterparts GAs and PS in each scenario. The better performance of the DE and GAs is because of the fact that DE and GAs are population-based metaheuristics, while PS is a single-solution-based algorithm, while comparison shows DE is relatively better than GAs. Robustness of the design schemes is established through estimation power signal parameters with different SNRs (70 db, 30 db and no noise), and results show that with the increase in noise variance, the accuracy of DE, GAs and PS algorithms degrades, yet the performance of both DE and GAs is found still reasonable in precision. Statistical results based on mean and STD values of MAE, RMSE and ER2 performance indices along with their global versions as well as illustrations of histograms, cumulative distribution functions and stacked bar validate the consistent accuracy of DE and GAs in each case. The superior performance of DE is further endorsed through ANOVA-based statistical test. Complexity analyses for all three algorithms show that by increasing the optimization variables in the signal model the complexity of the algorithms increases, while the gauges of complexity operators are greater for GAs than the rest. One may explore recently introduced variants of modern optimization procedures for the better performance such as firefly algorithm, backtracking search algorithm, fractional particle swarm optimization, fireworks and gravitational search algorithm.

It seems to be a potential research direction to study other metaheuristics based on the swarming optimization, fractional evolutionary PSO, scatter search and backtracking search optimization algorithm for power signals parameter estimation.