Firefly Optimization Heuristics for Sustainable Estimation in Power System Harmonics

Abstract

The sustainable power development requires the study of power quality while taking into account of electrical equipment is an important aspect because it highly compromises the overall efficiency including quality, reliability and continuity of power flow. The aim for smooth power flow is only accomplished if compatibility is met between all the instruments connected to the system. The odd harmonics both on amplitude and phase domain must be known in order to exactly cop up with their adverse effects on overall working of the system. In this regard, parameter estimation is performed in detail for diverse generation size (gs) and particle size (ps), besides for altered signal to noise ratio. Firefly optimization technique under different scenarios for both phase and amplitude parameters accurately estimated the power signal harmonics and proved its robustness under different noise levels. The MSE values achieved by FFO are 6.54 × 10⁻³, 1.04 × 10⁻⁵ and 1.35 × 10⁻⁶ for 20 dB, 50 dB and 80 dB respectively for gs = 200 in case study 1. While the respective results in case study 2 are 7.33 × 10⁻³, 6.67 × 10⁻⁶ and 6.59 × 10⁻⁹ for gs = 1000. Whereas no significant effect in performance is seen with the change in ps values.

1. Introduction

By the arise of current era, power electronics equipment’s have become the most important constituent of electrical systems including variable resistance speed drives, smart computers, auto function protection devices and so on. The proper functionality of these devices mostly depends on the smooth voltage and current waveform which is only achieved by attenuating the harmonics involved in the input voltage and current waveforms [1,2]. Research has been carried out using different methods to analyze the periodic signal contaminated by harmonics and sub harmonics. Numerous techniques for study gathered the attention of harmonics distortion [3,4,5,6].

The research community proposed various schemes for powers signals estimation [7,8,9] and a number of papers have been presented in the literature that pivots the estimation of harmonics contamination [10]. These techniques include fast harmonics estimation using EA-AWNN by Sachin K. Jain et al. [11], exponential modulation integral observer by Hin Say Lam et al. [12], compressive sensing by Palczynska [13], Bayesian approach used by Gabriele D Antona et al. [14], ESPIRIT associated filter bank scheme by Santos [15], Kalman filter [16,17], least mean square based algorithms [18,19], and recursive least square strategy [20].

The concept of population search based optimization through swarm/evolutionary heuristics has been emerged over the recent years with excellent performance in solving various engineering and applied sciences problems [21,22,23,24,25]. The metaheuristics are also exploited for optimization of power systems [26,27,28,29,30], as well as harmonics estimation, such as, neuro evolutionary approach [31], evolutionary technique [32], grey wolf optimizer [33], particle swarm optimizer (PSO) [34,35,36], artificial bee colony [37,38,39], biogeography based optimization [40], fractional order PSO [41] and cuckoo search optimization [42].

The swarm intelligence based firefly optimization (FFO) algorithm has shown excellent performance in solving nonlinear and multimodal problems [43,44,45]. The FFO has been effectively applied to solve point tracking of PV systems [46,47,48], job scheduling [49], fraud detection [50], material sciences [51], energy management [52], flying adhoc networks [53], automatic data clustering [54,55], resonant accelerometers [56], multi robot foraging [57], load balancing [58] and many others. The brilliant performance of the FFO for these spectrum of challenging problems motivates the authors to exploit the optimization strength of the FFO for power systems harmonics estimation. In this paper the domain of work is further extended using a nature inspired heuristic technique based on FFO algorithm that focuses on the estimation of harmonics pollution. The main contributions/salient features of the proposed study are:

• The swarm intelligence of firefly optimization approach is exploited for sustainable power development through effective harmonics estimation arising in industrial loads.
• The phase and amplitude variables related to the first, third, fifth and eleventh harmonics are effectively estimated using optimization knacks of firefly algorithm.
• The FFO shows stupendous performance in both examples of harmonics estimation with enhanced accuracy for increased particle size.
• The proposed scheme of FFO is robust, convergent and stable for different signal to noise scenarios in case varying particle and generation size.

The rest of the article is organized as: the harmonics estimation scheme is presented in Section 2, firefly based swarm intelligence approach is given in Section 3, the experimental simulation results are provided in Section 4, and concluding remarks of the investigation are presented in Section 5.

2. Materials and Methods

The multi frequency sinusoidal signal in terms of amplitude $\lambda$, angular frequency $\omega$ and phase $\mu$ is defined as:

$$g\left(t\right)={\lambda }_1\sin \left({\omega }_{1}t+{\mu }_1\right)+{\lambda }_2\sin \left({\omega }_{2}t+{\mu }_2\right)+\cdots +{\lambda }_M\sin \left({\omega }_{M}t+{\mu }_M\right).$$

(1)

Considering the noise $\epsilon \left(t\right)$ in the (1) and writing in compact form

$$g(t)={\displaystyle \sum _{m=1}^M{\lambda }_m\sin \left({\omega }_{m}t+{\mu }_m\right)}+\epsilon \left(t\right).$$

(2)

The signal presented in Equation (2) is sampled with h period, $t_k=kh$ and $g\left(t_k\right)=g\left(k\right)$ provides

$$g\left(k\right)={\displaystyle \sum _{m=1}^M{\lambda }_m\sin \left({\omega }_{m}k+{\mu }_m\right)}+\epsilon \left(k\right).$$

(3)

Applying the trigonometric identity, sin (α + β) = sinαcosβ + cosαsinβ to (3) yields

$$g\left(k\right)={\displaystyle \sum _{m=1}^M\left[{\lambda }_m\left\{\sin \left({\omega }_{m}k\right)\cos {\mu }_m+\cos \left({\omega }_{m}k\right)\sin {\mu }_m\right\}\right]}+\epsilon \left(k\right).$$

(4)

Supposing

$$x_m={\lambda }_m\cos {\mu }_m\mathrm{and}{y}_m={\lambda }_m\sin {\mu }_m,$$

(5)

Using (5) in (4)

$$g\left(k\right)={\displaystyle \sum _{m=1}^M\left[x_m\sin \left({\omega }_{m}k\right)+y_m\cos \left({\omega }_{m}k\right)\right]}+\epsilon \left(k\right).$$

(6)

Defining the intermediate parameter vector $\Theta$ in (7), the corresponding information vector $\Omega$ in (8) and the identification model in (9)

$$\Theta =\left[x_1,y_1,x_2,y_2,\dots ,x_m,y_m\right],$$

(7)

$$\Omega \left(k\right)=\left[\sin \left({\omega }_{1}k\right),\cos \left({\omega }_{1}k\right),\sin \left({\omega }_{2}k\right),\cos \left({\omega }_{2}k\right),\dots ,\sin \left({\omega }_{m}k\right),\cos \left({\omega }_{m}k\right)\right]$$

(8)

The actual parameters, i.e., amplitude $\lambda$ and phase $\mu$ are obtained by using the intermediate parameters through the following relations

$${\lambda }_m=\sqrt{{\left(x_m\right)}^2+{\left(y_m\right)}^2},{\mu }_m={\tan }^{-1}\frac{y_m}{x_m}.$$

(9)

The criterion function based on the difference between actual $g\left(k\right)$ and estimated $g^{\prime }\left(k\right)$ signal in mean square sense is defined as

$$\Psi \left(k\right)=\mathrm{mean}{\left[g\left(k\right)-g^{\prime }\left(k\right)\right]}^2={\left[g\left(k\right)-{\Omega }^T\left(k\right){\Theta }^{\prime }\right]}^2$$

(10)

3. Optimization Scheme: Firefly Algorithm

In 2008 Yang et al., put forward an algorithm that was inspired on the basis of the flaring attribute of fire flies with clear explaination using following three rules;

(i)

Fire flies are unisex, so the probability of attraction on the basis of sex is zero.

(ii)

Since brightness is the key characteristic of a firefly. So the brighter firefly will tend to fly towards the less bright firefly. The relationship between the attractiveness and brightness is inverse with respect to the distance, i.e., brightness will decrease if the distance is increased and vice versa. In this manner, relationship maintained randomness between the firefly movements.

(iii)

Brightness is associated key factor in construction of the problem specific objective function.

The dissimilarity in light intensity and attractiveness of the fireflies are two important factors of the interest. Attractiveness in the firefly measure on brightness quality, i.e., the attractiveness is defined on the basis of brightness. This brightness B of the fire fly at a specific position x is directly to the maximization of objective function f and is mathematically expressed as follows:

$$B(x)\propto f(x).$$

(11)

In this scenario, the attractiveness A is relative to the eyes of the observer or distance $d_{ij}$ between the firefly i^th and j^th. In mathematical form, brightness B depends upon the distance d, i.e., it exponential function representation is given as:

$$B=B_o{e}^{-\gamma d}.$$

(12)

In the above expression B_o is the original brightness and γ is the coefficient of light absorption. In attractiveness representive with A in terms of the exponential function is given as:

$$A=A_o{e}^{-r{d}^2},$$

(13)

where A_o is the attractiveness at d = 0. Replacing exponent rd by rd^m (m > 0). Between two fireflies i and j, the distance between x_j and x_j are the Cartesian distance

$$r_{ij}=\left|\right|x_i+x_j|{|}_2.$$

(14)

The motion of the firefly i is fascinated by firefly j and is set on by

$$x_i=x_i+A_0{e}^{-\gamma r^2{}_{ij}}(x_j-x_i)+\alpha {\epsilon }_i.$$

(15)

In above expression, ε_i is the vector of randomization and can be pinched by Gaussian distribution and in most cases we take A^o = 1, α є [0, 1] and γ = 1.

The difference in the attractiveness is characterized by γ and its value is very important to determine the speed of convergence and also FFO behavior. Theoretically, we have Γ є [0, ∞] but in practical/generally we assume γ є [0, 1]. If γ = 0 the attractiveness is considered constant A = A_o. This means that the brightness B remains constant and flashing firefly can be visualized in any dominion. If the firefly is moving in a misty area, the attractiveness is almost zero. This corresponds to an exceptional case of PSO. Thus if the nested loop for j^th. firefly is eliminated and A^j is put back by the current g - best, the potential efficiency of FFO becomes equal to PSO in this special case. Now if γ ―› ∞, we have A(r) ―› ∂(r), which is Dirac ∂ -function and the attractiveness is totally blur, which means it becomes zero again or fireflies become short sighted. No other fireflies are seen in very foggy region and each firefly moves in a random way. For that reason, this leads to the completely random search method. The flow chart of the FFO is provided in Figure 1, while the graphical abstract of the proposed study is given in Figure 2. The electrical circuit given in part 1 of Figure 2 is the general circuit producing the basic signal and then the general representation of the harmonics corrupted signal is provided. While part 2 provides the mathematical representation of the problem and fitness function formulation. The part 3 of the Figure 2 describe the optimization procedure and last part reflects the results obtained from the FFO scheme.

4. Results and Discussion

Fire fly optimization algorithm is applied in this research area for accurate estimation of power signal under two scenarios for different generations and results are stored in tables provided in this section along with graphs. All the simulation work is carried out on Matlab with sampling frequency of 2 kHz. White Gaussian noise is introduced for three levels i.e., $\epsilon$ = 20 dB, 50 dB and 80 dB to evaluate the robustness and accuracy of the system for real time scenarios. For each scenario the investigation is carried out for four generation size (gs) and also three particle sizes (ps) for monitoring the behavior of system under different conditions. The values taken are gs = 100, 200, 300 and 400 for example no 1 with ps = 20, 50 and 100. The parameters of the algorithm are selected analytically with great care based on the experience and understanding of the optimization problem to ensure maximum performance.

Case study 1: The harmonic signal taken in first study is

$$h(t)=\left[\begin{array}{l}1.5\sin \left(2\pi f_{1}t+1.396\right)+0.5\sin \left(2\pi f_{3}t+1.047\right)+0.2\sin \left(2\pi f_{5}t+0.785\right)\\ +0.15\sin \left(2\pi f_{7}t+0.628\right)+0.1\sin \left(2\pi f_{11}t+0.523\right)\end{array}\right]$$

(16)

The total number of variables taken are ten including five phase and five amplitude parameters as;

$$\begin{array}{ll}\zeta & =\left[{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4,{\mu }_5\right]\\ & =\left[1.50,0.50,0.20,0.15,0.10,1.396,1.047,0.785,0.628,0.523\right]\end{array}$$

(17)

Case study 2: The harmonic signal considered in second study is

$$h(t)=\left[\begin{array}{l}1.2\sin \left(2\pi f_{1}t+1.309\right)+0.8\sin \left(2\pi f_{3}t+0.959\right)+0.2\sin \left(2\pi f_{5}t+0.785\right)\\ +0.18\sin \left(2\pi f_{7}t+0.698\right)+0.1\sin \left(2\pi f_{11}t+0.523\right)\end{array}\right]$$

(18)

The five amplitude and five phase parameters is

$$\begin{array}{ll}\zeta & =\left[{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5,{\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4,{\mu }_5\right]\\ & =\left[1.20,0.80,0.20,0.18,0.10,1.309,0.959,0.785,0.698,0.523\right]\end{array}$$

(19)

The results of case study 1 in terms of parameter estimates through FFO along with the MSE value are given in

Table 1. Outcomes of the FFO in case of case study 1 for gs = 100.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.1548	0.7979	0.2117	0.1976	0.1361	1.3176	0.9549	0.6284	0.6519	0.2675	1.24 × 10−2
	50	1.2190	0.7730	0.1823	0.2007	0.0881	1.3418	0.9370	0.9831	0.7175	0.6905	1.07 × 10−2
	100	1.1697	0.7873	0.2086	0.1428	0.0891	1.2897	0.9727	0.8105	0.6601	1.0891	1.02 × 10−2
50	20	1.1786	0.7907	0.1833	0.1951	0.0959	1.2982	0.9921	0.6189	0.7859	0.8905	2.26 × 10−3
	50	1.1833	0.8191	0.1811	0.1766	0.1237	1.2945	0.9622	0.8468	0.6749	0.4560	9.98 × 10−4
	100	1.1867	0.7947	0.2165	0.1675	0.0980	1.3033	0.9600	0.7815	0.5739	0.4477	5.99 × 10−4
80	20	1.2156	0.7610	0.1899	0.1772	0.0531	1.3378	0.9549	0.8896	0.7150	1.6492	6.02 × 10−3
	50	1.1965	0.7865	0.1757	0.2092	0.1092	1.3167	0.9645	0.7428	0.8010	0.5862	1.21 × 10−3
	100	1.2214	0.8028	0.1937	0.1757	0.1032	1.2901	0.9372	0.7409	0.6959	0.6160	6.89 × 10−4
		1.2000	0.8000	0.2000	0.1800	0.1000	1.3090	0.9590	0.7850	0.6980	0.5230	0

,

Table 2. Outcomes of the FFO in case of case study 1 for gs = 200.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.2032	0.7803	0.1926	0.1729	0.1062	1.2966	0.9901	0.8137	0.6196	0.3942	7.34 × 10−3
	50	1.1799	0.7868	0.2064	0.1793	0.0992	1.2912	0.9598	0.8849	0.7747	0.2160	8.80 × 10−3
	100	1.1879	0.8048	0.2065	0.1669	0.1016	1.2980	0.9482	0.7472	0.6806	0.4741	6.54 × 10−3
50	20	1.1987	0.7994	0.1989	0.1793	0.0997	1.3044	0.9581	0.7853	0.6957	0.5059	1.29 × 10−5
	50	1.1999	0.8009	0.1989	0.1804	0.0981	1.3057	0.9583	0.7866	0.6970	0.5187	1.13 × 10−5
	100	1.2000	0.7997	0.1994	0.1788	0.0998	1.3056	0.9591	0.7874	0.7005	0.5141	1.04 × 10−5
80	20	1.2016	0.8009	0.2000	0.1792	0.1007	1.3060	0.9582	0.7783	0.6946	0.5268	3.56 × 10−6
	50	1.1997	0.7998	0.1996	0.1797	0.1003	1.3061	0.9594	0.7878	0.7007	0.5131	1.08 × 10−6
	100	1.2004	0.7986	0.2001	0.1801	0.1000	1.3062	0.9587	0.7826	0.6998	0.5192	1.35 × 10−6
		1.2000	0.8000	0.2000	0.1800	0.1000	1.3090	0.9590	0.7850	0.6980	0.5230	0

,

Table 3. Outcomes of the FFO in case of case study 1 for gs = 300.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.1996	0.7891	0.1962	0.1824	0.0951	1.3113	0.9201	0.6863	0.6163	0.7443	8.62 × 10−3
	50	1.2018	0.8010	0.2145	0.1910	0.1041	1.3183	0.9547	0.7064	0.6876	0.4976	8.28 × 10−3
	100	1.1992	0.7987	0.1847	0.1881	0.1046	1.3081	0.9496	0.8449	0.6778	0.6463	6.84 × 10−3
50	20	1.1996	0.7999	0.1995	0.1799	0.0997	1.3061	0.9593	0.7874	0.6975	0.5251	7.44 × 10−6
	50	1.2000	0.8005	0.1994	0.1792	0.1001	1.3060	0.9587	0.7846	0.7013	0.5249	8.17 × 10−6
	100	1.1995	0.7997	0.2001	0.1799	0.1001	1.3061	0.9596	0.7834	0.6994	0.5300	7.13 × 10−6
80	20	1.2000	0.8000	0.2000	0.1799	0.1000	1.3060	0.9590	0.7850	0.6977	0.5233	1.68 × 10−8
	50	1.2000	0.8000	0.2000	0.1800	0.1000	1.3059	0.9590	0.7851	0.6981	0.5229	1.36 × 10−8
	100	1.2000	0.8001	0.2000	0.1799	0.1000	1.3060	0.9590	0.7851	0.6980	0.5232	1.31 × 10−8
		1.2000	0.8000	0.2000	0.1800	0.1000	1.3090	0.9590	0.7850	0.6980	0.5230	0

and

Table 4. Outcomes of the FFO in case of case study 1 for gs = 400.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.2090	0.7946	0.2090	0.1866	0.0887	1.3073	0.9423	0.8180	0.6076	0.9990	7.86 × 10−3
	50	1.2002	0.8059	0.2040	0.1556	0.1129	1.3144	0.9341	0.7539	0.6662	0.5639	8.32 × 10−3
	100	1.2129	0.8061	0.1857	0.1926	0.1040	1.3080	0.9661	0.7048	0.6267	0.4264	7.69 × 10−3
50	20	1.2001	0.7998	0.2004	0.1796	0.0992	1.3065	0.9581	0.7875	0.6984	0.5223	7.61 × 10−6
	50	1.1997	0.8000	0.2005	0.1802	0.1002	1.3060	0.9584	0.7842	0.6975	0.5155	6.85 × 10−6
	100	1.1999	0.8004	0.2001	0.1801	0.1003	1.3060	0.9586	0.7860	0.6968	0.5254	5.89 × 10−6
80	20	1.2000	0.8000	0.2000	0.1800	0.1000	1.3060	0.9590	0.7850	0.6981	0.5231	7.09 × 10−9
	50	1.2000	0.8000	0.2000	0.1800	0.1000	1.3060	0.9590	0.7850	0.6980	0.5230	6.57 × 10−9
	100	1.2000	0.8000	0.2000	0.1800	0.1000	1.3060	0.9590	0.7849	0.6980	0.5231	7.87 × 10−9
		1.2000	0.8000	0.2000	0.1800	0.1000	1.3090	0.9590	0.7850	0.6980	0.5230	0

for gs = 100, 200, 300 and 400 respectively. While the respective results in case of Example 2 are presented in

Table 5. Outcomes of the FFO in case of case study 2 for gs = 500.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.4797	0.4980	0.2219	0.1534	0.0854	1.4040	1.0635	0.7878	0.4316	0.6992	9.73 × 10−3
	50	1.4924	0.5094	0.1924	0.1296	0.0978	1.3982	1.0474	0.7031	0.5590	0.5366	7.37 × 10−3
	100	1.4650	0.4977	0.1999	0.1549	0.0968	1.3915	1.0443	0.7970	0.6036	0.5288	7.96 × 10−3
50	20	1.5003	0.4999	0.2003	0.1501	0.1002	1.3966	1.0470	0.7865	0.6284	0.5202	8.14 × 10−6
	50	1.5000	0.5004	0.2006	0.1497	0.1001	1.3958	1.0462	0.7859	0.6314	0.5256	7.23 × 10−6
	100	1.4998	0.4993	0.1997	0.1501	0.1005	1.3958	1.0477	0.7835	0.6300	0.5251	7.33 × 10−6
80	20	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5229	7.93 × 10−9
	50	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7849	0.6279	0.5229	8.00 × 10−9
	100	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6281	0.5232	7.27 × 10−9
		1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5230	0

,

Table 6. Outcomes of the FFO in case of case study 2 for gs = 1000.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.5121	0.5080	0.2013	0.1518	0.1017	1.3837	1.0544	0.7881	0.5550	0.4114	7.74 × 10−3
	50	1.4865	0.5093	0.1888	0.1561	0.1067	1.3986	1.0470	0.7636	0.5171	0.5950	7.18 × 10−3
	100	1.4870	0.4935	0.2092	0.1558	0.0952	1.3938	1.0710	0.7631	0.6316	0.3378	7.33 × 10−3
50	20	1.4999	0.4999	0.2002	0.1498	0.0997	1.3958	1.0465	0.7853	0.6402	0.5186	9.35 × 10−6
	50	1.5000	0.5000	0.1996	0.1497	0.1005	1.3962	1.0457	0.7846	0.6333	0.5209	7.64 × 10−6
	100	1.5004	0.5000	0.2000	0.1503	0.1001	1.3960	1.0468	0.7865	0.6279	0.5218	6.67 × 10−6
80	20	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6279	0.5232	8.84 × 10−9
	50	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6281	0.5228	7.82 × 10−9
	100	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6279	0.5231	6.59 × 10−9
		1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5230	0

,

Table 7. Outcomes of the FFO in case of case study 2 for gs = 1500.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.4952	0.4994	0.1583	0.1452	0.0646	1.3884	1.1014	0.8158	0.4447	1.3259	1.13 × 10−2
	50	1.5074	0.4941	0.1758	0.1622	0.0920	1.3972	1.0556	0.8331	0.6290	0.6939	7.44 × 10−3
	100	1.4872	0.5005	0.2167	0.1631	0.1039	1.3942	1.0575	0.8208	0.6091	0.3720	6.45 × 10−3
50	20	1.5000	0.5007	0.1998	0.1507	0.1001	1.3959	1.0477	0.7800	0.6237	0.5194	8.55 × 10−6
	50	1.5005	0.5002	0.2001	0.1500	0.1006	1.3961	1.0468	0.7842	0.6258	0.5296	8.26 × 10−6
	100	1.4998	0.4999	0.1999	0.1498	0.1000	1.3961	1.0480	0.7828	0.6286	0.5245	7.36 × 10−6
80	20	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6282	0.5230	7.85 × 10−9
	50	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7849	0.6281	0.5231	7.67 × 10−9
	100	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6281	0.5231	7.59 × 10−9
		1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5230	0

and

Table 8. Outcomes of the FFO in case of case study 2 for gs = 2000.

$\mathit{\epsilon }$	ps	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	${\mathit{\lambda }}_4$	${\mathit{\lambda }}_5$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	${\mathit{\mu }}_3$	${\mathit{\mu }}_4$	${\mathit{\mu }}_5$	$\mathbf{\Psi }$
20	20	1.5137	0.5157	0.2028	0.1598	0.0820	1.4024	1.0210	0.5951	0.6559	1.1507	1.00 × 10−2
	50	1.5007	0.5144	0.1893	0.1648	0.1178	1.3930	1.0380	0.7978	0.6760	0.3082	7.38 × 10−3
	100	1.4934	0.4925	0.1921	0.1622	0.0968	1.3956	1.0293	0.8280	0.7062	0.7294	7.37 × 10−3
50	20	1.5011	0.4997	0.2001	0.1512	0.0994	1.3959	1.0466	0.7878	0.6249	0.5296	8.12 × 10−6
	50	1.5000	0.5003	0.1995	0.1499	0.1000	1.3960	1.0452	0.7824	0.6235	0.5258	8.19 × 10−6
	100	1.4998	0.5000	0.2000	0.1506	0.1000	1.3962	1.0479	0.7853	0.6287	0.5294	7.47 × 10−6
80	20	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5234	7.03 × 10−9
	50	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5232	6.98 × 10−9
	100	1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5231	7.67 × 10−9
		1.5000	0.5000	0.2000	0.1500	0.1000	1.3960	1.0470	0.7850	0.6280	0.5230	0

for gs = 500, 1000, 1500 and 2000. The FFO achieves the MSE values of 6.54 × 10⁻³, 1.04 × 10⁻⁵ and 1.35 × 10⁻⁶ for 20 dB, 50 dB and 80 dB respectively for gs = 200 in case study 1. While the respective results in case study 2 are 7.33E × 10⁻³, 6.67 × 10⁻⁶ and 6.59 × 10⁻⁹ for gs = 1000. These results indicate the convergent and robust performance of the FFO scheme for harmonics parameter estimation.

The learning curves along with convergence plots under different noise levels for case study 1 are given in Figure 3, Figure 4, Figure 5 and Figure 6 for gs = 100, 200, 300 and 400 respectively while for case study 2, the learning curves and convergence plots are provided in Figure 7, Figure 8, Figure 9 and Figure 10 for gs = 500, 1000, 1500 and 2000 respectively. The Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 consist of seven subfigures (a–g). The learning curves corresponding to the first, third, fifth, seventh and eleventh harmonic amplitude are given in the subfigures (a–c) for noise 20 dB, 50 dB and 80 dB respectively. While, the respective learning curves corresponding to the first, third, fifth, seventh and eleventh harmonic phase are given in the subfigures (d–f). The overall convergence plots based on the fitness function (11) are provided in the subfigures (g). Looking at the tables and plots it is quite obvious that FFO gives better accuracy if we decrease the signal to noise ratio up to 80 dB. Also from the data tabulated in the Tables, it is clear that while increasing the generation size more fitness is achieved.

5. Conclusions

The swarm intelligence of firefly optimization, FFO, algorithm is exploited for sustainable power systems performance through effectively estimating the harmonics parameters required for mitigating the adverse effects of harmonics on power systems efficiency. The phase and amplitude variables related to the first, third, fifth and eleventh harmonics are successfully estimated through the proposed optimization scheme. The FFO showed robust behavior against different noise conditions, however, increasing the noise level results in relatively decreased value of mean square error based cost function. The accuracy of the FFO scheme enhances for increased particle size, i.e., ps, however, increasing the number of generations, i.e., gs, has very little impact on accuracy level. The accurate and reliable performance of the FFO makes it an attractive alternative to be exploited in micro grids optimization [59,60,61,62]. Moreover, the application of some new algorithms like, Political optimization [63], Harris Hawk optimization [64], Equilibrium optimizer [65], slime mould algorithm [66] and marine predators algorithm [67] to solve parameter estimation problem of power system harmonics looks promising future research direction.