Keywords

1 Introduction

In the modern power system, the use of microprocessor-based controllers and power electronic-based devices is enlarged. This leads to power quality problems and harmonic pollution [1]. The harmonics components will result in some adverse effects such as malfunction of relays and heating of components. To reduce harmonic pollution, it is necessary to determine harmonic parameters. The harmonic parameters include amplitude (A), phase angle (φ) and frequency (f). With these estimated parameters, the harmonics are eliminated by connecting some devices to the power system [2, 3].

Several techniques such as conventional, optimization, neural networks (NN) and hybrid algorithms have been implemented for harmonics estimation and tested. The discrete Fourier transform (DFT) [4] is one of the old conventional fast feasible algorithm. But, it does not give actual results and unstable for a few undesirable conditions. The fast Fourier transform (FFT) [5] is an advanced extension of the discrete Fourier transform (DFT). The FFT is the best technique for a static signal, and it does not perform well for time-varying signals. Kalman filtering method [6] is a simple way to estimate parameters. But it needs prior statistics of the signal. The least square techniques RLS [7] and LMS [8] are the best algorithms used so far. The values obtained by these estimation algorithms are closer to the actual values. But, these have certain drawbacks in terms of inaccuracies due to the existence of some variable parameters and noise. Some optimization techniques PSO, generic algorithm (GA) and artificial bee colony (ABC) are presented in [9,10,11,12,13]. These are the most effective algorithms for interharmonics, signals with noise and dynamic signals estimation. The estimation by these algorithms is carried out in one phase. So far the hybrid techniques [14,15,16,17,18,19] are considered, and these techniques are derived by combining the conventional least square algorithms with evolutionary algorithms and NN-based algorithms. The main objective of integrating the least square algorithms with other techniques is to reduce the error and to improve the accuracy of the estimation. The main drawback of these hybrid approaches is the estimation carried out in two phases. The amplitudes of harmonics are estimated by the least square technique, and the fundamental and other harmonic frequency components are estimated by other techniques.

In this paper, harmony search (HS) algorithm is implemented to estimate the parameters of the harmonics directly based on curve fitting approach by search-based mechanism. This algorithm is the best choice in terms of accuracy, computational time and convergence.

2 Problem Formulation

The general representation of distorted voltage and/or current signals of known fundamental frequency f is

$$x\left( t \right) = \mathop \sum \limits_{h = 1}^{N} A_{h} \sin (\omega_{h} t + \varphi_{h} ) + \mu \left( t \right)$$
(1)

In Eq. (1), N = total number of harmonics, \(\omega_{h} = h.2\pi f_{0}\) (\(f_{0}\) = fundamental frequency), \(\mu \left( t \right)\) = Additive noise. The discrete time version of the above equation can be written as

$$x\left( k \right) = \mathop \sum \limits_{h = 1}^{N} A_{h} \sin (\omega_{h} kT_{s} + \varphi_{h} ) + \mu \left( k \right)$$
(2)

In Eq. (2), \(T_{s}\) = Sampling period. The parameters of the harmonics have been estimated from the discrete time signal. Let the estimated parameters be \(A_{h}^{\prime } \,{\text{and}}\,\varphi_{h}^{\prime }\). The distorted signal with estimated parameters is represented as

$$x\left( k \right)^{\prime } = \mathop \sum \limits_{h = 1}^{N} A_{h}^{\prime } \sin (\omega_{h} kT_{s} + \varphi_{h}^{\prime } )$$
(3)

For estimation of exact parameters, an error is evaluated using Eq. (4) given by

$${\text{Error}},d\left( k \right) = x\left( k \right) - x\left( k \right)^{\prime }$$
(4)

The objective function is framed with the help of error signal to estimate the parameters of harmonic signal given by

$$J = \hbox{min} \mathop \sum \limits_{k = 1}^{N} d\left( k \right)^{2}$$
(5)

In this paper, the objective function is minimized to estimate accurate parameters of harmonics by the harmony search algorithm whose details are provided in below section.

3 Harmony Search (HS) Optimization Algorithm

Harmony search optimization (HS) is a population-based algorithm proposed by Geem from music improvisation concept [20,21,22,23]. The first step in this algorithm is to initialize the optimization process for the objective function \(f\left( {x_{1} , \ldots ,x_{n} } \right)\) with parameters such as range of variables \(a_{i} \le x_{i} \ge b_{i} ,i = 1,2, \ldots ,n,\) harmony memory (HM) vector size (HMS), harmony memory rate coefficient (HMRC) and pitch adjusting rate (PAR). Next is to initialize the vector of harmony memory (HM). For this, randomly pick HMS number of vectors \(\left( {x_{1} , \ldots ,x_{n} } \right)\) and put them in the HM vector corresponding to the dominant values \(f\left( {x_{1} , \ldots ,x_{n} } \right)\):

$${\text{HM}} = \left[ {\begin{array}{*{20}c} {x_{1}^{1} , \ldots ,x_{n}^{1} } & | & {f\left( {x^{1} } \right)} \\ \vdots & | & \vdots \\ {x_{1}^{\text{HMS}} , \ldots ,x_{n}^{\text{HMS}} } & | & {f\left( {x^{\text{HMS}} } \right)} \\ \end{array} } \right].$$

The next step is to update agents of HS \(\varvec{x}^{\prime } = \varvec{x}_{1}^{\prime } , \ldots ,\varvec{x}_{\varvec{n}}^{\prime }\). For each \(i = 1,2, \ldots ,n\) the element \(x_{i}^{\prime }\) is chosen. With the probability level to HMCR, from the total numbers \(x_{i}\) obtained in the vector of HM and with the probability level to 1—HMCR, randomly from the presumed limit \(a_{i} \le x_{i} \ge b_{i}\). If, in the prior step, the element \(x_{i}^{\prime }\) is chosen from the vector of HM then, with the probability level to PAR, adjust the element \(x_{i}^{\prime }\) in this way, \(x_{i}^{\prime } \to x_{i}^{\prime } + \alpha\) (control the pitch of the note), for \(\alpha = bw.u\), where \(bw\) indicates the bandwidth part of the limit of the variables and u is the randomly chosen number from range ∈ [−1, 1] and with the probability level to 1—PAR we have nothing to do. If \(f\left( {x^{\prime}} \right) < f\left( {x^{\text{HMS}} } \right)\) then replace the element \(x^{\prime}\) inside the vector of harmony memory HM instead of the element \(x^{\text{HMS}}\) and modify vector HM corresponding to the advancing values of the objective function. Updating HM vector steps are repeated until the termination criterion is satisfied.

4 Simulation Results

To analyze the performance of the harmony search algorithm for the estimation of harmonic parameters, two case studies with different signals have been taken.

Signal 1: Signal corrupted with noise along with DC component with decaying nature is considered as signal 1. In this case, a stationary signal having harmonics of order fundamental, 3rd, 5th, 7th, 11th is generated in MATLAB. The fundamental frequency (f1) is 50 Hz, 3rd harmonic frequency f3 = 150 Hz, 5th harmonic frequency f5 = 250 Hz, 7th harmonic frequency f7 = 350 Hz, eleventh harmonic frequency f11 = 550 Hz. The amplitude value of the fundamental harmonic component had taken a higher value compared to other harmonic components.

$$\begin{aligned} x\left( t \right) & = 1.5.\sin \left( {\omega t + 80} \right) + 0.5\sin \left( {3\omega t + 60} \right) \\ & \quad + 0.2\sin \left( {5\omega t + 45} \right) + 0.15\sin \left( {7\omega t + 36} \right) \\ & \quad + 0.1\sin \left( {11\omega t + 30} \right) + 0.5\exp \left( { - 5t} \right) + \mu \left( t \right) \\ \end{aligned}$$

where \(\mu \left( t \right)\) = random noise (Tables 1, 2 and Fig. 1).

Table 1 Optimal amplitude values for  signal-1obtained by HS
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Table 2 Optimal amplitude and phase values for  signal-1 obtained by HS
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Fig. 1
figure 1

Estimation plots of signal 1 using PSO and HS

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Signal 2: In this case, a stationary signal having harmonics of order fundamental, 3rd, 5th, 7th, 11th, is generated in MATLAB (Table 3).

Table 3 Optimal amplitude and phase values for  signal-2 obtained by HS
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$$\begin{aligned} x\left( t \right) & = 1.2\sin \left( {2\pi f_{1} t + 75} \right) + 0.2\sin \left( {2\pi f_{3} t + 55} \right) \\ & \quad + 0.2\sin \left( {2\pi f_{5} t + 45} \right) + 0.15\sin \left( {2\pi f_{7} t + 40} \right) \\ & \quad + 0.1\sin \left( {2\pi f_{11} t + 30} \right) + \mu \left( t \right) \\ \end{aligned}$$

5 Conclusions

In this paper, direct curve-based harmonic estimation concept is presented for distorted power system signals, and these harmonic components are accurately estimated using HS algorithm. This method provided direct amplitude and phase values of power system signals of different harmonic contents. The comparisons with PSO showcase the advantages of the HS in estimation.