Minimizing Harmonic Distortion in Power System with ...

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Minimizing Harmonic Distortion in Power System with Optimal Design of

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Hybrid Active Power Filter using differential evolution

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Partha P Biswas1, P. N. Suganthan1, Gehan A J. Amaratunga2 1School

Nanyang Technological University, Singapore [email protected], [email protected]

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of Electrical and Electronic Engineering

2Department

of Engineering, University of Cambridge, UK [email protected]

Abstract: Hybrid active power filter (HAPF) is an advanced form of harmonic filter combining

advantages of both active and passive filters. In HAPF, selection of active filter gain, passive inductive and capacitive reactances, while satisfying system constraints on individual and overall voltage and current harmonic distortion levels, is the main challenge. To optimize HAPF parameters, this paper proposes an approach based on differential evolution (DE) algorithm called L-SHADE. SHADE is the success history based parameter adaptation technique of DE optimization process for a constrained, multimodal non-linear objective function. L-SHADE improves the performance of SHADE with linearly reducing the population size in successive generations. The study herein considers two frequently used topologies of HAPF for parameter estimation. A single objective function consisting of both total voltage harmonic distortion (VTHD) and total current harmonic distortion (ITHD) is formulated and finally harmonic pollution (HP) is minimized in a system comprising of both non-linear source and non-linear loads. Several case studies of a selected industrial plant are performed. The output results of L-SHADE algorithm are compared with a similar past study and also with other well-known evolutionary algorithms. Keywords: Hybrid active power filter · Total harmonic distortion · Power quality · Harmonic pollution · L-SHADE algorithm

Nomenclature 𝑅𝑆ℎ , 𝑋𝑆ℎ

:

Transmission system resistance and inductive reactance in ohms at harmonic ‘h’

𝑅𝐿ℎ , 𝑋𝐿ℎ

:

Load resistance and inductive reactance in ohms at harmonic ‘h’

𝐺𝐿ℎ , 𝐵𝐿ℎ

:

Load conductance and susceptance in mho at harmonic ‘h’

K

:

Controllable feedback gain of hybrid active power filter in ohms

XL, XC

:

Fundamental inductive and capacitive reactance in ohms of the passive filter

IS

:

Root mean square (RMS) value of supply current in amperes

IL

:

RMS value of load current in amperes

VS

:

RMS value of supply voltage (line-to-neutral)

VL

:

RMS value of the load voltage (line-to-neutral)

PL

:

Load active power per phase

1.

Introduction

In pursuit of high efficiency in energy usage, more flexibility and controllability, new semiconductor technologies are often being introduced in power system. Unfortunately, these devices draw non-sinusoidal current which has major impact on power quality, an important qualitative indicator to both the electric utility company and the end user. Nonlinear loads are typically characterized by two types of harmonic sources, the harmonic current source and the harmonic voltage source. With increased penetration of non-linear loads, the degradation of power factor (PF) of the load, increase in transmission-line losses, and hence reduction of transmission network efficiency etc. are all expected [2]. Importantly, the level of harmonic distortion in distribution networks significantly rises. Passive power filters (PPF), active power filters (APF) or hybrid active power filters (HAPF) can all be used to reduce or eliminate harmonics from power system network. The disadvantage of PPF is its inability to modify the 1

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compensation characteristics following the dynamic changes of nonlinear loads. Further, small allowable design tolerances in the values of inductor (L) or capacitor (C) require extra care as little changes in these values modify the filter resonant frequency. An APF, typically consisting of a three phase pulsewidth modulated (PWM) voltage source inverter, overcomes these disadvantages of PPF. In active filter, voltage-source inverter is usually used rather than current source inverter because the former has higher efficiency, lower cost, and smaller size compare to current source inverter. The shunt active filter acts as a harmonic compensator and injects the current in anti-phase with the distortion components present in the line current, while the series active filter acts as a harmonic isolator [3]. However, using APF alone in circuit asks for high power rating of the converters. Consequently, HAPF combining elements of both active and passive filter has been developed to solve the problem of high size and cost associated with the standalone active filter. In spite of the benefits HAPF offers, passive power filter (PPF) is still a popular mean for compensation and correction. Extensive research studies have been carried out on PPF and various optimization algorithms like particle swarm optimization (PSO) [4], ant direction hybrid DE [5], sequential quadratic programming [6], bacterial foraging [7] etc. have all been applied for design of PPF. However, a limited number of literatures could be traced on application of meta-heuristic optimization techniques on HAPF design. One probable reason is design complexity, specifically in finding optimum gain K which is either pre-defined or decided by trial and error method [1]. The complexity escalates when the source also injects harmonics into the system alongwith the loads. Authors in [8], [9] and [10] suggested design techniques of HAPF without adding the possible scenario when source non-linearities could be present. In this paper, HAPF configuration of ‘APF in series with shunt passive filter’ under circumstance of both load and source non-linearities is examined for optimization of the parameters employing algorithm L-SHADE, the linear population size reduction technique of success history based parameter adaptation for DE [11,12]. DE and its variants have emerged as one of the most competitive and versatile family of the evolutionary computing algorithms and have been successfully applied to solve numerous real world problems [13]. Classical DE requires only 3 control parameters – scale factor (F), crossover rate (CR) and population size (Np). As the selected control parameters remain same throughout iterations in classical DE, initial values are to be carefully chosen. However, in adaptive DE like SHADE, F and CR values are adapted based on some form of learning from successes/failures of past generations. In L-SHADE, population size is continually reduced following a linear function. At each successive generation, certain percentage of individuals with worst fitness values are removed from the population. So, it can be inferred that performance of L-SHADE hardly depends on initial selection of control parameters provided a moderate size of Np is selected. The aforementioned features make L-SHADE superior in fast and accurate convergence to an optimal solution. L-SHADE exhibited the best competitive performance among non-hybrid algorithms at the CEC 2014 competition on real parameter single-objective optimization [14]. For the HAPF problem under study, a single objective function of optimization is judiciously formed in applying L-SHADE algorithm which can minimize both VTHD and ITHD in single run satisfying defined constraints. Comparative study is carried out with results obtained applying other evolutionary algorithms like genetic algorithm (GA), particle swarm optimization (PSO), artificial bee colony (ABC), differential evolution (DE), success history based adaptive differential evolution (SHADE) and also a past study performed under similar conditions adopting Fortran feasible sequential quadratic programming (FFSQP) [1]. The study with L-SHADE algorithm is further extended to ‘combined series APF and shunt passive filter’ configuration as this type of HAPF is also used to compensate reactive power, harmonics and unbalanced loads at medium voltage level of the power system [15].

2.

Hybrid active power filters – circuit analysis

Two popular topologies of HAPF, used for compensation in industrial power systems, are shown in Fig. 1(a) and 1(b) without indicating the interface transformer. In first configuration of ‘APF in series with shunt passive filter’, the active filter improves the performance of passive filter by injecting harmonic current into it to cancel the load harmonics. Additionally, fundamental system voltage is dropped across the passive filter, thereby reducing the voltage rating of the APF [16]. Second configuration is for ‘combined series APF and shunt passive filter’, where series APF offers high impedance to supply harmonics and forces the harmonic current to flow to passive filter, thus allowing lower current rating of the APF. In power system, point of common coupling (PCC) is usually taken as the point closest to the user where the system owner or operator could offer service to another user [17]. Here it is identified as the point where other linear loads are connected to the system. The passive filter, represented with inductive and capacitive reactances XL and XC respectively, is a set of tuned filters or simply a single tuned filter according to the system requirements. Fig. 2 indicates the single-phase equivalent circuit applicable for both HAPF configurations at fundamental frequency [18]. Subscript ‘1’ to a parameter in the diagram denotes the value of that parameter at fundamental frequency. The single-phase equivalent circuits of the two configurations differ at harmonic frequencies as can be observed in Fig. 3 and Fig. 4 representing ‘APF in series with shunt passive filter’ and ‘combined series APF and shunt passive filter’ respectively. The difference is because of the location of active power filter and the way it reacts 2

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to supply harmonics. Active power filter is considered as a controlled voltage source (say 𝑉𝐴𝐹 ) and its performance is realized by injecting a voltage harmonic waveform at its terminals proportional to the harmonic component of the supply current, i.e. 𝑉𝐴𝐹 = 𝐾𝐼𝑆ℎ . Filter gain K is designed to offer zero impedance at fundamental frequency. It implies the active filter component acts as a virtual harmonic resistor [1]. This study focuses on optimization of K, XL and XC under scenarios with non-linearities arising out of both the source and the load. Source harmonic voltage and current non-linearities are accounted in 𝑉𝑆ℎ and 𝐼𝑆ℎ respectively, and those of load in 𝑉𝐿ℎ and 𝐼𝐿ℎ . Thevenin voltage source representing the utility supply voltage and the harmonic current source representing the nonlinear load are [1], 𝑣𝑆 (𝑡) = ∑ 𝑣𝑆ℎ (𝑡)

(1)

ℎ

𝑖𝐿 (𝑡) = ∑ 𝑖𝐿ℎ (𝑡)

(2)

ℎ

10 Source

Source

VSh

VSh

RSh

RSh

ISh

XSh

XSh PCC

ISh

ILh

VLh

PCC

VLh

Active power filter

XL

XC

ILh XC

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Active power filter

Linear loads

Non-linear loads

Linear loads

(a) Config.1: APF in series with shunt passive filter

XL

Non-linear loads

(b) Config.2: Combined series APF and shunt passive filter

Fig. 1: Circuit configurations of HAPF jXS1

RS1

VL1

IS1

jXL

RL1

-jXC

jXL1

+

VS1

IL1 -

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Fig. 2: Single-phase equivalent circuit at fundamental frequency (h =1) jXSh

RSh

VLh

ISh jhXL

RLh +

-jXC/h

VSh

+

ILh jXLh

KISh

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-

Fig. 3: Single-phase equivalent circuit for config.1 ‘APF in series with shunt passive filter’ at harmonic frequencies (h ≥2)

3

jXSh

RSh

KISh

VLh

-

+ ISh

jhXL

RLh

-jXC/h

jXLh

+ ILh

VSh -

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Fig. 4: Single-phase equivalent circuit for config.2 ‘combined series APF and shunt passive filter’ at harmonic frequencies (h ≥2)

The h-th harmonic source impedance is 𝑍𝑆ℎ = 𝑅𝑆ℎ + 𝑗𝑋𝑆ℎ

(3)

𝑍𝐿ℎ = 𝑅𝐿ℎ + 𝑗𝑋𝐿ℎ

(4)

𝑌𝐿ℎ = 𝐺𝐿ℎ − 𝑗𝐵𝐿ℎ

(5)

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The h-th harmonic load impedance is

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Analysis of equivalent circuit in Fig.3 for config.1 ‘APF in series with shunt passive filter’ yields following equations for compensated utility supply current and load voltage respectively at harmonic ‘h ≥2’

Hence, the load admittance is

𝐼𝑆ℎ =

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𝑉𝐿ℎ =

C+jD 𝐸+𝑗𝐹 C+jD

(6) (7)

Analysis of equivalent circuit in Fig.4 for config.2 ‘combined series APF and shunt passive filter’ yields following equations for compensated utility supply current and load voltage respectively at harmonic ‘h ≥2’ 𝐼𝑆ℎ =

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𝐴+𝑗𝐵

𝑉𝐿ℎ =

𝐴+𝑗𝐵 C+jD′ 𝐸+𝑗𝐹′ C+jD′

(8) (9)

where, 𝐴 = 𝑉𝑆ℎ 𝑅𝐿ℎ − 𝐼𝐿ℎ 𝑋𝐿ℎ (ℎ𝑋𝐿 −

𝑋𝐶 ) ℎ

𝑋𝐶 𝑋𝐶 ) + 𝐼𝐿ℎ 𝑅𝐿ℎ (ℎ𝑋𝐿 − ) ℎ ℎ 𝑋𝐶 𝐶 = 𝑅𝑇𝐿ℎ + 𝐾𝑅𝐿ℎ − (𝑋𝐿ℎ + 𝑋𝑆ℎ ) (ℎ𝑋𝐿 − ) ℎ 𝑅𝑇𝐿ℎ = 𝑅𝑆ℎ 𝑅𝐿ℎ − 𝑋𝑆ℎ 𝑋𝐿ℎ 𝑋𝐶 𝐷 = 𝑋𝑇𝐿ℎ + 𝐾𝑋𝐿ℎ + (𝑅𝐿ℎ + 𝑅𝑆ℎ ) (ℎ𝑋𝐿 − ) ℎ 𝑋𝑇𝐿ℎ = 𝑅𝐿ℎ 𝑋𝑆ℎ + 𝑅𝑆ℎ 𝑋𝐿ℎ 𝑋𝐶 𝑋𝐶 𝐸 = 𝑉𝑆ℎ [𝐾𝑅𝐿ℎ − 𝑋𝐿ℎ (ℎ𝑋𝐿 − )] + 𝐼𝐿ℎ 𝑋𝑇𝐿ℎ (ℎ𝑋𝐿 − ) ℎ ℎ 𝑋𝐶 𝑋𝐶 𝐹 = 𝑉𝑆ℎ [𝐾𝑋𝐿ℎ + 𝑅𝐿ℎ (ℎ𝑋𝐿 − )] − 𝐼𝐿ℎ 𝑅𝑇𝐿ℎ (ℎ𝑋𝐿 − ) ℎ ℎ 𝑋 𝐶 𝐷′ = 𝑋𝑇𝐿ℎ + 𝐾𝑋𝐿ℎ + (𝐾 + 𝑅𝐿ℎ + 𝑅𝑆ℎ ) (ℎ𝑋𝐿 − ) ℎ 𝑋 𝑋𝐶 𝐶 𝐹 ′ = 𝑉𝑆ℎ [𝐾𝑋𝐿ℎ + (𝐾 + 𝑅𝐿ℎ ) (ℎ𝑋𝐿 − )] − 𝐼𝐿ℎ 𝑅𝑇𝐿ℎ (ℎ𝑋𝐿 − ) ℎ ℎ 𝐵 = 𝑉𝑆ℎ (𝑋𝐿ℎ + ℎ𝑋𝐿 −

(10) (11) (12) (13) (14) (15) (16) (17) (18) (19)

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1 2 3 4 5 6 7 8 9 10 11

A careful review of equations (6) and (8) reveals that compensated utility supply harmonic current, 𝐼𝑆ℎ is inversely related to gain K. The active filter acts as an ‘obstructing resistor’ which impedes the harmonic current produced by source non-linearities 𝑉𝑆ℎ . Further it also acts as a ‘damping resistor’ to harmonic current 𝐼𝐿ℎ and can entirely attenuate the resonance between the shunt passive filter and the source impedance [18]. On the other hand, compensated PCC voltage 𝑉𝐿ℎ has an additional proportional K term in the numerator alongwith inverse term in the denominator as can be seen in equations (7) and (9). So, the objective of optimization is to find a suitable value of gain K such that the value is high enough to reduce both 𝐼𝑆ℎ (i.e. ITHD) and 𝑉𝐿ℎ (i.e. VTHD), however not so high that it adversely affects 𝑉𝐿ℎ . Formulae for calculation of other system parameters are: Compensated load displacement power factor (DPF),

12 13 14 15

𝐷𝑃𝐹 =

𝑃𝐹 = Transmission loss is given by,

19

Transmission efficiency is calculated as,

𝑃𝐿 𝑉𝐿 𝐼 𝑆

=

√(𝐼𝑆1 2 +∑ℎ≥2 𝐼𝑆ℎ 2 )(𝑉𝐿1 2 +∑ℎ≥2 𝑉𝐿ℎ 2 )

𝑃𝐿𝑂𝑆𝑆 = 𝐼𝑆1 2 𝑅𝑆1 + ∑ℎ≥2 𝐼𝑆ℎ 2 𝑅𝑆ℎ

𝜂=

𝑉𝑇𝐻𝐷 =

𝑃𝐿 𝑃𝐿+𝑃𝐿𝑂𝑆𝑆

(21)

(22)

(23)

√∑ℎ≥2 𝑉𝐿ℎ 2 𝑉𝐿1

(24)

Compensated ITHD for the utility supply current,

24

43 44

(20)

Compensated VTHD at the load terminal,

22

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𝐼𝑆1

𝐺𝐿1 𝑉𝐿1 2 +∑ℎ≥2 𝐺𝐿ℎ 𝑉𝐿ℎ 2

20

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𝐺𝐿1 𝑉𝐿1

Compensated load power factor (PF),

17 18

23

=

where, subscript ‘1’ denotes the fundamental component

16

21

𝑃𝐿1 𝑉𝐿1 𝐼𝑆1

𝐼𝑇𝐻𝐷 =

√∑ℎ≥2 𝐼𝑆ℎ 2 𝐼𝑆1

(25)

Approximated formula for Harmonic Pollution (HP) as per [19], 𝐻𝑃 = √𝑉𝑇𝐻𝐷 2 + 𝐼𝑇𝐻𝐷 2

3.

(26)

L-SHADE algorithm

Differential Evolution (DE), introduced in the year 1996 by Storn and Price, is a powerful population-based stochastic optimization algorithm where probabilistic operators like recombination and mutation facilitate the population members to evolve and improve their fitness. It has been observed that the settings of control parameters which are: the scaling factor (F), the crossover rate (CR) and the population size (Np), and also the chosen mutation / crossover strategies affect the performance of DE [20,21]. Therefore, finding suitable mechanisms for on-line adjustment of control parameters during the search process has drawn attention of many researchers. Subsequently, a novel mutation strategy, JADE [22] and a success history based parameter adaptation scheme, SHADE [11] have been proposed. Recent updates are available in [23]. The present study focuses on L-SHADE [12] optimization algorithm, an extension of SHADE with linear population size reduction (LPSR) strategy, where population size continually decreases according to a linear function. This section provides a brief description of the algorithm. 3.1 Initialization DE optimization process starts with creating an initial population (Np vectors with d dimensions) of candidate solutions of randomly generated values within the feasible bounds (between maximum & minimum) of the decision 5

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vector. Initialization of j-th component of the i-th decision vector is done as:

14 15 16 17 18

Following initialization, mutation operation of DE produces a mutant/donor vector 𝑣𝑖 corresponding to each (𝑡) population member or target vector 𝑥𝑖 in the current t-th generation. Out of a few strategies for mutation, the one adopted in L-SHADE is ‘current-to-pbest/1’:

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(0)

where 𝑟𝑎𝑛𝑑𝑖𝑗 [0,1] is a uniformly distributed random number lying between 0 and 1 and superscript ‘0’ represents initialization; ‘Np’ is the population size and ‘d’ is the dimension of the decision vector. Therefore, i = 1,2,…,Np and j = 1,2,…,d. (0)

The HAPF optimization problem here has 3 decision variables 𝐾, 𝑋𝐶 & 𝑋𝐿 . Therefore, each decision vector 𝑥𝑖

(0)

for i = 1,2,….,Np is formulated as a 3 dimensional vector (i.e. d = 3) with each element (i.e. decision variable) 𝑥𝑖,𝑗 bounded by its range of defined maximum and minimum values. 3.2 Mutation (𝑡)

(𝑡)

𝑣𝑖

(𝑡)

(𝑡)

(𝑡)

(𝑡)

(𝑡)

(𝑡) 𝑅1𝑖

= 𝑥𝑖 + 𝐹𝑖 . (𝑥𝑝𝑏𝑒𝑠𝑡 − 𝑥𝑖 ) + 𝐹𝑖 . (𝑥

(𝑡) ) 𝑅2𝑖

(28)

−𝑥

(𝑡)

The indices 𝑅1𝑖 & 𝑅2𝑖 are mutually exclusive integers randomly chosen from the population range; 𝑥𝑝𝑏𝑒𝑠𝑡 is the top 𝑁𝑝 × 𝑝 (𝑝 ∊ [0,1], and the product is rounded up to nearest integer value) best individuals of current generation. (𝑡) (𝑡) The positive scaling factor 𝐹𝑖 is for scaling the difference vectors at t-th generation. If an element 𝑣𝑖,𝑗 of the mutant vector goes outside the search range boundaries [𝑥𝑚𝑖𝑛,𝑗 , 𝑥𝑚𝑎𝑥,𝑗 ], it is corrected as: (𝑥𝑚𝑖𝑛,𝑗 + 𝑥𝑖,𝑗𝑡 )/2 if 𝑣𝑖,𝑗 < 𝑥𝑚𝑖𝑛,𝑗 ( )

(𝑡)

25 26 27 28

(27)

𝑥𝑖,𝑗 = 𝑥𝑚𝑖𝑛,𝑗 + 𝑟𝑎𝑛𝑑𝑖𝑗 [0,1](𝑥𝑚𝑎𝑥,𝑗 − 𝑥𝑚𝑖𝑛,𝑗 )

𝑣𝑖,𝑗 = {

(𝑡)

(29)

(𝑥𝑚𝑎𝑥,𝑗 + 𝑥𝑖,𝑗𝑡 )/2 if 𝑣𝑖,𝑗 > 𝑥𝑚𝑎𝑥,𝑗 ( )

(𝑡)

3.3 Parameter adaptation (𝑡)

(𝑡)

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At each generation t, each individual uses its own parameters 𝐹𝑖 and 𝐶𝑅𝑖 to generate a new trial vector. Adaptation of these control parameters are performed using following operation:

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Through crossover operation, donor vector 𝑣𝑖 mixes its elements with the target vector 𝑥𝑖 to form the (𝑡) (𝑡) (𝑡) (𝑡) trial/offspring vector 𝑢𝑖 = (𝑢𝑖,1 , 𝑢𝑖,2 , … . . , 𝑢𝑖,𝑑 ). More commonly adopted binomial crossover operates on each variable whenever a randomly generated number between 0 and 1 is less than or equal to the adapted crossover rate (𝑡) 𝐶𝑅𝑖 . For an element, the scheme is expressed as:

(𝑡)

(𝑡)

𝐹𝑖 = 𝑟𝑎𝑛𝑑𝑐(µ𝐹𝑟 , 0.1) (𝑡) (𝑡) 𝐶𝑅𝑖 = 𝑟𝑎𝑛𝑑𝑛(µ𝐶𝑅𝑟 , 0.1) (𝑡)

(𝑡)

where 𝑟𝑎𝑛𝑑𝑐(µ𝐹𝑟 , 0.1) is the value sampled from Cauchy distribution with location parameter µ𝐹𝑟 and scale (𝑡) (𝑡) parameter 0.1; 𝑟𝑎𝑛𝑑𝑛(µ𝐶𝑅𝑟 , 0.1) generates a value following the Normal distribution with mean µ𝐶𝑅𝑟 and (𝑡) variance 0.1. The value sampled from Cauchy distribution at generation 𝑡 is truncated to 1 if 𝐹𝑖 ≥ 1 or (𝑡) regenerated if 𝐹𝑖 ≤ 0. The location parameter of scale factor and mean value of crossover rate are initialized to (0) (0) 0.5 i.e. µ𝐹𝑟 = µ𝐶𝑅𝑟 = 0 and subsequently updated in each generation following weighted Lehmer mean, the detail is in reference [11,12]. 3.4 Crossover (𝑡)

(𝑡)

(𝑡)

48

(30)

(𝑡) 𝑢𝑖,𝑗

={

(𝑡)

𝑣𝑖,𝑗 if 𝑗 = 𝑗𝑟𝑎𝑛𝑑 or 𝑟𝑎𝑛𝑑𝑖,𝑗 [0,1] ≤ 𝐶𝑅𝑖 , (𝑡)

𝑥𝑖,𝑗 otherwise

(31)

6

1 2 3 4 5 6 7 8

where 𝑗𝑟𝑎𝑛𝑑 is a natural number randomly chosen in the range [1, d], with d being the dimension of decision vector. 3.5 Selection Selection process works on ‘survival of the fittest principle’. The selection operator performs fitness checks to determine whether the target (parent) or the trial (offspring) vector survives to the next generation at t+1. The operation can be mathematically represented as: (𝑡)

(𝑡+1)

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𝑥𝑖

={

u𝑖 (𝑡) if 𝑓(u𝑖 (𝑡) ) ≤ 𝑓(𝑥𝑖 ), (𝑡)

𝑥𝑖 otherwise

(32)

where f(.) is the objective function to be minimized. 3.6

Linear population size reduction (LPSR)

It has been observed that selection of scaling factor F and crossover rate CR impact the DE performance. SHADE implements adaptation of these parameters based on their success history. L-SHADE improves the performance of SHADE by introducing a linear function for dynamic reduction of population size Np. After each generation t, the population size in the next generation t+1 is computed by – 𝑁𝑝(𝑡 + 1) = 𝑟𝑜𝑢𝑛𝑑 [(

𝑁𝑝𝑚𝑖𝑛 −𝑁𝑝𝑖𝑛𝑖 𝑁𝐹𝐸𝑚𝑎𝑥

) . 𝑁𝐹𝐸 + 𝑁𝑝𝑖𝑛𝑖 ]

(33)

𝑁𝑝𝑚𝑖𝑛 is set to 4 because mutation strategy adopted in L-SHADE requires minimum 4 individuals. NFE is the current number of fitness evaluations and 𝑁𝐹𝐸𝑚𝑎𝑥 is the maximum number of fitness evaluations. If 𝑁𝑝(𝑡 + 1) < 𝑁𝑝(𝑡), the [𝑁𝑝(𝑡) − 𝑁𝑝(𝑡 + 1)] worst ranking individuals are deleted from the population [12]. An initial population size of 𝑁𝑝𝑖𝑛𝑖 =100 has been considered for the HAPF problem under consideration. Maximum 50000 (𝑁𝐹𝐸max ) fitness evaluations are performed for the objective function.

4.

Application and approach using L-SHADE algorithm

Zobaa [1] in applying FFSQP considered the two stage optimization technique of minimizing VTHD first while satisfying constraints for ITHD, load PF and on the optimized parameters themselves. The resulting sets of decision variables are then used to minimize HP. The above approach may not always lead to optimal solution as can be seen from case study results discussed in subsequent sections of this literature. The HAPF filter parameters to be optimized are 𝐾, 𝑋𝐶 & 𝑋𝐿 and each of these variables is bound within following respective range in terms of Ohmic values. • 0 ≤ 𝐾 ≤ 20 • 0 ≤ 𝑋𝐶 ≤ 10 • 0 ≤ 𝑋𝐿 ≤ 1 To ensure that VTHD and ITHD are within allowable limits i.e. for 𝑉𝑇𝐻𝐷 ≤ 𝑉𝑇𝐻𝐷𝑙𝑖𝑚 and 𝐼𝑇𝐻𝐷 ≤ 𝐼𝑇𝐻𝐷𝑙𝑖𝑚 , the objective function for optimization is formulated as: 𝐻𝑃𝐴𝑃𝑃 = 𝑎𝑏𝑠(𝑉𝑇𝐻𝐷𝑙𝑖𝑚 − 𝑉𝑇𝐻𝐷) + 𝑎𝑏𝑠(𝐼𝑇𝐻𝐷𝑙𝑖𝑚 − 𝐼𝑇𝐻𝐷)

(34)

where, 𝑉𝑇𝐻𝐷𝑙𝑖𝑚 = limitation on VTHD prescribed by IEEE 519-2014 [17] based on system voltage level 𝐼𝑇𝐻𝐷𝑙𝑖𝑚 = limitation on ITHD prescribed by IEEE 519-2014 [17] based on system short circuit ratio The objective of optimization while meeting individual harmonics within IEEE standard limits: Maximize ‘𝐻𝑃𝐴𝑃𝑃 ’ subject to 𝑃𝐹 = 𝑃𝐹𝑔𝑜𝑎𝑙 ± 𝜀 where, 𝑃𝐹𝑔𝑜𝑎𝑙 is the desired power factor and ε is a very small error value (