Journal of Electrical and Computer Engineering

Volume 2017, Article ID 7621413, 10 pages

https://doi.org/10.1155/2017/7621413

## Predicting Harmonic Distortion of Multiple Converters in a Power System

Department of Electrical and Electronics Engineering, Electrical Systems and Optics Research Group, University of Nottingham, Nottingham, UK

Correspondence should be addressed to P. M. Ivry; gn.ude.udn.liam@yrvieyerp

Received 5 March 2017; Accepted 24 April 2017; Published 5 June 2017

Academic Editor: Raj Senani

Copyright © 2017 P. M. Ivry et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Various uncertainties arise in the operation and management of power systems containing Renewable Energy Sources (RES) that affect the systems power quality. These uncertainties may arise due to system parameter changes or design parameter choice. In this work, the impact of uncertainties on the prediction of harmonics in a power system containing multiple Voltage Source Converters (VSCs) is investigated. The study focuses on the prediction of harmonic distortion level in multiple VSCs when some system or design parameters are only known within certain constraints. The Univariate Dimension Reduction (UDR) method was utilized in this study as an efficient predictive tool for the level of harmonic distortion of the VSCs measured at the Point of Common Coupling (PCC) to the grid. Two case studies were considered and the UDR technique was also experimentally validated. The obtained results were compared with that of the Monte Carlo Simulation (MCS) results.

#### 1. Introduction

Controllable small distribution networks containing closed assemblage of distributed generators, Renewable Energy Systems (RES), storage systems, and loads are becoming increasingly common in Electrical Power Systems (EPS). The optimization of RES, such as wind turbines and photovoltaics, has been successful thus far. This can be attributed to the advancement and use of power electronic converters, which help to achieve higher power and voltage level operation in wind turbines and photovoltaics.

In recent years, the Voltage Source Converter (VSC) is popularly used and has enjoyed more attention than other converters due to its better controllability and fast switching responses [1, 2]. Nevertheless, VSCs generate harmonic voltages and currents that are transmitted to the rest of the grid. These harmonics may cause malfunction or damage of the power system and equipment on the system.

On an EPS, harmonics can significantly increase and become difficult to predict where variations are present in certain factors like the operating conditions (output power) or system parameters (grid voltage background distortion) [3]. Furthermore, in a case where many VSCs are connected to the EPS, the net effect of the harmonics on the EPS’s Power Quality (PQ) becomes more challenging to predict as harmonics do not add up arithmetically. This and the inherent stochastic nature of harmonics necessitate the use of statistical techniques in predicting the cumulative harmonic distortion level of power converters in an EPS. The use of statistical techniques was also suggested by the IEEE Probabilistic Aspects Task Force on Harmonics for quantifying harmonic distortions [4].

Statistical techniques, such as the Monte Carlo Simulation (MCS) [5], have been extensively used as a common approach in predicting the harmonic distortion level of power converters [6, 7]. However, it requires tens of thousands of simulations to obtain an accurate prediction and this affects the feasibility of using such approach for systems containing large number of VSCs.

In some previous studies [8–14], an analytical approach was used to predict the level of harmonics generated by power converters. The studies represented harmonic vectors as phasors having random amplitudes and angles. The probability density functions (pdfs) of the phasors were obtained and represented in the rectangular coordinates for the convenience of adding phasors. In cases of a large number of harmonic sources/loads the harmonic phasor’s pdfs are then vectorially summed. The studies predicted harmonics in terms of low harmonic orders such as the 3rd, 5th, 7th, and 11th harmonic orders. This may be due to the type of power converter utilized (6/12 pulse converters) and their associated harmonics generated. In a VSC system, the majority of the harmonics appear at the switching frequency and multiplies of the switching frequency [1, 2]. Another way to quantify the harmonics would be to use the Total Harmonic Distortion (THD) [2].

Furthermore, an analytical approach usually entails assumptions to handle the complex interaction of a large number of random harmonic quantities [15, 16] and practical converter systems were usually simplified and represented by mathematical formulas to accommodate the approach as seen in [8–14]. However, other methods which do not require these simplifications or assumptions in designing the systems/converters or in generating random occurrences can be deployed. They include the Monte Carlo Simulation (MCS) [5, 17, 18], Unscented Transform (UT) [19, 20], Point Estimate Method (PEM) [21], and Univariate Dimension Reduction (UDR) [22, 23]. For instance, in Probabilistic Load Flow (PLF) studies, some of these methods have been utilized. The UDR which has proved successful for problems with complex and large number of statistical variation will be utilized in the probabilistic harmonic analysis because of its ability to drastically reduce computational cost, time, and burden. In this study, practical VSC models are simulated with a full switching model.

In VSC design and utilization, the interfacing inductor value depends on the VSC switching frequency to achieve adequate attenuation of harmonics and it is usually chosen with a tradeoff between harmonic attenuation capability and filter cost. Furthermore, various formulas [24–28] can be used in estimating the inductance value with each giving a different filter size, thus contributing to the uncertainty. In most RES, output power is variable. For wind turbine systems, the output power is a function of the wind speed, while, for photovoltaics, output power depends mainly on incident sun rays on the photovoltaic cells. This gives rise to further uncertainty as wind speed/sun light varies with time, day, season, and place. All these uncertainties have to be properly accounted for to ensure harmonic distortion within the EPS is within prescribed limits.

This paper presents a method for predicting current harmonics at the PCC of an EPS in the presence of uncertainties in the filter parameter and operating power of multiple VSCs. The level of harmonic distortion is quantified using statistical evaluators such as the mean and standard deviation. The Univariate Dimension Reduction (UDR) for 3 and 5 points’ approximation is utilized. It is utilized as an alternative to the Monte Carlo Simulation (MCS) in predicting the harmonics distortion level of multiple connected VSCs because of its significant reduction of computational cost, time, and burden associated with the MCS. The results obtained and the accuracy of the UDR for each of the cases were compared with the MCS and presented in the sections below. The results for the UDR technique were also validated using a laboratory experiment.

#### 2. Unscented Transform and Univariate Dimension Reduction

##### 2.1. Unscented Transform (UT)

The UT works by approximating a nonlinear mapping by a set of selected points called sigma points. The sigma points are developed using the moments of the distribution functions pdf, and the weighted average of the sigma points produces the expectation of the mapping [29]. The UT has been utilized in nonlinear problems in electromagnetic compatibility [19, 20, 30] and medical statistics amongst other fields [22]. This method can be used in approximating a continuous distribution function with pdf as a discrete distribution using deterministically chosen points called sigma points and weights such that the moments of both distributions are equal [31]. This is mathematically represented in

contains the location of the abscissas at which the function is to be evaluated while are the weighting coefficients which when multiplied by give an approximation to the integral of . represents the moments of the expectation where implies the mean value and implies variance.

The Gaussian quadrature technique is applied to solve (1) such that the integration points for integrating correspond to the desired sigma points . Hence, for a function assumed as a polynomial, with pdf (weighting function) , the nonlinear mapping for the expectation is given as

The sigma points of the distribution function can be obtained as the roots of its associated orthogonal polynomial when (1) or (2) is integrated using Gaussian quadrature. This method can be easily applied, as most common distributions have known classical orthogonal polynomials associated with them. The associated orthogonal polynomials for some distribution functions can be found in [32].

##### 2.2. Univariate Dimension Reduction (UDR)

The procedure discussed in the section above is only directly applicable when obtaining individual sigma points and weights. For problems involving more than one variable , the simplest technique is based on tensor product of the individual sigma points , giving the number of evaluations as Unfortunately, the technique is plagued by the* curse of dimensionality* problem as the number of variables increases. For example, using 5 sigma points in (3), 5 variables will require 3,125 evaluations while, for 10 variables, . The dimension reduction technique is thus employed in this work.

The dimension reduction [22, 29, 33, 34] is an approximate technique for estimating the statistical moments of an output function. The technique involves an additive decomposition of an -dimensional function involving -dimensional integral into a series sum of -dimensional functions such that . It provides a means of efficiently combining the sigma points and weights for a large number of variables such that the number of evaluation points can be minimized [22, 29]. For the method is referred to as the Univariate Dimension Reduction (UDR) while it is called bivariate dimension reduction for .

The UDR method has been utilized in stochastic mechanics [23] and probabilistic load flow studies [22, 35, 36]; however, it has been less applied to harmonic analysis. The UDR method is briefly described below while a detailed mathematical derivation of the techniques can be found in [33].

With the UDR, the main function, , is decomposed into a summation of one-dimensional functions such thatwhere is the mean of the th random variable.

The resultant function in (4) can easily be integrated as only one randomly distributed variable is present at every instance while the others are held constant at their mean values. The moments of the function are approximately the same as those of the decomposed function as represented in The same procedure is applied for obtaining the higher order moments.

The number of evaluations () required for an -dimensional function using estimation (sigma points) for the UDR is given in [33]For problems where all random variables are symmetrical and (sigma points) is odd, the estimation points for the UDR can be further reduced to (7) while still maintaining the same level of accuracy. For clarity, this will be referred to as reduced UDR (rUDR).

This is clearly a substantial saving in computational time over the alternative of evaluations.

#### 3. Uncertainty Representation

The appropriate design and sizing of the system filter have a great impact on the amount of distortion seen at the grid side of the converter. If not designed properly, it can affect the systems stability. When an RES is integrated to the grid, one of the main concerns is to reduce the harmonic current injected into the grid [24]. In designing harmonic filters for converters, the maximum ripple current should be less than 20% of the rated current [24]. This current is a function of the inductance , the switching frequency, and the DC link voltage. However, to achieve a decrease of the ripple current, the inductance is the most flexible parameter since increasing the switching frequency affects the system efficiency. The value of the inductance can be obtained usingwhere is the switching frequency, is the dc voltage, and is the peak ripple of the maximum rated load current.

In deciding the filter inductance value, another solution is to choose the value relative to the total system inductance. It was suggested in [25, 28] that the total system inductance should be about 10% of the base inductance value. Since no fixed value can be used but that which is determined at the discretion of the designer, the interfacing inductor can be viewed as a stochastic variable such that the optimal size cannot be clearly determined. The value of the inductance will be in a range, howbeit as a percentage of . The value of the inductance can thus be represented using the uniform distribution such that an optimal value can be easily predicted. Hence, (9) can be substituted into (8) to account for this variation.and is calculated aswhere is a factor limiting to be less than .

The VSC harmonic filter is designed using (8)–(14). The value for the filter capacitor () can be obtained using where is calculated by limiting the total reactive power required by the capacitor to be within 15% of the total rated active power.

It is important to consider resonant frequency of an LC or LCL filter as they usually require a damping element to maintain stability and to ensure optimum operation [24].

The resonant frequency is usually within .

For an LC filter, the resonant frequency is calculated fromFor an LCL filter, it isThe damping resistor is calculated using [24]Figure 1 shows the bode plot of the designed filter and highlights the effectiveness of the filter.