Harmonic distortion in residential distribution networks: literature review

Figure 2 shows the citation network for the issue harmonic distortion in residential distribution networks. For the sake of space, only some lines of research are described. However, most of the articles of the network are considered in the analysis and discussion, including the isolated ones.

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Figure 2 Chronological citation network related to harmonic distortion in residential distribution networks. 

Several investigations on harmonic distortion in residential distribution networks were performed in the 1990s as is shown in Figure 2. The main reason was the incorporation of a new nonlinear load: CFLs. For instance, in the work by ^{Pileggi et al. (1993)}, residential feeders were simulated with CFLs. That study demonstrates the inverse linear relationship between linear load connected to the network and measured THDV (Total Harmonic Distortion in Voltage). This effect is due to damping of harmonic distortion caused by shunt linear loads. According to Figure 2, the research by Pileggi et al. (1993) led to the work by ^{Emanuel et al. (1994}, 1995)battery chargers, PC's, TV's and electronically ballasted lights. The analysis is based on the determination of the most harmonic susceptible busses and their response to each harmonic frequency. A new expeditive method that takes into account the background harmonic voltage phasor, and an equivalent bus impedance was developed and used to compute the maximum non-linear loads that yields VTHD = 5%, (Voltage Total Harmonic Distortion.

The evolution of a line of research is remarkable, since the same authors reported progress in their work. At first, the distortion caused by CFLs was studied (^{Pileggi et al., 1993}). Afterwards, the distortion caused by a variety of nonlinear loads was analyzed (^{Emanuel et al., 1994})battery chargers, PC's, TV's and electronically ballasted lights. The analysis is based on the determination of the most harmonic susceptible busses and their response to each harmonic frequency. A new expeditive method that takes into account the background harmonic voltage phasor, and an equivalent bus impedance was developed and used to compute the maximum non-linear loads that yields VTHD = 5%, (Voltage Total Harmonic Distortion. Finally, a forecast of THDV in the previously studied networks was estimated, for the next two decades (Emanuel et al. , 1995).

Figure 2 shows that research conducted by ^{Emanuel et al. (1994)}, led to the work by ^{Mansoor et al. (1995a)}, where the harmonic interaction and diversity effect in harmonics produced by a large number of single-phase electronic loads were investigated. An analytic model was developed for a single-phase full-wave rectifier, typically utilized in residential loads. From several case studies, the authors concluded that the constant harmonic current source model leads to overestimation of distortion, since the effect of harmonic interaction (which leads to attenuation or amplification) is omitted. This finding is of paramount importance in the analysis of harmonics in residential distribution networks. However, the model complexity hinders the application to scenarios of mass penetration of nonlinear loads. On this basis, several works investigated the harmonic interaction. For in-stance, ^{Nassif et. al. (2009}, 2012) proposed the equivalent lamp index to characterize the harmonic attenuation on CFLs. Further developments were achieved by Nassif et al. (2010) proposing a method for obtaining suitable nonlinear load models for harmonic load-flow calculations based on extensive field measurements.

Following the line of research derived from ^{Mansoor et al. (1995a}), Salles et al. (2012) proposed a probabilistic bottom-up approach based on Monte Carlo simulation to assess the collective impact of residential nonlinear loads. Residence models were constructed by using probable operating states (on/off) of typical home appliances. Thereby, the impact on harmonic distortion in the secondary distribution network was assessed. Afterwards, aggregated models were obtained to assess the impact on the primary distribution network. The methodology takes into ac-count variability over time of harmonics and the bottom-up approach facilitates the investigation on the influence of market trends and regulations. Nonetheless, the harmonic interaction was not taken into account and extensive input data are required. Jiang et al. (2012) presented applications of the described methodology. For instance, the effect of CFLs on a distribution network was forecasted, according to market trends. Results revealed a 10% growth rate in THDV for the following two years after the study.

Bearing in mind the same line of research, ^{Silva et al. (2015)} presented a methodology to assess unbalanced distribution networks, considering load variation. Despite the simple application and the accounting of cancellation without diversity factors; it provides little understanding of the diversity effect and the harmonic interaction. Concluding this line of research, according to Figure 2, appeared ^{Tang et al. (2017}) with a method to establish harmonic current responsibilities at the PCC of residential networks.

In other line of research, the data collected by ^{Emanuel et al. (1991}, 1993) in extensive surveys, was used by ^{Cavallini et al. (1995)} to propose a deterministic-stochastic approach to calculate the Probability Density Function (PDF) resulting from the sum of harmonic currents in distribution network buses. The novelty of this work lies in considering the variation patterns of harmonics over time from a deterministic component, coupled to a component which takes into account the randomness of distortion. A similar approach was proposed by ^{Stankovic & Marengo (1998)}, using Markov chains to model the stochastic component.

According to Figure 2, a line of research was initiated by ^{Dwyer et al. (1995)}. In that work, the impact of CFLs on harmonic distortion was studied through simulations in time domain to analyze the effect of partial cancellation caused by diversity. Furthermore, simulations in frequency domain were carried out to investigate the propagation of harmonic currents. The authors concluded that a massive penetration of CFLs may cause the violation of harmonic distortion limits allowed by the standards. Afterwards, ^{Mazin et al. (2011)} presented a method to determine the contribution of a residence to harmonic distortion, by using measurements. Findings from this study show that voltage distortion at the Point of Common Coupling (PCC) was mainly due to the system contribution.

In the early 1980s, one of the first concerns regarding the impact of PEVs on power quality of the electric utility appeared, in effect, harmonics generated by PEV battery chargers. In the work by ^{Orr et al. (1982a)}, an analysis of harmonic currents generated by a cluster of PEV battery chargers was carried out. Chargers with a very simple topology consisting of a step-down transformer and a full-wave rectifier were assessed. A Monte Carlo simulation method was proposed, considering battery and charger parameters; start time of charging; initial SOC of batteries; and the number of connected PEVs to a single bus. Results demonstrate that harmonic amplitudes are highly dependent on the start time and the number of PEVs. Orr et al. (1982b, 1984) complemented their work, firstly, by studying the harmonic currents of different types of PEVs, and secondly, with an im-proved approach to predict the harmonic currents affecting the substation transformer, considering the random distribution of PEVs along low voltage feeders. On the basis of these works, ^{Jiang et al. (2014)} introduced more detail in modeling the probabilistic aspects related to PEVs. Moreover, other works analyzed the effect of charging stations (J. ^{Wang et al., 2014}), and fast charging of PEVs on harmonic distortion (^{Lucas et al., 2015}).

According to Figure 2, other line of research derived from the work by ^{Orr et al. (1982a)}, led to the research by ^{Staats et al. (1997b)}. In that work, an analytic method to predict the injected harmonic currents by a concentration of PEV battery chargers was proposed. Such method is quite similar to the one presented by Orr et al. (1982a) in the initial considerations, therefore, it has similar drawbacks such as assuming all PEVs connected to a single bus. Besides, the method was based on a single type of battery charger. Despite these disadvantages, the method was very detailed in the modeling of start time of charging and initial SOC as PDFs. Thus, partial harmonic cancellation due to diversity in phase angles is taken into account. In addition, theoretical calculations without iterations can be achieved, reducing computation time drastically. Based on the same method, besides Monte Carlo simulation, further developments were achieved by Staats et al. (1998) by computing the statistics of harmonic voltages at each bus in a residential network. From the results, the probability that THDV will not exceed the limit established by standards was evaluated.

Based on previous works, some researches focused on the investigation of the impact of PEVs on distribution transformers. For instance, ^{Staats et al. (1997a)} determined the relation between PEVs penetration and substation transformer derating, based on the transformer hot-spot temperature and life expectancy, while also considering the effect of harmonic currents. Following this line of research, ^{Gómez & Morcos (2003)} developed a computer tool to determine the life consumption of distribution trans-formers based on the thermal-aging rate and loss-of-life of the transformer, and the characteristics of connected PEV battery chargers. The impact of PEV chargers on the increase in load on transformers, cables, circuit breakers and fuses was also investigated. On a similar basis, Jouybari et al. (2012) studied the effect of PEV battery charging on transformer derating. To that end, the authors utilized the statistical models and battery charging characteristics investigated by Staats et al. (1997b) to represent randomness in input parameters. Then, a harmonic load-flow was carried out, using a backward/forward sweep-based method. Finally, the effects of harmonics on transformer K-factor and life-span were analyzed.

Analysis of the literature review

From the literature review, the stages to carry out the assessment of harmonic distortion in distribution networks by modeling and simulation were identified. The impact of multiple distributed small harmonic sources was emphasized. In the following sections, stages, models and methods for harmonic propagation analysis are described.

Harmonic source modeling

The electrical model, as well as the operating state model, should be defined to adequately represent harmonic sources Salles et al. (2012). Magnitude and phase angle of harmonic currents drawn by a nonlinear load depend on the electrical model. The operating state model represents time variation of harmonic sources, allowing for modeling uncertainty. Figure 3 shows a classification of harmonic source models.

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Figure 3 Classification of harmonic source models. 

Electrical model

In Figure 4, the models of harmonic sources in frequency domain are depicted. The constant source model (see Figure 4(a)) is represented as a complex current with fixed magnitude and phase angle obtained from the typical harmonic spectrum of the nonlinear load (^{Bonner et al., 1996}). Moreover, the decoupled Norton equivalent is formed with a constant current source, connected in parallel to an admittance, as is shown in Figure 4(b). The constant current source represents harmonics due to the fundamental component of voltage. The admittance allows to consider the attenuation/ amplification, but only for voltages and currents of the same harmonic order (^{Nassif et al., 2010}). The coupled Norton equivalent is shown in Figure 4(c), where the connection of a constant current source and an admittance is depicted, similarly as the decoupled Norton equivalent. In addition, a voltage dependent current source in parallel, represents the deviations caused by the interaction of harmonic currents and voltages of different order (^{Yong et al. 2010}). Finally, in the parametric model, components of the nonlinear load (semiconductors, resistors, capacitors, inductors, etc.) are characterized. Hence, detailed knowledge of the load is required (^{Mansoor et al., 1995a}).

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Figure 4 Models for harmonic sources in frequency domain: (a) constant current source; (b) decoupled Norton; and (c) coupled Norton. 

Modeling of nonlinear loads may also be classified by the way for estimating the model parameters. For instance, the modeling based on measurements (top-down) yields nonlinear load parameters from extensive measurements (^{Nassif et al., 2010}). Moreover, the approach base on components (bottom-up) al-lows to obtain aggregate models, from individual components (Salles et al., 2012). On the other hand, the analytic approach yields expressions from the theoretical examination of the non-linear load operation (^{Mansoor et al., 1995a}). Finally, there are hybrid approaches which combines various types of modeling. For instance, ^{Yong et al. (2010)} combine measurements and analytic examination to obtain the model for full-wave rectifier-based loads.

Operating state model

The operating state may be deterministic when the nonlinear load is considered a source of constant parameters, which represents the steady-state performance. On the other hand, the probabilistic approach models the variation over time of harmonic currents. Thus, the uncertainty in the operation of a nonlinear load may be modeled as a PDF.

Distribution network modeling

The scope of the distribution network model depends on the target of the harmonic analysis and the availability of network data. In the literature, four main aspects have been taken into account. The first one is the unbalance. In this regard, considering unbalance implies modeling of multiphase networks, including impedances of each phase and the neutral conductor. Otherwise, only the single-phase representation of the network is modeled. The second main aspect is the model of primary and secondary distribution. The target of the harmonic analysis defines whether both, the primary distribution network (between ~ 1 kV and ~ 69 kV), and the secondary distribution network (less than ~ 1 kV), should be modeled. The third aspect is the distribution transformer model, given the fact that several harmonic studies focus on assessing the impact on distribution transformers. Therefore, specific models for transformers are developed to represent suitably their frequency response. The last aspect is the linear load models, which may be aggregated or detailed, and they should properly represent the effect on the network frequency response.

Harmonic sources penetration modeling

The level of penetration of harmonic sources may be deterministic, when it is approximately estimated, based on knowledge of the specific network. In the case of new nonlinear loads, certain levels of penetration, which are of particular interest, can be defined. For instance, the scenario in which the THDV limit (usually 5%) is exceeded in one or more buses is of particular interest. Moreover, probabilistic approaches model the level of penetration based on statistics from trends in the use of specific nonlinear loads, in a particular place. For future scenarios, these probabilistic models are used to forecast the level of penetration, based on market trends of the new nonlinear load to be studied.

Analysis and simulation of the harmonic propagation

Figure 5 presents a classification of methods for analysis and simulation of harmonic propagation in power systems. In this section, a brief description of each method of Figure 5 is presented.

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Figure 5 Classification of methods for the simulation of harmonic propagation. 

Time domain methods

In time domain, the conventional solution (brute force) and the fast periodic steady-state solution are distinguished (^{Medina et al., 2013}). In these methods, voltage and current waveforms are obtained by integrating the differential equations that describe the steady-state of the system. Then, harmonic components of the waveforms are computed, by using Fourier analysis.

The computational effort in the conventional solution increases exponentially with the size of the system. In the fast periodic steady-state approach, the convergence of the solution is accelerated, by applying the Newton-Raphson method to obtain the steady-state.

Frequency domain deterministic methods

In the direct method, the harmonic currents are calculated at each bus, from the fundamental voltage and the characteristics of nonlinear loads. Then, bus harmonic voltages are computed, from the system admittance matrix, and the harmonic currents. For the iterative harmonic analysis, nonlinear loads are modeled as voltage dependent current sources. Harmonic voltages at buses are calculated in each iteration. Then, new harmonic cur-rents are computed with the obtained harmonic voltages. The process is repeated until the solution converges (^{Medina et al., 2013}).

The complete harmonic load flow was introduced by ^{Xia & Heydt (1982)}, based on the Newton-Raphson algorithm. The solution is obtained by using the set of nonlinear equations of the conventional load-flow, together with the equations related to nonlinear loads. In a further improvement, namely, the multi-phase harmonic analysis, the three phases and neutral conductors of the network are modeled. Hence, uncharacteristic harmonics caused by unbalance are considered. The dynamic harmonic domain is another derivation of the harmonic load-flow, which includes interharmonics and unbalanced networks (^{Medina et al., 2013}).

The coupled admittance matrix is a non-iterative approach that takes advantages of the coupled admittance model of electronic devices (^{Sun et al., 2007}). Being this model linear, the equations of harmonic voltages and currents, associated with nonlinear loads and with the network are solved simultaneously. On the other hand, the backward/ forward sweep, proposed by ^{Teng & Chang (2007)}, is applied in two steps. Firstly, a backward sweep allows to calculate branch harmonic currents in a radial network. Then, a forward sweep is used to compute harmonic voltages at buses, from previously obtained harmonic currents.

Frequency domain probabilistic methods

Probabilistic methods can be classified as analytic or based on Monte Carlo simulation. In analytic methods based on the deterministic direct approach, magnitude and phase angles of harmonic currents are represented with Joint Probability Density Functions (JPDFs). Afterwards, JPDFs of harmonic voltages are computed (^{Carpinelli et al., 2001}). Other analytic methods are achieved by linearizing the complete harmonic load-flow around mean values of harmonic voltages and currents. On the other hand, the Monte Carlo-based approaches involve the simulation of multiple scenarios of harmonic penetration (^{Caramia et al., 1994}). In these methods, values of random variables are generated from PDFs. Then, a deterministic harmonic load-flow is run and the results are stored. The procedure is repeated until accurate results are obtained. Finally, the statistics of the output variables are computed.

Frequency domain possibilistic methods

A direct method of possibilistic harmonic load-flow was pro-posed by ^{Romero et. al. (2011b}, 2012). The method uses possibility functions to represent input variables, and results are re-ported as possibility functions of harmonic voltage magnitudes. Another possibilistic method was presented by Dobrc et al. (2015). This hybrid approach is based on fuzzy logic and Monte Carlo simulation. Its application is focused on distribution net-works.

Hybrid methods (time-frequency)

In hybrid methods, network impedances and invariant linear loads are modeled in frequency domain, whereas nonlinear and variant loads are modeled in time domain. Harmonic voltage at buses is calculated by using an iterative procedure (^{Medina et al., 2013}).

Table 2 shows the characteristics of each methodological contribution to the assessment of harmonic distortion in residential distribution networks and to PEVs harmonic impact assessment.

Table 2 Characteristics of the methodologies for harmonic distortion assessment in distribution networks 

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