Fair Power Factor Billing Under Unbalanced and Nonsinusoidal Voltage Supply

Utilities apply an additional fee for medium and large customers with low power factors. However, unfair financial charges may occur in installations subjected to voltage unbalance and harmonic distortion. The objective of this paper is to determine the fairest PF definitions and their corresponding measurement algorithms in the case in which a constant impedance load or an induction motor is supplied with unbalanced and nonsinusoidal voltages. Fairness is defined considering that the meter (built based on a particular definition and measurement method) under nonideal supply should lead to very close values as if it was submitted to an ideal balanced sinusoidal supply. We performed computational simulations to emulate several conditions in which a balanced customer (modeled as a constant impedance load or an induction motor) is charged due to a voltage supply no longer balanced and sinusoidal. We also performed experimental tests with an induction motor subjected to a wide range of unbalanced nonsinusoidal supply conditions to ratify the conclusions drawn from the simulations. Based on the simulation results and the experimental tests, we indicate some power factor definitions and measurement methods that are not significantly affected by voltage unbalance and harmonic distortions. These indicated PF definitions provide the fairest billing for conditions with unbalanced nonsinusoidal voltages.

Under unbalanced and distorted voltage conditions, different power factor (PF) meters may give different readings for the same condition. In [1], it is reported an experience in which an industry's PF dropped from 0.95 to 0.88 after the meter replacement. The new meter would have led to a 4% surcharge in the customer's bill had the technician not noticed this discrepancy. One thing that stands out is that these meters were approved for commercialization and usage. As pointed out by [1], ''Utilities must be able to install any meter in any electrical environment (sinusoidal or nonsinusoidal) with full confidence that they will all give the same readings for the same load. Anything less is unacceptable.'' To achieve such level of confidence, it is necessary to understand the possible causes for measurement divergences.
According to [2], the discrepancies shown in [1] were due to different PF definitions and measurement methods implemented in the commercial meters. It can be noted that although the arithmetical, geometrical (vectorial), positive sequence, and effective PFs discussed in IEEE Standard The associate editor coordinating the review of this manuscript and approving it for publication was Md. Rabiul Islam.
1459 [3] give the same value in balanced sinusoidal conditions, it turns out that they may provide different values in unbalanced nonsinusoidal conditions. The problem of finding a 'correct' general PF definition and measurement method is not a trivial task, and an indication of the most appropriate for each application is still lacking. Most of the countries studied by [4] ' 'have not yet established rules for reactive power billing under nonsinusoidal conditions and keep applying old regulation to systems where voltage and current are not sinusoidal anymore.'' As field research, reference [5] investigated the arithmetic, geometric, effective, fundamental, and modified fundamental PF applied to an arc furnace of a real industry in Taiwan. The authors showed that the selection of different definitions can cause an impact up to 1% in the customer's bill. Although this percentage may seem small, it represents a huge impact for large customers with expensive monthly revenues. However, the study of [5] is restricted to industries that use arc furnaces, and it does not consider different supply conditions and measurement methods.
On the other hand, reference [4] compared the fundamental reactive power (Q 1 ) and the non-active power (N ) under several supply conditions and with different measurement VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ methods. Digital simulations were performed to find out which definition is the fairest for billing purposes of a residential load (at 9 pm and 12 pm) under nonsinusoidal voltage supply. According to [4], the definition less sensible to the voltage harmonic distortion is the fairest one, which in their simulations was Q 1 . It is also pointed out that Q 1 can be measured with simple digital techniques. The analysis, however, was restricted to a residential load supplied by nonsinusoidal voltages, and it did not consider unbalanced conditions. Among all PF definitions, the most important seems to be the effective PF (PF e ). In fact, it has been investigated by several researchers [3], [6]- [11] and it is recommended by IEEE Standard 1459-2010 [3] for unbalanced sinusoidal conditions. For unbalanced nonsinusoidal conditions, references [9], [10], [12], [13] ratify PF e as a measure of the energy transmission efficiency. Furthermore, reference [6] showed that PF e values of balanced constant impedance loads are not significantly affected by voltage unbalance.
From the foregoing discussion, it can be noticed that evaluation of one important load has not been done yet. According to [14]: ''there are 85 million large electrical motors in the community market. These consume 65-70% of the energy used within industry.'' Therefore, the evaluation of PF billing fairness for balanced three-phase induction motors (TIMs) under unbalanced nonsinusoidal supply voltages is an important issue.
The objective of this paper is to seek the fairest PF definitions and their corresponding measurement algorithms for metering the PF of a balanced customer (modeled as a constant impedance load or an induction motor) submitted to a wide range of unbalanced and nonsinusoidal voltage conditions. Here, the fairness notion is in the sense that the meter (built based on a particular definition and measurement method) under nonideal supply should lead to very close values as if it was submitted to an ideal balanced sinusoidal supply. First, the sensitivity of the main PF definitions (the fundamental geometric, fundamental arithmetic, etc.) with respect to the voltage unbalance is evaluated. Once the fairest PF definitions are identified, the effect of the measurement algorithm on the final value is then assessed. This paper has the following contributions: i) It shows that a PF definition that leads to a fair billing for a constant impedance load may not provide a fair billing for a TIM in unbalanced and nonsinusoidal voltage conditions (UNVC). ii) It determines the fairest power factor (PF) definitions and their corresponding measurement algorithms for metering the PF of any of the two loads, a balanced constant impedance and a TIM, submitted to a wide range of UNVC. This paper is organized as follows. For a prospect of current standards, representative PF charging policies applied in the real world are revisited in Section II. In Section III, some definitions and measurement methods are listed. Considering the fairness perspective, these PF definitions are evaluated employing computational simulations in Section IV and experimental tests in Section V. In Section VI, measurement techniques for the fairest definitions are evaluated under several unbalanced and nonsinusoidal voltage conditions. At last, conclusions are drawn, and suggestions for future works are given in Section VII.

II. PROSPECT OF PF CHARGING POLICIES
It is a common practice for utilities to adopt mechanisms to encourage customers to control their reactive energy consumption. One of these mechanisms is the PF charging, which consists of financially penalising customers with PF below the established limit. In order to provide a basic overview of the existing charging policies, standards taken from 6 utilities, 9 states, and from 5 countries are investigated and presented in this section. The standards were selected based on different policies adopted worldwide, as presented in [4] and [15]. Results are shown in Table 1.
In Table 1, four types of charging policies are observed. BC-Hydro, for example, applies a surcharge percentage proportional to the PF deviation from the minimum accepted value [16]. Hydro-Quebec, on the other hand, charges the maximum power demand, defined as the greatest value between the maximum real power and 90% of the maximum apparent power [17]. The third and fourth policies are the charging of exceeding reactive power and energy, respectively.
From Table 1, it can be noticed that there is a PF limit/threshold below which charges are applicable, regardless of the policy adopted. In the case of BC-Hydro [16] and Brazilian utility companies [29], the PF should also be used to compute the charge that will be applied.
All investigated standards have explicit statements establishing that residential customers are not chargeable [16]- [30]. This can also be inferred from Table 1 based on the minimum demand 1 above which PF evaluation is applied. The minimum demand threshold (among the investigated standards) found in this investigation was 35 kW, which is not compatible with residential and small customers. Such demand level is typical of medium and large units (in general, industries) supplied by three-phase voltages. Therefore, a three-phase PF definition compatible with the characteristics of TIMs (industries main load [14]) might be more suitable for billing. Nonetheless, it was not found an explicit statement in the investigated documents about which of the existing PF definitions should be used.
Considering the foregoing discussion, it can be concluded that the PF is used as a trigger in all four policies, and, in some of them, it is also used in the computation of the charge. This initial analysis highlights that only medium and large customers are subjected to PF billing. For this reason, the usage of TIMs for evaluating different PFs may be more suitable than the constant impedance load model usually adopted in PF studies. Additionally, it is pointed out the lack of an explicit standardization of which definition should be used for revenue purposes in the investigated standards.

III. POWER FACTOR DEFINITIONS AND MEASUREMENT METHODS
In this section, we recall some definitions and measurement methods presented by IEEE standard 1459-2010 [3]. The advantages and disadvantages of the measurement methods, as well as the signal processing requirements, can be found in [3].
In sinusoidal balanced systems, the energy transmission efficiency (or line utilization) can be measured by the PF which can be computed in terms of the total active power (P) and apparent power (S) as [3], [31] where P and S are computed by the classical theory of alternate current (AC) polyphase systems [3], [31]. Alternatively, the reactive power (Q) can be used to compute the PF as Under these conditions, low transmission efficiency is related uniquely to the reactive power flow. The presence of voltage unbalance and/or harmonic distortion contribute to lower the transmission efficiency. In this case, there is no consensus on the meaning and definition of reactive power, apparent power and PF, as can be observed in [3], [6], [8]- [12], [32]- [39]. Some of the several PF definitions proposed in the literature are provided in [3]. These definitions can be divided according to the phenomenon they address: i) harmonic distortion, ii) voltage unbalance, and iii) the concurrence of i) and ii).
In nonsinusoidal single-phase systems, the single-phase 'true' apparent power 2 (S T ) can be decomposed accordingly to Fryze [33] or to Budeanu [34]. In both cases, S T is computed as where V T and I T refer to the true RMS values of voltage and current, respectively. According to Fryze, S T is decomposed as where N is the non-active power. In Budeanu's approach, S T is broken into where Q B and D B are Budeanu's reactive and distortion power, respectively. At this point, divergences regarding PF starts. For instance, some meters employ (3) in (1), while others compute the PF via (2) considering the reactive power equal either to N or to Q B . In [3], decomposition of S T in terms of the fundamental apparent power (S 1 ) and the nonfundamental apparent power (S N ) is presented. With S 1 , the fundamental power factor (PF 1 ) is defined as where P 1 is the fundamental active power.
In balanced nonsinusoidal systems, all phases have the same PF value if the same definition is employed. In this case, the load PF is exactly equal to the PF measured in any of the phases. The only cause for possible divergences is due to different single-phase definitions (usage of N or Q B for instance).
In sinusoidal unbalanced systems, any definition used to compute the PF per phase will give exactly the same results. However, phases will have different PF values. Some VOLUME 10, 2022 approaches to obtain the load PF are to compute it arithmetically or geometrically, which are respectively given by where Q is the three-phase reactive power, computed by the sum of each phase reactive power, and S X is the apparent power for phase X = A, B, C. It is noteworthy that different reactive and apparent power definitions can be employed in (7) and (8), which can result in different values under nonsinusoidal conditions. Alternatively, the load PF can be adopted as the positive sequence PF (PF + ) given by Since the positive sequence system is completely balanced, S + can be computed as S + = S +a or S + = S +g in which and Setting S + = S +a or S + = S +g in (9) leads respectively to the positive sequence arithmetical power factor PF +a = P + S +a (12) and to the positive sequence geometrical power factor The quantities PF +a and PF +g are mathematically equivalent (PF +a = PF +g ) but may have different values depending on the chosen measurement method. 3 Since this section deals only with definitions, the positive sequence will be simply denoted by PF + .
It is also possible to employ the effective power factor (PF e ) computed as where V e and I e are the effective voltage and current respectively. In terms of symmetrical sequence components, they are computed as 3 The apparent power S + X , X = A, B, C can be measured based on the RMS values of positive sequence voltage and current of each phase, whereas S +g requires the measurement of active and reactive power. The methods for measuring the voltage and current RMS values are different from those used to measure the active and reactive power. As a result, the measurements of PF +a and PF +g may have different values.
where V + , V − , and V 0 are the positive, negative, and zero sequence voltage true RMS values, respectively. Analogously, I + , I − and I 0 refers to the currents sequence components true RMS values. The variables ξ and ρ are used to consider the effects of zero sequence components in the system. If ξ and ρ are unknown, it is recommended to set them to one [3].
The possible combinations of single-phase quantities with the aggregation method to compute the load PF results in the definitions shown in Table 2. Each row is a method to aggregate single-phase values and to calculate the load PF. The columns refer to definitions under nonsinusoidal conditions. It is worth mentioning that the effective power factor cannot be combined with different definitions because it is already uniquely defined for both unbalanced and nonsinusoidal conditions.
The effectiveness of a PF measure depends on the definition employed and on the implemented measurement method. Therefore, in the following subsections, we select some measurement algorithms to be presented.

A. ACTIVE POWER MEASUREMENT METHODS
It is known that the total three-phase active power is obtained by adding up the total active power of each phase. According to [3], the total active power, in balanced nonsinusoidal conditions, can be decomposed into fundamental active power (P 1 ) and total harmonic active power (P H ).
In the general case, where voltage unbalance and harmonic distortion can be present, these last two decompositions can be superimposed resulting in 6 components: P + 1 , P − 1 , P 0 1 , P + H , P − H and P 0 H . The total active and positive sequence powers are measured, respectively, by [3] P ≈ where v X (t) and i X (t) are the voltage and current from phase X = A, B, C; v + A (t) and i + A (t) are the positive sequence voltage and current of phase A; T is the signal period, and k is the total number of cycles measured.

B. REACTIVE POWER MEASUREMENT METHODS
The methods of voltage integration and of 90 • displacement of voltage, based on voltage and current instantaneous samples, are defined as follows.
1) 90 • DISPLACEMENT OF VOLTAGE [40] Reactive power per phase is computed applying a 90 • shift via time delay to the voltage of that phase as The reactive power is estimated using a 90 • phase shift in sinusoidal signals obtained from integration as follows According to [4], Q X ,displacement and Q X ,integration can be considered approximately equal to Q 1X under common distortion levels.
Besides the two mentioned measurement algorithms, there is a wide range of algorithms inspired by different techniques. They are based on voltage and current instantaneous samples, on phasor measurements, on the fast Fourier transform (FFT), and on the wavelet transform [40], [42]- [44]. In this work, we present only the simpler measurement algorithms (the 90 • shift and voltage integration), which are sufficient for the precision evaluation of the fairest PF definitions, as corroborated by the experiments in Section VI.

C. POSITIVE SEQUENCE POWER MEASUREMENT ALGORITHMS
It is known that performing FFT is computationally costly and that physical filters have limitations [45]- [47]. Therefore, this section shows a simple method for measuring the fundamental positive sequence geometrical PF (PF +g 1 ). First, the phase A voltage instantaneous positive sequence component is extracted with the proposed algorithm of [48] as follows, The instantaneous positive sequence current i + A (t) is extracted analogously. Then, it is assumed that P + 1 is approximately equal to P + , which is computed by (19). The fundamental positive sequence reactive power (Q + 1 ) is estimated with the 90 degree displacement of voltage (20) or the voltage integration (21) methods substituting v X (t) and i X (t) by 3v + A (t) and 3i + A (t). Having computed P + 1 and Q + 1 , the value of PF +g 1 is obtained by One of the loads employed is a constant impedance, following the literature trend [3], [6], [8]- [12]. A delta connection is used, and the reference PF is set by declaring each individual impedance branch with an angle of cos −1 (0.95).
The second load selected was a TIM because, in the studied countries, PF billing applies for medium and large consumers. Since the TIM represents approximately 65-70% of industries' total consumption [49], it is important to observe how PF definitions affect this specific load. The simulated TIM is delta connected, 220 V (line-to-line), 7.5 kW, 60 Hz and has six poles. It is modeled accordingly to [50], [51] with the electrical parameters shown in Table 3. A constant slip of 1.5% is adopted, representing medium loading. Additionally, to emulate real operating conditions, the motor reactive power is partially compensated by delta connected capacitors so that its PF ref is equal to 0.95.
Several supply conditions are evaluated to identify the effects of each voltage parameter on the existing PF definitions. All combinations of V + ∈ [0.9, 1.1] pu in steps of 0.05 pu and of V − and V 0 ∈ [0, 0.03] pu in steps of 0.005 pu are considered. These magnitudes were chosen so that the positive sequence is never greater or less than 10% of the nominal voltage and so that the maximum voltage unbalance factor (VUF), given by is below 3.5%, which is slightly higher than the VUF usual limits in existing regulations [28], [52]. The angles θ 0 and θ − used in the database range from 0 to 330 • in steps of 30 • . The angle θ + is selected as the angular reference. Thus, it is set to zero in all cases. The voltage harmonics described in Table 4 are superimposed proportionally to each of the system unbalanced voltages, resulting in a total harmonic distortion (THD) of 3.74%. According to [4], the third, fifth and seventh harmonics components (with their magnitude and angle values presented in Table 4) are typical for voltage distortions in power systems. Proceeding in this way, a total of 35280 voltage conditions are simulated. It is noteworthy that the simulated conditions result in VUF values in the range of 0 up to 3.5% approximately. These values of VUF are close to those found in some distribution VOLUME 10, 2022 systems, and to the limits established by some standards. In fact, a voltage unbalance between 1% and 3% is present in 32% of the distributions systems in the United States, according to [52]. In terms of standards, the European Standard EN 51060, for example, establishes that the VUF should be lower than 2% in 95% of 10 minutes periods, according to [52].

A. PER PHASE POWER FACTOR
In the lack of a PF definition for billing purposes, a per phase approach may seem reasonable since the customer can act and compensate each phase individually.

1) CONSTANT IMPEDANCE LOAD
A set of different supply conditions was first applied to the constant impedance load, which was held constant by keeping its magnitude constant. Since the load is connected in delta, line voltages (measured between phases) were employed to compute PF 1X , X = A, B, C.
From the results obtained, it is possible to conclude that, for all investigated conditions, the computed PF 1X , X = A, B, C was constant and equal to the reference value of 0.95. The same conclusion was found by [6] using different voltage conditions.

2) THREE-PHASE INDUCTION MOTOR
Afterwards, the dataset was applied to the TIM. It should be highlighted that the motor is considered to operate a mechanical load with constant speed. As a result, the motor slip is set to 1.5% for all VU conditions. Since the desired mechanical output is constant and the motor is not modified, except for the voltages applied in it, the load can be characterised as constant from the customer's perspective. Consequently, its PF should also be constant. Fig. 1 shows a scatter plot of PF 1A as a function of the VUF for all 35280 conditions of the database. Many of these conditions result in the same pair of VUF and PF values, leading to the superimposition of points in the graph. Similar results are obtained for the remaining phases, which are not shown to avoid graph cluttering. From Fig. 1 Although the results for phases ''B'' and ''C'' are analogous to those presented for phase ''A'' in Fig. 1, it was observed that each phase had a different behaviour for each VU condition. Two different conditions that yields 1% VUF can result in different PF 1X , X = A, B, C values, as shown in   Table 5. In order to assess the VUF associated with conditions that cause charges in at least one of the phases, the minimum fundamental PF between phases (PF 1min ) for each condition is computed and shown in Fig. 2.
Similarly to PF 1A shown in Fig. 1, there are conditions with the same VUF that leads to different values of PF 1min as shown in Fig. 2. The range of possible values also increases with the VUF. From Fig. 2, it can be observed that for a VUF of 1% there are some conditions conducting to values of PF 1X , X = A, B, C lower than 0.92 in at least one of the phases. For VUF equal or greater than 1.5%, all simulated conditions yielded at least one phase PF 1X , X = A, B, C lower than the commonly acceptable value of 0.92. In both situations, the VUF is still within the usually acceptable limit of 2% [52], and the utility has no obligation to improve the supply quality. Nonetheless, the existing unbalance in the grid may result in PF charges if a per phase charging is applied or if PF 1min is utilized as the three-phase PF definition.
Summarizing, both the per phase approach as well as PF 1min are highly sensitive to the VU. For a given VUF, there are several conditions yielding different PF values. Some conditions can even result in leading PF, which is unacceptable in some regulations as shown in Section II. Moreover, when the VUF is equal or greater than 1.5%, at least one of PF 1X , X = A, B, C is less than 0.92. What stands out is that the customer's load was held constant with PF ref equal to 0.95 under balanced conditions. Therefore, these approaches are not suitable for billing purposes neither for constant impedance loads nor for TIMs.

B. THREE-PHASE POWER FACTOR
Given that PF computed per phase is not suitable for billing purposes, three-phase definitions are investigated for a constant impedance load and for the induction motor.
In [4], [37], [53], fairness aspects of some three-phase definitions have been discussed under unbalanced and nonsinusoidal conditions. The issues regarding Budeanu's power definitions are extensively pointed out in [53] and [37]. Investigating nonsinusoidal conditions, the authors of [4] concluded that usage of fundamental quantities, instead of quantities based on the non-active power, are fairer for billing purposes. Therefore, the definitions presented in the columns of Table 2 related to Budeanu's reactive power and nonactive power are also not suitable for billing purposes. With this, PF g 1 , PF a 1 , PF + 1 and PF e are evaluated in this study considering the same 35280 conditions used in simulations of Section IV-A. For the computation of PF e , the values of ξ and ρ in (16) and in (17) are set to one, following the recommendation given in [3] when they are unknown.

1) CONSTANT IMPEDANCE LOAD
For the constant impedance load submitted to VUF up to 3.4%, it was observed that PF g 1 and PF + 1 are always equal to the PF ref , while PF a 1 and PF e presented more variation around PF ref but still with standard deviation less than 10 −4 . Fig. 3    The authors are aware that unbalanced loads may cause voltage unbalance. In these simulations, however, the loads were perfectly balanced and kept constant so that the customer had no responsibility for the supply quality.

V. EXPERIMENTAL EVALUATION OF POWER FACTOR DEFINITIONS
In this section, the sensitivity to the voltage unbalance of different PF definitions is evaluated experimentally with a motor operating under unbalanced and distorted voltages. The experimental setup consists of a controlled voltage source (California Instruments CSW 11110), a speed sensor, a 4 kW DC generator, a variable resistor, and a 1,5 kW three-phase squirrel cage induction motor (4 poles, 60 Hz, rated 220 V and connected in delta).
The CSW 1110 supplies the motor with unbalanced and nonsinusoidal voltages. The motor drives the DC generator, which feeds the variable resistor. The DC generator and the variable resistor, working together, emulate a mechanical load in the motor shaft. The variable resistor is adjusted so that the motor operates with nominal power. A speed sensor is used to check if the motor speed was constant throughout the experiment. The connections between the equipment are shown in Figure 4.
To confirm that the conclusions drawn from the simulations study are not particular for the set of supply conditions and the motor considered, the experimental PFs were evaluated at different supply conditions (another collection of VUF values, harmonic spectra, and distortion levels) applied to another TIM operating close to nominal power. In the construction of the set of unbalanced supply conditions, the following values are taken into consideration: V + ∈ [0.87, 1.06] pu in steps of 0.02 pu, VUF ∈ [0, 4]% in steps of 0.5% and θ − ∈ [0 • , 120 • ] in steps of 24 • . All these values are combined, forming a set of 442 voltage conditions. For the harmonics, it is considered the pairs of harmonics: (third, fifth), (third, seventh), and (fifth, seventh). Individual distortion levels of 2%, 5%, and 8% are considered for each pair of harmonics. In total, 9 harmonic conditions are considered. The final set of supply conditions is obtained by the combination of the voltage unbalance and harmonic distortion values forming a dataset of 3978 unbalanced and distorted voltage conditions (each condition is applied to the motor for 30 seconds). Amongst all conditions, 901 led to a practically constant slip of the motor (slip between 2.4% and 2.6%). They were selected to elaborate Fig. 5, which shows PF g 1 , PF e , PF a 1 and PF + 1 as a function of VUF.
In Fig. 5, it can be seen that the TIM 3 PFs have a more scattered pattern compared to the correspondent PFs in Fig. 3. This scattering shows that the points obtained experimentally do not superimpose as perfectly as in Fig. 3, meaning that more supply conditions lead to results with the same VUF but different PF values. In Fig. 5, it can be seen that PF e and PF a 1 definitions presented the greatest scattering bands for any fixed VUF. In other words, PF e and PF a 1 are the most vulnerable definitions to the unbalanced and distorted conditions. Although all PF definitions present some sensitivity to the presence of unbalances and distortions, one can see that the general behavior of the PFs with respect to VUF is similar in Figures 5 and 3. It can be observed that, when the VUF increases, the scattering band of PF e and PF a 1 shifts down, the PF g 1 band shifts down slightly, and that the position of the band of PF + 1 is the most stable presenting the least variation with respect to changes of VUF. In particular, considering the worst case, it can be seen that, for any VUF, the lower values of PFs are such that PF + 1 ≥ PF g 1 ≥ PF a 1 ≥ PF e . Based on these results, it can be concluded that the experimental investigation and the simulation study are in accordance. Besides, these results indicate that the most recommendable PF definitions for revenue purposes are PF + 1 and PF g 1 .

VI. EVALUATION OF THE EFFECTS OF MEASUREMENT ALGORITHMS ON THE FAIREST POWER FACTOR DEFINITIONS
This section evaluates the accuracy of the measurement methods discussed in Section III-B considering unbalanced and nonsinusoidal voltage and current signals in conditions with noise and also in conditions without noise. Here, a meter is simulated as an algorithm that measures a specific PF definition. For evaluation, the algorithms should obtain measurements for PF g 1 and PF + 1 submitted to the same supply and load conditions simulated in Section IV. It is important to note that PF + 1 must be distinguished as PF +a 1 and PF +g 1 since they may provide different values, depending on the measurement algorithm applied.

A. ACCURACY OF MEASUREMENT ALGORITHMS: WITHOUT NOISE
In order to evaluate the measurement accuracy as a function of the VU, the error ( PF) is computed as the difference between the PF measurement (PF meas. ) and the PF ref. . In particular, the error for PF g 1 is PF the error for PF +a 1 is PF +a and the error for PF +g 1 is where PF g 1,meas. is computed substituting P and Q by P 1 and Q 1 in (8); PF +a 1,meas. is computed substituting P, S A , S B and S C by P + 1 , S + 1A , S + 1B and S + 1C in (7); and PF +g 1,meas. is computed substituting P and Q by P + 1 and Q + 1 in (8). In this simulation, voltage and current waveforms are generated for each VU condition. Then, each signal is sampled 64 times per cycle. A window of calculations of 1s is employed. In the case of positive sequence components, values in the first cycle are not taken into account since the estimation process is only possible after two-thirds of a cycle.
The harmonic active power is often negligible with respect to the total fundamental active power [12], [54]. So, P 1 is measured applying (18). Similarly, P + 1 is measured with (19). The fundamental reactive power is computed by the 90 • displacement of voltage (20) and by the voltage integration (21) methods. The computation of PF +a 1 can be simplified taking into account that where V + 1 and I + 1 are the fundamental positive sequence voltage and current, respectively. To measure V + 1 and I + 1 , the instantaneous positive sequence components are estimated and the fundamental component is obtained applying the FFT. Proceeding this way, the values of PF +a 1 obtained in the simulation were lower, in module, than 0.0003 and were approximately constant with regard to the VUF. Fig. 6 shows PF It can also be observed that for VUF equal to zero, both techniques had deviations close to zero. On the other hand, for a 3% VUF, the methods applied to obtain PF +g 1 presented errors from −2×10 −3 up to +2×10 −3 , depending on the VU condition. As a matter of fact, PF +g 1 reading presented different errors for the same VUF (greater than zero) depending on the condition. This phenomenon is noted to be intensified proportionally to the increase of VUF. This behaviour was not verified for PF g 1 . On the other hand, PF g 1 readings deviate negatively from its reference value reaching up to −3 × 10 −3 for VUF equal to 3.4%. For VUF greater than 2.5%, PF g 1 readings performed worst than PF +g 1 .

B. ACCURACY OF MEASUREMENT ALGORITHMS: WITH NOISE
In this subsection, it is evaluated how the measurement methods perform with noisy voltage and current signals. For this purpose, an additive white Gaussian noise with standard deviation of 0.2 pu was considered in the signals used in the previous subsection.  The values of PF +a 1 obtained with noise were approximately constant (lower, in module, than 0.0003) with regard to the VUF. Therefore, the computation of PF +a 1 with the FFT is shown to be robust to the presence of noise in the voltage and current signals. Fig. 8 shows, as a function of the VUF, the measurement errors of the TIM PF  Fig. 7. Therefore, the usage of the 90 • displacement of voltage and the voltage VOLUME 10, 2022 integration for measuring PF g 1 is shown to be robust to the presence of noise in the voltage and current signals.
Despite the results of PF +g 1 evaluations with the 90 • displacement of voltage or the voltage integration methods being sensitive to noise, the maximum and minimum errors are similar to those obtained under noiseless conditions (of order 10 −3 ). Therefore, both PF g 1 and PF +g 1 can be accurately measured by the investigated methods considering voltage and current signals with noise up to 0.2 pu.

VII. CONCLUSION
This paper evaluated three-phase PF definitions and their applicable measurement algorithms under several unbalanced and nonsinusoidal voltage conditions.
Simulations showed that all investigated definitions and algorithms yielded fair PF measurements for the balanced constant impedance load. For the motor, billing of PF per phase leads to unfair PF values. For VUF equal or greater than 1.5%, at least one of the phases had a PF value lower than the commonly accepted value of 0.92. It is noteworthy that, in this case, the customer may be penalised due to the voltage unbalance, while the utility has no obligation to improve the supply quality. Among the investigated threephase definitions, only PF g 1 and PF + 1 had stable PF readings regardless of the unbalanced or/and nonsinusoidal voltages in simulation and experimental tests.
Additionally, PF g 1 and PF +g 1 have the property of being accurately measurable even considering 20% of noise in the voltage and current signals, and employing simple methods such as the 90 • displacement and voltage integration methods. In other words, the measured values of PF g 1 and PF +g 1 for balanced customers (modeled as constant impedance or as induction motors) have little sensibility on the measurement algorithm and are not affected even if the utility delivers unbalanced nonsinusoidal voltages.
The results obtained in this paper showed that the voltage unbalance and harmonic distortion can reduce the customer's PF e to values below the acceptable limits. Therefore, the PF e definition is not the most adequate for the scenarios investigated in this work, even though it is recommended by IEEE Standard 1459-2010. This unexpected conclusion is one of the paper's main contributions and shows that finding the most appropriate PF definition demands more investigation, especially considering situations in which the voltage and the load are both unbalanced.
Considering that PF billing policies impact customers, utilities, and energy meter companies, we suggest, for future works, analysis of the PF and reactive power definitions and measurement methods considering several combinations of supply conditions (unbalanced, nonsinusoidal, etc.) and nonlinear or/and unbalanced loads. Field investigation of commercial meters and of industries bills is also an interesting topic for further research.