—Growing penetrations of distributed energy re- sources (DERs) increase the power injection variability in dis- tribution systems, which can result in power quality issues such as voltage unbalance. To measure unbalance, organizations such as IEC, NEMA and IEEE deﬁne phase unbalance in their power quality standards. However, the deﬁnitions in these different standards are not consistent, and voltages that are considered acceptable by one standard may violate good practices deﬁned by another standard. To address this issue, this paper provides analytical comparisons of the most common voltage unbalance deﬁnitions, which are supplemented with numeri- cal simulations. The analytical relationships suggest that it is possible to approximately bound the symmetrical-component- based voltage unbalance factor (which depends on the magnitude and relative phase angle) by limiting the line-to-line voltage unbalance, whereas applying line-to-ground voltage unbalance deﬁnitions neglects all information about phase angle offsets. I. I NTRODUCTION Increasing penetration of distributed energy resources (DERs) are resulting in greater variability of the net load in distribution systems. This challenges distribution system operators’ ability to maintain acceptable power quality and may exacerbate problems such as voltage unbalance, which is the focus of this paper. Unbalanced operating conditions can damage power system equipment, such as three-phase induction motors [1], and increase network losses [2]. Distribution system operators strive to maintain balanced voltages by equally distributing load demands on all three phases of distribution feeders [3]. Variability associated with DERs can cause time-varying changes in the per-phase loading. Thus, high penetrations of DERs can challenge traditional approaches for balancing the net loading among the three phases in distribution systems. Organizations such as IEC, NEMA and IEEE have proposed power quality standards regarding voltage unbalance. How- ever, the associated deﬁnitions of voltage unbalance are not consistent. The IEC standard [4] uses a deﬁnition based on the positive and negative sequences from the symmetrical compo- nent transformation [5], which requires measurement of both voltage magnitudes and relative phase angles. Standards from NEMA [6] and IEEE [7] deﬁne phase unbalance using line- to-line and line-to-ground voltage magnitudes, respectively. This work was funded by the U.S. Department of Energy’s Ofﬁce of Energy Efﬁciency and Renewable Energy (EERE) under Solar Energy Technologies Ofﬁce (SETO) Agreement Number 34235.There are a number of papers analyzing the differences among the voltage unbalance deﬁnitions. In [8] and [9], three deﬁnitions of voltage unbalance (IEEE 1159, IEEE 141-1993, and NEMA) are numerically compared. While the overall conclusion of these papers is that voltage unbalance deﬁnitions agree reasonably well for unbalances below 5%, the numerical comparisons show that there are still differences. This implies that operating points which satisfy one power quality standard might violate other standards, which motivates a more thorough analysis of the relationships among various voltage unbalance deﬁnitions. Several approximations for the IEC deﬁnition based only on line-to-line voltage magnitude measurements (thus avoiding angle measurements) are ana- lyzed in [10] and [11]. However, the resulting expressions are highly non-linear and no clear error bounds are provided. Relative to existing literature, we do not just numerically evaluate the differences and similarities among voltage unbal- ance deﬁnitions (which is useful if we want to use one deﬁ- nition as an approximation for the others). Instead, our focus is on providing bounds on the maximum difference among the voltage unbalance deﬁnitions. This is helpful for understanding how to use one voltage unbalance deﬁnition (which may, e.g., be easier to measure in real time) to infer safe bounds on the voltage unbalance as deﬁned by another deﬁnition. In particular, we are interested in how measurements of line- to-ground and line-to-line voltage magnitudes, which allow us to compute unbalance using IEEE and NEMA deﬁnitions, can be used to enforce limits on the voltage unbalance factor deﬁned by the IEC standard. This is a practically important task, as access to measurements such as relative voltage angles are frequently unavailable. Furthermore, while previous papers have relied solely on numerical simulations, we derive bounds based on analytical expressions and then verify those bounds numerically. In summary, the main contributions of this paper are: 1) derivations of analytical relationships among the voltage unbalance deﬁnitions, 2) empirical evaluations characterizing these relationships’ accuracy, and 3) assessment of the condi- tions under which these relationships are most inaccurate. The rest of the paper is structured as follows: Section II introduces various voltage unbalance deﬁnitions. Section III provides analytical analyses and numerical experiments that characterize voltage unbalances according to these deﬁnitions. Section IV concludes the paper. 978-1-7281-0407-2/19/$31.00 © 2019 IEEE II. V OLTAGE UNBALANCE DEFINITIONS V oltage unbalance arises from asymmetric operation leading to phase voltages with unequal magnitudes, phase shifts that are not equal to 120 degrees, or a combination of both [5]. This section presents three common deﬁnitions for voltage unbalance from IEC [4], NEMA [6], and IEEE [7]. Other unbalance deﬁnition based solely on line-to-line voltage mag- nitudes are used by instrument manufacturers for power quality measurements as an approximation to the IEC deﬁnition [11]. These approximations are not discussed in this paper as they do not represent fundamentally different deﬁnitions, but are rather approximations of one of the considered deﬁnitions. We use the following notation to represent phasors: V= V\, where Vdenotes the voltage phasor as a complex number,Vis the voltage magnitude, and \is the phase angle. Denote the complex conjugate of VasV. Letj=p1. A. IEC deﬁnition (VUF) Unbalanced voltages can be described using symmetrical components. The deﬁnition of the V oltage Unbalance Factor (VUF) frequently considered to be the “true” deﬁnition of voltage unbalance is based on the relative magnitudes of the negative and positive sequence voltages. The VUF deﬁnition adopted by the IEC [4] is VUF [%] =jVnj jVpj100; (1) whereVp=Va+aVb+a2Vc 3; Vn=Va+a2Vb+aVc 3: In (1),VpandVnare the positive and negative sequence voltage phasors, respectively; a= 1\120; andVa,Vb, andVcare the three-phase unbalanced line-to-ground voltage phasors. The use of sequence components VpandVnposes some practical difﬁculties due to the associated complex algebra [8] and the need to track both the voltage magnitudes and phase angles for each phase. One of the major challenges in using the IEC deﬁnition is the inability to directly measure the sequence components using installed RMS meters as they provide no information about the relative phase angles of the voltages [9]. For low- and medium-voltage systems, IEC standard 61000-2-2 [4] requires that the voltage unbalances, as deﬁned in (1), are less than 2%. B. NEMA deﬁnition (LVUR) Motors make up a large portion of the loads connected to distribution systems. Motor manufacturers use the NEMA deﬁnition of voltage unbalance [6], which is also referred to as the Line V oltage Unbalance Rate (LVUR). The NEMAdeﬁnition calculates unbalance using the magnitudes of line- to-line voltages Vab;VbcandVca: LVUR [%] =Vmax L Vavg L100; (2) whereVavg L=Vab+Vbc+Vca 3; Vmax L = maxfjVabVavg Lj;jVbcVavg Lj;jVcaVavg Ljg: Per NEMA MG-1 [6] and ANSI C84.I [12], power systems must be operated such that the maximum voltage unbalance, as deﬁned in (2), does not exceed 3% under no-load conditions. C. IEEE deﬁnition (PVUR) The IEEE deﬁnition is a recommended guideline to measure unbalance for the electrical design of industrial facilities [7]. It is commonly referred to as Phase V oltage Unbalance Rate (PVUR) and is described using the line-to-ground voltage magnitudes Va;VbandVc: PVUR [%] =Vmax P Vavg P100; (3) whereVavg P=Va+Vb+Vc 3; Vmax P = maxfjVaVavg Pj;jVbVavg Pj;jVcVavg Pjg: Motors are affected by phase-voltage unbalances that exceed 2% and are most likely to overheat when operating near full load [7]. By ignoring the phase angle information, the IEEE deﬁnition does not detect voltage unbalance that is due to phase angle asymmetries. III. C OMPARISON OF VOLTAGE UNBALANCE DEFINITIONS This section provides an in-depth comparison of the differ- ent voltage unbalance deﬁnitions by deriving analytical bounds that are veriﬁed through numerical simulations. To facilitate our analysis, we divide voltage unbalance into three categories: (a) V oltage magnitude unbalance, (b) V oltage angle unbalance, (c) V oltage magnitude and angle unbalance. Considering phase Aas the reference phase, we analyze the impacts of varying the voltage phasors associated with either phaseBalone or both phases BandC. Fig. 1 on the next page illustrates the voltage triangle with unbalance in phase B for each of these categories. With balanced voltage magnitude indicated by V, the magnitude unbalance is denoted using the multiplicative factor xand the angle unbalance is denoted using . The angle in Fig. 1 is determined by xand . Starting with the simplest voltage unbalance cases involving only magnitude or angle unbalance, we utilize small angle approximations to derive analytical expressions for VUF, PVUR and LVUR. We then use the analytical expressions to characterize the relationships and derive bounds on the maximum difference among the various voltage deﬁnitions. In a second step, the validity of the bounds is tested using numerical simulations without any approximations involved. In the more general cases with either angle unbalance in more than one phase or combinations of magnitude and angle (a) V oltage magnitude unbalance (b) V oltage angle unbalance ∓θ (c) V oltage magnitude and angle unbalance Fig. 1: V oltage triangles for three cases of unbalance in phase B unbalance, the analytical expressions become quite complex, and hard to analyze and compare. Hence, we refrain from including analytical expressions for these cases and instead rely on numerical simulations alone. A. Voltage magnitude unbalance in phase B This section investigates the voltage magnitude unbalance case (i.e., = 0). We ﬁrst consider a voltage magnitude unbalance in one of the three phases as shown in Fig. 1(a). Two line-to-line voltages VLare affected by the unbalance of the phase- B-to-ground voltage magnitude, while the third line-to-line voltage remains the same as in the balanced case with a magnitude ofp 3V. The line-to-ground and line-to-line voltage phasors are Va=V\0; Vab=VL\(30); Vb=Vx\120;Vbc=VL\(90); Vc=V\120; Vca=p 3V\150: Analytical Comparison We start by providing analytical expressions for the three voltage unbalance deﬁnitions. 1) IEC deﬁnition (VUF): The magnitudes of the positive and negative symmetrical components VpandVnare Vp=jV\0+Vx\0+V\0j 3=V(x+ 2) 3; (4a) Vn=jV\0+Vx\120+V\120j 3 =V(1x) 3j1jp 3j 2=Vjx1j 3: (4b) The voltage unbalance according to the IEC deﬁnition is then VUF =jx1j=(x+ 2): (5) 2) NEMA deﬁnition (LVUR): The LVUR voltage unbalance deﬁnition (2) depends on the line-to-line voltage VL. To expressVLin terms of x, the sine triangle rule is applied to the triangle4OAB in Fig. 1(a): VL sin 120=V sin(30)=Vx sin(30): (6)Using (6) and the trigonometric angle sum identities yields two equations for VLin terms ofxand: VLsin(30) =Vsin 120 )VL cosp 3 sin =p 3V; (7a) VLsin(30) =Vxsin 120 )VL cosp 3 sin =p 3Vx: (7b) By adding (7a) to (7b) and applying the small angle approx- imations sinandcos1, we obtain the following relation between VLandx, VL=p 3Vx+ 1 2 cos p 3Vx+ 1 2 : (8) The average line-to-line voltage Vavg L and maximum devi- ation Vmax L are Vavg L=VL+VL+p 3V 3p 3Vx+ 2 3 ; (9a) Vmax L =jp 3VVLavgjp 3Vjx1j 3 ; (9b) which allows us to obtain the following expression for LVUR: LVURjx1j=(x+ 2): (10) 3) IEEE deﬁnition (PVUR): The PVUR deﬁnition (3) is computed using the average voltage magnitude, Vavg P, Vavg P=V+V+Vx 3=V(x+ 2) 3; and the deviation from the average, Vmax P, Vmax P =jVxV(x+ 2) 3j=2Vjx1j 3: This gives rise to the following expression for PVUR, PVUR = 2jx1j=(x+ 2): (11) Comparing the three deﬁnitions based on (5), (10), and (11), we observe the following relationships: LVURVUF =PVUR= 2: (12) As indicated by the sign, the calculations of LVUR involve small angle approximations, whereas VUF and PVUR are exact calculations. We observe that for a given deviation jx1jin voltage magnitude, the expressions for Vn,Vmax L andVmax P in the numerators of (1), (2) and (3), respectively, are invariant to over-voltage ( x >1) or under-voltage ( x <1) conditions. Conversely, the expressions for Vp,Vavg L andVavg P in the denominators of (1), (2) and (3), respectively, are directly proportional to x. Hence, the unbalance is greater for under- voltage cases compared to the over-voltage cases, which is consistent with the observations in [10]. Numerical Validation To validate the expressions derived above, we compute the voltage unbalances that result from varying the phase- Bvoltage magnitude. The multiplicative factorxis varied from 0:8to1:2in steps of 0:01to investigate both under-voltage and over-voltage conditions. The left part of Fig. 2 illustrates the variation of VUF (on the y-axis) with respect to both PVUR and LVUR (on the x-axis) for voltage magnitude unbalance in phase Bonly. The dashed black line represents the situations where VUF and LVUR have the same value, while the dashed red line represents the situation where PVUR is a factor of 2larger than VUF. The orange and green dots correspond to values obtained for LVUR and PVUR, respectively. Consistent with our analytical results, we observe that PVUR is exactly twice as large as VUF. The values for VUF and LVUR are almost identical, although there are minor variations due to the inaccuracy of the small angle approximation. For the range of xvalues considered, the relative difference jLVURVUFj VUF is less than 2%. B. Voltage magnitude unbalance in phases BandC We express the voltage unbalance in a similar way as the voltage unbalance for just phase B, except that we consider two deviations in the magnitudes, i.e., Vb=x1VandVc= x2V. Analytical Comparisons Due to the complexity of the expres- sions, we only include a high-level discussion of the case with ∗ √𝟑 𝟐∗ ≥ √𝟑∗ Fig. 2: V oltage magnitude unbalance for 0.80 x1.20voltage magnitude unbalances in both phases BandC. Similar to the case with magnitude deviation in only one phase, the denominators of all unbalance deﬁnitions simplify to the same expressions when we have magnitude deviations in two phases. To assess the relative magnitudes of the voltage unbalance deﬁnitions for a given voltage unbalance, we therefore only need to assess the differences in the numerators of (1), (2) and (3) . 1) IEC deﬁnition (VUF): The numerator for VUF in (1) is the magnitude of the negative sequence voltage Vn: jVnj=V 3q x2 1+x2 2+ 1x1x2x1x2: (13) 2) NEMA and IEEE deﬁnitions (LVUR and PVUR): The numerators for LVUR and PVUR in (2) and (3), respectively, depend on which voltage magnitude is furthest from the average. We hence need to consider three cases, Vmax LVmax P 2=V 6maxfjy1j;jy2j;jy3jg; (14) wherey1=x12x2+ 1; y 2=x1+x22; y3=2x1+x2+ 1 Comparing the three deﬁnitions, we observe that the de- nominator for LVUR is still approximately equivalent to the denominator for PVUR, suggesting the following relationship between PVUR and LVUR: PVUR LVURVmax P Vmax L2)PVUR2LVUR: (15) When comparing LVUR and PVUR with VUF, we observe that there is not a one-to-one mapping because we need to consider the three different cases corresponding to whether y1,y2, ory3is the largest. Since both jVnjandVmax P are non-negative, we consider the square of the ratio between VUF and PVUR, i.e., VUF PVUR2 =jVnj2 (Vmax P)2=x2 1+x2 2+ 1x1x2x1x2 (maxfjy1j;jy2j;jy3jg)2: By identifying the largest among jy1j;jy2jandjy3j, simplify- ing the expressions, and taking the square root of both sides, we obtain an upper bound on the ratio of VUF to PVUR, jVnj Vmax P1p 3)VUF1p 3 PVUR: (16) The deﬁnitions of y1,y2, andy3imply that the unbalance ratio seen in (16) with unbalance in two phases is always larger than the situation where either x1= 1 orx2= 1. This implies that the case with magnitude unbalance only in phase B(12) is a lower bound on the more general case with magnitude unbalance in two phases. Combining this observation with (16) yields the following relationship between PVUR and VUF: 1 2 PVURVUF1p 3 PVUR: (17a) By considering (15), we obtain an approximate upper bound on the ratio of VUF to LVUR, LVUR .VUF.2p 3 LVUR (17b) As indicated by the .sign, the calculations of Vmax L (numerator of LVUR) in (14) involves small angle approxi- mations. Numerical Validation Similar to the case with unbalance in only one phase, we run experiments where we let x1andx2 vary from 0:8to1:2in steps of 0:01. The results are shown in the right part of Fig. 2 with a similar legend as the left part. In addition to the black and red lines that are used to identify the situation where LVUR =VUF andPVUR = 2VUF respectively, we also add lines to represent the upper bounds VUF = (2p 3)LVUR (solid blue line) and VUF = (1p 3) PVUR (solid purple line). Almost all points fall within the predicted range. However, LVUR is greater than VUF for approximately 17% of the considered points. These points fall below the dashed black line in the grey shaded region as shown in Fig. 2. This situation corresponds to cases where two of the line-to- line voltages were signiﬁcantly higher than the third line-to- line voltage, i.e., extreme cases of under-voltage in two phases and over-voltage in the third phase, resulting in inaccuracy of the small angle approximation. However, the errors in our bounds are small: within the ranges of voltage magnitudes corresponding to 0:8x1; x21:2, we never see violations larger than 2% relative to the bounds in (17). C. Voltage angle unbalance in phase B We next consider the single voltage angle unbalance case with angle displacement and balanced voltage magnitude V as shown in Fig. 1(b). The voltage phasors in this case are Va=V\0; Vab=Vab\(30 =2); Vb=V\(120 );Vbc=Vbc\(90 =2); Vc=V\120; Vca=p 3V\150: Analytical comparison 1) IEC deﬁnition (VUF): To derive the expression for VUF, we use the small angle approximation to ﬁnd the magnitudes of the sequence voltages, VpandVn: 2) NEMA deﬁnition (LVUR): To determine LVUR, we use the sine triangle rule and the small angle approximation for to obtain the line-to-line voltage magnitudes, VabandVbc: VabV(p 3 2); V bcV(p 3 2): (18) We then derive approximations of the average line-to-line voltageVavg L and the maximum deviation Vmax L, which provide an approximate expression for LVUR: Vavg Lp 3V;Vmax LV 2)LVURp 3 2 3:(19) √𝟑 𝟐∗ ≥ Fig. 3: V oltage angle unbalance for -20 20 3) IEEE deﬁnition (PVUR): Since the line-to-ground volt- age magnitudes are balanced, Vmax P = 0. Hence,PVUR = 0for any unbalance that only affects the voltage angles. Based on the analytical expressions above, we observe that PVUR does not provide any information about phase angle unbalance. The approximate relationship between LVUR and VUF is LVUR .VUF.2p 3 LVUR: (20) Numerical Validation The relationship in (20) is veriﬁed by numerical experiments. With phase Aselected as a reference, the phase angle displacement in phase Bis varied in the range20 20in steps of 1. The results of these computations are shown in the left part of Fig. 3 for unbalance in phaseB. The experiment conﬁrms our the analytical results, withPVUR = 0 regardless of the angle unbalance and LVUR(p 3 2)VUF . D. Voltage angle unbalance in Phases B and C To extend the above analysis, we assess how voltage angle unbalance in two phases affects the relationships among the deﬁnitions. We introduce two phase angle deviations, 1and 2for phasesBandC, respectively. Due to the complexity of the resulting analytical expressions, we only detail the numerical results for this case. Numerical Validation We perform a similar numerical study as above, but consider angle unbalance in both phases B andC. We use phase Aas a reference, and consider all combinations of phase angle deviations in phases BandC in the range20 1; 220, in steps of 1. The results are shown in the right part of Fig. 3. We observe that PVUR, which includes no information about phase angle unbalance, is still always zero for all combinations of angle unbalance. As with voltage magnitude unbalance, introducing phase angle unbalance in both phases means that there is no longer a direct correspondence between LVUR and VUF. Interestingly, we can numerically verify that VUF is still effectively bounded by the same relations as seen in (17b) or (20). E. Voltage magnitude and angle unbalance Finally, we consider the general case where the unbalance involves asymmetries in both the voltage magnitudes and angles. We ﬁrst investigate the case with both magnitude and angle unbalance in phase B, which has an unbalanced voltage magnitudeVxand the angle displacement deﬁned by as shown in Fig. 1(c) and expressed below: Va=V\0; Vab=Vab\(301); Vb=Vx\(120 );Vbc=Vbc\(902); Vc=V\120; Vca=p 3V\150: In addition, we consider the situation where we have similarly deﬁned unbalances for both magnitudes and angles in both phasesBandC. Due to the complexity of the expressions, we derive our bounds based on numerical validation only. Numerical Comparison To perform our numerical experi- ment, we choose Vaas reference voltage phasor at 1\0and calculate the voltage unbalance according to each deﬁnition as we vary the deviations in both the voltage magnitudes (from 0:8to1:2in steps of 0:01) and angles (from 20to20in steps of 1). The results are shown in Fig. 4. The left part of this ﬁgure shows the case with deviations only in phase B and the right part depicts the case with unbalances in both phasesBandC. As previously discussed, PVUR does not consider voltage angle unbalances. Hence, Fig. 4 shows that there is a large variation of PVUR corresponding to any given value of VUF. The relationship between LVUR and VUF is similar to the previous cases, yielding the same bounds as in (17b) or (20). The few points where VUF < LVUR correspond to the extreme condition of under-voltage in two phases and over- voltage in the third phase. For the ranges of voltage magnitude deviations 0:8x1; x21:2and phase angle deviations 20 1; 220, we never observe violations greater than5% relative to the bounds in (17b). ∗ √𝟑 𝟐∗ Fig. 4: V oltage magnitude and angle unbalances for 0:8x1:2and20 20. IV. C ONCLUSION This paper analyzes the relationships among the three most prevalent deﬁnitions of voltage unbalance. The “true” voltageunbalance deﬁnition used by the IEC (VUF) is based on negative and positive sequence voltages. Thus, VUF captures both voltage magnitude and angle unbalance, but can be difﬁcult to measure because it requires measurements of the relative phase angles. Deﬁnitions from NEMA and IEEE based on the line-to- line voltages (LVUR) and line-to-ground voltages (PVUR), respectively, are frequent substitutes used in the design of electrical machines and industrial plants. The goal of our paper is to extend existing comparisons among these deﬁnitions with a more comprehensive analysis that considers a range of voltage unbalance conditions, using both analytical derivations and numerical simulations. To this end, we derive and numerically demonstrate rela- tionships among the different deﬁnitions. In summary, PVUR does not provide any information about phase angle unbalance and has no clear relationship with VUF. Conversely, LVUR provides information about unbalances in both voltage mag- nitudes and phases. Using the small angle approximation, we bound the relationships between the IEC deﬁnition (VUF) and the NEMA deﬁnition (LVUR) as LVUR .VUF.2p 3 LVUR: For a reasonable ranges of voltage magnitude and angle deviations ( 0:8to1:2p.u. voltage magnitude and 20to20 voltage angle), the violations of these bounds are less than 5% (relative to the bounds). Our ongoing work is investigating how these deﬁnitions can be formulated in order to provide the most tractable constraints in three-phase optimal power ﬂow problems that include minimization of voltage unbalance.