Reduced Reference Frame Transformation Assessment in Unbalanced Three-phase Four-Wire Systems

Received 17 February 2023, accepted 2 March 2023, date of publication 8 March 2023, date of current version 15 March 2023. Digital Object Identifier 10.1109/ACCESS.2023.3254299 Reduced Reference Frame Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems FRANCISCO CASADO-MACHADO , JOSÉ L. MARTÍNEZ-RAMOS , (Senior Member, IEEE), MANUEL BARRAGÁN-VILLAREJO , AND JOSÉ MARÍA MAZA-ORTEGA , (Member, IEEE) Department of Electrical Engineering, University of Seville, 41092 Seville, Spain Corresponding author: Francisco Casado-Machado ([email protected]) This work was supported in part by the Ministerio de Ciencia e Innovación (MCIN) /Agencia Estatal de Investigación (AEI)/ 10.13039/501100011033 through the ‘‘European Regional Development Fund (ERDF) A way of making Europe’’ under Grant PID2021-124571OB-I00; and in part by the CERVERA Research Program of Centro para el Desarrollo Tecnológico y la Innovación (CDTI), Industrial and Technological Development Centre of Spain, through the Project HySGrid+ under Grant CER-20191019. ABSTRACT Three-phase four-wire systems require a reference frame similar to that used in 3-phase 3-wire system configurations that allow them to be easily analysed, modelled and controlled. A new reference frame called Reduced Reference Frame (RRF) was presented by the authors in a previous work to satisfy these objectives. This is based on the locus of the voltage space vector which is used to perform a transformation applied to 3-phase 4-wire systems. This paper uses the proposed RRF to demonstrate the benefits of this transformation under different conditions of load or voltage imbalance. To this end, the different types of imbalances considered are studied by comparing the results in the classical αβγ axes, the so-called mno-transform specifically developed for 3-phase 4-wire systems and the RRF coordinates. In addition, a theoretical identification of some types of imbalances is done comparing the simulation results and the fundamentals of the RRF. INDEX TERMS Coordinate transformation, reference frame, three–phase four–wire system, unbalanced power system. I. INTRODUCTION The integration of distributed energy resources such as photovoltaic plants, energy storage systems, or electric vehicles in low-voltage networks will require more detailed studies of three-phase four-wire systems. These will be deployed in a three-phase or single-phase manner depending on the power to be managed by each resource, which could lead to a greater imbalance of these networks. An example of this situation can be found in [1], where a year-long campaign was carried out to measure the currents and voltages on the low-voltage side of more than a hundred transformers integrated in a network with high PV penetration in Great Britain. The report shows an average voltage imbalance in the transformers of more than 0.5%, with some of them reaching maximum imbalances of more than 5% in certain moments. In this situation, The associate editor coordinating the review of this manuscript and approving it for publication was Yizhang Jiang . the neutral wire current may be high with maximum values above the rated current during some periods. In addition, in electrical systems such as microgrids, it is common to find load and generation imbalance conditions, due to the single-phase character of the consumers/generators and to the different demand/generation of each of them [2], [3]. This situation becomes more pronounced for islanded microgrids dominated by power converters, where it is especially interesting to have the ability to control the voltage and currents of these power converters in an unbalance way [2], [3]. This way, it could be possible to improve the general performance of the microgrid [4]. As a consequence, it is essential to define an adequate reference frame that allows to analyze, model, and control three-phase four-wire systems with a high level of imbalance. The classical reference frames based on the Clarke et al. [5] and Park [6] transforms have proven to be successful in threephase three-wire systems. However, for three-phase four-wire VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 24591 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems systems, a more general approach than the one proposed by Akagi et al. in [7] is required for a better understanding of how the unbalanced magnitudes may affect the active and reactive instantaneous powers [8]. In recent years, several transformations have emerged to address this issue. Of special interest are those based on Singular Value Decomposition proposed by V. Choqueuse in [9] or those based on the Frenet-Serret frame proposed by Granjon and Phua [10] and more recently Milano et al. [11]. Others authors use tensor algebra [12], while others apply the geometric or Clifford algebra to analyze three-phase four wire systems [13], [14] or multiphase power systems in time-domain [15], [16]. Other authors apply transformation methods based on the voltage space vector and its time derivative to define a stationary mno reference frame, whose orientation depends on the system imbalance [17]. In addition, the active and reactive power terms can be related to the corresponding voltages and currents in the mno reference frame. Nevertheless, this frame is undefined for some particular voltage conditions, i.e. a voltage with just zero-sequence component or null voltage in the phase a[17]. A similar approach was proposed by Tan et al. [18], representing unbalanced signals in a non-orthogonal reference frame by means of a space vector for 3-phase 3-wire systems. This strategy was recently extended for 3-phase 4-wire systems in [19] and [20]. Despite the extension of the transform, it is not defined if any of the phase magnitudes of the system is null or the phase difference between two of them is 0 or πradians [21]. To cover all possible imbalances within 3-phase 4-wires systems, the authors recently presented a new reference frame called Reduced Reference Frame (RRF) [22] whose orientation depends on the trajectory, or locus, defined by the space vector of the analysed three–phase abc magnitude. This allows to formulate a power theory for computing the instantaneous active and reactive power similar to the one proposed by Akagi et al. for three-phase balanced systems in [7]. Additionally, the axes that define these coordinates are oriented taking advantage of the locus geometry. Specifically, they are oriented on the semi-major and semi-minor axes of its geometry. This brings a series of benefits once the space vector of the measured signals are expressed in terms of the RRF: (i) a dimensional reduction of the space vector from three to two coordinates and (ii) both coordinates are sinusiodal signals delayed a quarter of cycle and their amplitudes are related through the locus eccentricity. Moreover, this locus eccentricity allows to compute one RRF magnitude from the other one. In addition, this parameter can be used to classify the imbalances of 3-phase 4-wire systems depending on the locus shape. The RRF advantages have been also addressed in dynamic scenarios and protection systems. Particularly, the dynamics of the space vectors related to first-order 3-phase 4-wire circuits with different imbalances has been analyzed in [23]. Its application for line protection has been reported in [24]. This time-domain protection technique detects the fault type FIGURE 1. Three–phase four–wire system under study. based on the variation of the angle between the space vectors related to voltages and currents. The objective of this paper is to show and to interpret several case studies of unbalanced 3-phase 4-wires systems using the RRF proposed by the authors in [22]. For this purpose, a three-phase four-wire circuit composed of three voltage sources feeding a three-phase load is analysed working under load and voltage imbalances. The voltage and current in the load will be transformed using the classical Clarke transform, the mno axes and the recently proposed RRF coordinates. This will allow to demonstrate the benefits of the RRF approach compared to the Clarke and mno transformations in terms of unbalance classification, computation of the power terms and the relationships between the transformed magnitudes. The paper is organized as follows: Section II presents the most relevant concepts of the RRF transform. Section III presents several case studies for evaluating the performance of the RRF under unbalanced voltages and loads. Finally, Section IV presents the most important conclusions of this work. II. REDUCED REFERENCE FRAME TRANSFORMATION The theoretical fundamentals and mathematical formulation of the RRF transformation are fully described in [22]. This section presents the most important concepts on which the RRF transform is based to understand the results presented throughout the paper. Let us consider a three-phase four-wire system as shown in Fig. 1, where up to three linear independent line voltages va(t),vb(t)and vc(t)can be defined taking the voltage of the fourth wire as the voltage reference. Expressing the generic set of the linearly independent line voltages as: vph (t)=√2Vph cos ωt+θphph =a,b,c,(1) where ph is the tag of the wire, ωis the angular frequency, Vph and θph are the RMS value and phase angle of the voltage respectively. 24592 VOLUME 11, 2023 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems FIGURE 2. Voltage locus and 5v–plane. Note that Vph and θph can take any value. Thus, using the space vector representation proposed in [25] and [26], the three-dimensional vector vabc (t)is defined as: vabc (t)=[va(t),vb(t),vc(t)],(2) and this vector describes a three-dimensional trajectory over the time t. This trajectory can be easily interpreted rewriting (2) as: vabc (t)=c1cos (ωt)+c2sin (ωt),(3) since (2) is also the homogeneous solution of the harmonic oscillator extended to vabc (t): d2vabc (t) d t2+ω2vabc (t)=0.(4) where c1,c2are constant vectors given by the initial conditions: c1=vabc(t)t=0 ,(5) c2=d dt vabc(t) ωt=0 .(6) From (3) it is derived that the trajectory of vabc (t)is in the plane defined by c1and c2. Both, the orientation of that plane (hereinafter 5v–plane) and the locus or the shape of the vabc (t)trajectory (hereinafter λv–locus), depend on the degree of the voltage imbalance. Figure 2shows a general case of voltage imbalance where the λv–locus described by vabc (t)in the 5v–plane and vectors c1and c2are illustrated. In [22], three classes associated with the locus shape were defined: circular (Class I), elliptical (Class II) and linear (Class III). Note that Class II (elliptical shape) is the most general case while the other two are degenerated versions of it. The proposed RRF takes advantage of the locus-shape symmetry and geometrical properties. For this purpose, the RRF orientation is defined over the ellipse semi-major and semi-minor axes which are denominated in the transformation as xand ydirections respectively. The orthonormal direction to the 5v–plane is known as odirection defining the xyo axes of the RRF. As illustrated in Fig. 3, the RRF frame is defined by the set of orthonormal vectors {ex,ey,eo}. Note that eois pointing outward the 5v–plane. FIGURE 3. Voltage locus and xyo axes. The components of vabc (t)can be represented in the RRF as follows: vxyo (t)=vx(t),vy(t),vo(t)=Text Rv vabc (t),(7) where Text Rv is the unitary rotation matrix Text Rv =  exa exb exc eya eyb eyc eoa eob eoc ,(8) and ex=[exa,exb,exc],ey=eya,eyb,eycand eo= [eoa,eob,eoc]are the coordinates of {ex,ey,eo}expressed in the canonical base {[1,0,0],[0,1,0],[0,0,1]}or the abc frame. The resulting components vx(t),vy(t)and vo(t)are characterised by the following properties: •Null value for the orthonormal component vo(t)since all the locus lies within the 5v–plane. This property allows to use the reduced version of the transformation: TRv =exa exb exc eya eyb eyc,(9) where only xy are required to be computed: vxy (t)=TRvvabc (t),(10) •The vx(t),vy(t)components are two sinusoidal functions π/2 radians out of phase (1/4 cycle), whose amplitudes are related with the eccentricity εvof the ellipse. Since in a three-phase four-wire system there are at most three lineal independent line currents ia(t),ib(t),ic(t), a similar procedure can be applied to the currents to obtain the λi–locus, the 5i–plane and the Text Ri transform. Note that the subscript iis used to denote that the RRF transform is related to currents. Once the voltages and currents have been defined in the same RRF, they can be used to formulate the active and reactive powers. In this work, the considered reference plane is 5v, therefore the voltages and currents in the RRF are given respectively by (7) and ixyo(t)=Text Rv iabc(t).(11) The power definition is independent of the selected reference frame, thus, using the abc frame, the active power of the system, P, is defined as the sum of the active power of each phase, P=Pa+Pb+Pc P= ℜ{VaI⋆ a}+ℜ{VbI⋆ b}+ℜ{VcI⋆ c},(12) VOLUME 11, 2023 24593 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems and the reactive power Q=Qa+Qb+Qc Q= ℑ{VaI⋆ a}+ℑ{VbI⋆ b}+ℑ{VcI⋆ c},(13) where Va,Vb,Vcare the phasors related to va(t), vb(t), vc(t), and I⋆ a,I⋆ b,I⋆ care the complex conjugate of the phasors related to ia(t), ib(t), ic(t). Using the RRF transformation:   Vx Vy Vo  =Text Rv ·  Va Vb Vc  ,(14)   Ix Iy Io  =Text Rv ·  Ia Ib Ic  ,(15) the powers are formulated as P=Px+Py+Po, P= ℜ{VxI⋆ x}+ℜ{VyI⋆ y}+ℜ{VoI⋆ o},(16) and Q=Qx+Qy+Qo, Q= ℑ{VxI⋆ x}+ℑ{VyI⋆ y}+ℑ{VoI⋆ o},(17) where Vx,Vy,Voare the phasors related to vx(t), vy(t), vo(t) respectively and I⋆ x,I⋆ y,I⋆ oare the complex conjugate of the phasors related to ix(t), iy(t), io(t) respectively. The instantaneous active and reactive power can be expressed according to [27] in abc and RRF axes as follows: p(t)=vabc (t)·iabc (t)=vxy (t)·ixyo (t),(18) q(t)= ∥qabc (t)∥=∥qxyo∥=∥vxy (t)×ixyo (t)∥(19) The power computation using the RRF presents a key advantage with respect to the abc frame, as the component vois always null. This allows to obtain a reduced expression for the active power P, reactive power Q and instantaneous active power, P=Px+Py= ℜ{VxI⋆ x}+ℜ{VyI⋆ y},(20) Q=Qx+Qy= ℑ{VxI⋆ x}+ℑ{VyI⋆ y},(21) p(t)=vxix+vyiy.(22) However, this reduction is not achieved for the instantaneous reactive power since ioaffects the voltages in xy axes: qxyo (t)=  vyio −vxio vxiy−vyix ,(23) q(t)=ri2 ov2 x+v2 y+vxiy−vyix2.(24) Despite a reduction in the reactive power terms is not achieved, its computation is simpler than in the abc frame since the vocomponent is always null. In the case of 5vand 5iare coincident, the component io would be null. As a consequence, the instantaneous reactive power is reformulated as q(t)= ∥qo(t)∥: qo(t)=vxiy−vyix.(25) This allows to represent the instantaneous active and reactive power in a compact form as follows: p(t) qo(t)=vxvy −vyvxix iy.(26) Equation (26) allows to adopt a complex formulation similar to the one presented in [8], but extending its application to voltages and currents including zero–sequence components: p(t)+j qo(t)=v⋆ xy(t)ixy(t),(27) where v⋆ xy(t)=vx(t)−jvy(t), ixy(t)=ix(t)+jiy(t). III. CASE STUDIES This section presents two case studies for evaluating the performance of the RRF transform in comparison with the classical αβγ frame and the recent mno coordinates proposed by Montanari and Gole [17]. For this, the 3-phase 4-wire circuit depicted in Fig. 1is analyzed with voltage and load imbalance conditions as reported in the following subsections. A. VOLTAGE IMBALANCE This case study considers a balanced load fed from an unbalanced voltage source. In this case, the line currents circulating through the load can be computed as follows:   Ia Ib Ic  =1 (R+jX)  1 0 0 0 1 0 0 0 1   Va Vb Vc  ,(28) with R+jX =Z=ZejϕZ=pR2+X2ejarctan X R.(29) where it is observed that each current phasor is proportionally scaled and delayed in an identical manner with respect to its voltage phasor. Applying the RRF transformation, defined in (7) and (8), to the voltage and current phasors in (28), leads to:   Vx Vy Vo  =Text Rv   Va Vb Vc  =  VxejϕVx Vyej(ϕVx−π/2) 0 ,(30)   Ix Iy Io  =Text Rv   Ia Ib Ic  =1 Z  Vxej(ϕVx−ϕZ) Vyej(ϕVx−π/2−ϕZ) 0 .(31) The instantaneous voltage vxyo(t) and current ixyo(t) are computed as:   vx(t) vy(t) vo(t) =  √2Vxcos ωt+ϕVx √2Vysin ωt+ϕVx 0 ,(32)   ix(t) iy(t) io(t) =  √2(Vx/Z)cos ωt+ϕVx−ϕz √2Vy/Zsin ωt+ϕVx−ϕz 0 .(33) 24594 VOLUME 11, 2023 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems Note that the values of vo(t) and io(t) are null, therefore, the voltage and current loci of the vectors v(t), i(t) are in the same plane. As a result, it can be stated that irrespective of the voltage imbalance, the 5vand 5iplanes are coincident when the load is balanced. In addition, Vyand Iylag π/2 with respect to Vxand Ix, respectively. This means that λvand λiare two ellipses with aligned semi-major and semi-minor axes. The eccentricity εvof λvis given by the amplitude of vx(t) and vy(t): εv=s1−Vy Vx2 .(34) Note that, considering (33), εv=εi. As a consequence, λvand λiare concentric since both loci have the same semi-major and semi-minor axes. This feature holds for balanced loads irrespective of whether the voltage is balanced or not. The formulation of the different power definitions provided by (20), (21) and (26) can be particularized as a function of the voltage, load impedance and eccentricity as follows: P=2−ε2V2 x/Zcos ϕZ,(35) Q=2−ε2V2 x/Zsin ϕZ,(36) p(t) q(t)=P+ε2V2 x/Zcos 2ωt+2ϕVx−ϕZ ∥2√1−ε2V2 x/Zsin ϕZ∥(37) Note that q(t) has a constant value only affected by Vx,ϕZ and ε. Whereas, p(t) is composed by a constant value Pand cosine function ˜p(t)with twice the angular frequency. These theoretical analysis can be complemented in a numerical way considering two test cases with different voltage imbalances: (i) voltage with positive and negative sequences (PNV case) and (ii) voltage with positive, negative and zero sequences (PNVZ case). The symmetrical components of the voltages for these two case studies are set as follows: VPNV 012 =V  0 1 k k2  ,(38) VPNZV 012 =V  k k0 1 k2 2  ,(39) where {V,k0,k2} ∈ Care constant values, and k∈[0,1) is a sweep variable. Note that in this way, it is possible to analyze how the variation of the voltage imbalance affects the transformation results. The numerical analysis considers {V,k0,k2}={224.4 √20 V,1−π 2,1−π 2}. Figure 4depicts in the phasor domain the voltages VPNV 012 and VPNZV 012 for k= {0,0.25,0.5,0.75,1}. These voltages are easily transformed to the abc frame through the Fortescue Transform [28], [29]: Vabc =[TF]−1V012 =  1 1 1 1α2α 1α α2 V012,(40) where α=ej2π 3. FIGURE 4. Symmetrical components of PNV and PNZV source imbalance. The numerical analysis presented in the next subsections assume a balanced load for the PNV and PNVZ cases: Ra= Rb=Rc=R=120 ,Xa=Xb=Xc=X=2πfL, with f=50 Hz and L=1.2 H. 1) PNV VOLTAGE IMBALANCE The PNV case aims to study how the negative sequence affects the voltage and current loci within the αβγ ,mno and xyo reference frames. In this case, kis the ratio between the positive and negative sequence components. The locus variation and voltages over time in the αβγ ,mno and xyo reference frames for k= {0,0.5,1.0}in the PNV case are shown in Fig. 5. The first row within Fig. 5presents the results in the αβγ and mno reference frames while the second row shows the voltage in the xyo reference frame. In this figure, the first column shows the voltage loci in each reference frame and the rest of the columns present the voltage components for the different unbalance parameter k. Note that for k=0, the voltage locus is circular in all the reference frames, being the 5-plane, the αβ-plane and mno-plane coincident. Moreover, the mno axes have the same orientation than the αβγ axes. This is because the objective of the mno transform is to contain the transformed-magnitude locus within the αβ-plane. It is interesting to note that, due to the circular voltage locus, it is possible to select the xyo to be coincident with the αβγ and xyo reference frames, since the semi-major and semi-minor axes have the same length. As a result, the time evolution of the voltage components is identical for all the reference frames, as shown in the second column of the Fig. 5. For k>0, the locus eccentricity increases approaching gradually to the linear locus. In these cases, a remarkable difference between the transformations appears since the αβγ and mno axis remains in the same position independently of the imbalance type. On the contrary, the xyo axis are oriented with the ellipse semi-major and semi-minor axes adapting to the specific unbalance voltage. As a result, the voltage components in the xyo axes remain with a phase difference of π/2 radians while those of αβγ and mno depend on the imbalance. Moreover, it is important to point out that the mno transform is not defined for the linear locus, i.e. k=1, as shown in the fourth column of Fig. 5. Finally, it is worth noting that, due to the absence of zero-sequence components, the voltages related to γand o VOLUME 11, 2023 24595 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems FIGURE 5. PNV voltage source imbalance. Voltage loci in αβγ ,mno and xyo reference frames. components are zero regardless of the negative sequence imbalanced defined by k. The current loci follows a similar trend than the voltage one since, according to (33), the current components are just scaled and delayed with respect to the voltages. Considering voltage and current loci in each of the transformation axes, the instantaneous values of p(t)and q(t), defined according to (37), are presented in Fig. 6for k= {0,0.5,1}. As expected, constant values of p(t)and q(t)are obtained for k=0 due to the balanced conditions of voltage and load. On the contrary, p(t)is no longer constant for k>0, which oscillates at 100 Hz. However, it is interesting to note that q(t)is constant irrespective of the negative sequence content. Moreover, q(t)is null for k=1 since vxy (t)and ixyo (t)are parallel each other (linear locus), being the cross product in eq. (19) equal to zero. 2) PNZV VOLTAGE IMBALANCE The PNZV case aims to study the influence of the zero sequence component in αβγ ,mno and xyo reference frames considering constant positive and negative sequence components. In this case, kis the ratio between the positive and zero sequence components. Fig. 7illustrates the voltages in each of the analyzed reference frames. Note that the figure contains three rows since the voltage loci within the mno reference frame is no longer equal to the corresponding loci on in αβγ axes. The results for k=0 leads to the conclusions pointed out in the previous subsection due to the absence of zero-sequence component. The zero-sequence component is responsible of a non-null γcomponent in the conventional αβγ reference frame as shown in Fig. 7for k>0. On the contrary, the mno and xyo FIGURE 6. PNV voltage source imbalance. Instantaneous power p(t), q(t) (solid lines). Active and reactive powers P,Q(dashed lines). k=0.0 (blue color), k=0.5 (orange color) and k=1.0 (green color). transformations are able to describe the imbalance voltage just with two components due to their capability of adapting to the voltage imbalance. With this regard, it can be stated the superior performance of the xyo reference frame with respect to the mno axes since the voltage components remain always orthogonal irrespective of the zero-sequence voltage content. The active and reactive instantaneous power for this PNVZ case are shown in Fig. 8. Evidently, the instantaneous active power maintains the oscillatory component at 100 Hz while the instantaneous reactive power remains constant. This is because, as shown in (33), voltage and current loci have the same eccentricity. 24596 VOLUME 11, 2023 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems FIGURE 7. PNZV source imbalance. Voltages in αβγ ,mno and xyo reference frames. FIGURE 8. PNZV source imbalance. Instantaneous power p(t), q(t) (solid lines). Active and reactive powers P,Q(dashed lines). k=0.0 (blue color), k=0.5 (orange color) and k=1.0 (green color). B. LOAD IMBALANCE This case considers balanced voltages and unbalanced loads in the system depicted in Fig. 1. The load current can be formulated in a matrix form as: Iabc =Yabc Vabc,(41) where Yabc is the admittance matrix. These currents can be represented in the RRF applying the voltage-based transformation matrix Text Rv as: Ixyo =Text Rv Yabc Text Rv TVxyo =Yxyo Vxyo.(42) The relationship between the admittance matrices, Y, formulated in the abc frame, symmetrical components and xyo coordinates are illustrated in Fig. 9. In case of a balanced voltage, the corresponding loci in the RRF is circular and Vxand Vyare signals delayed π/2 radians with equal amplitude: vx(t) vy(t)=√2Vxcos ωt+ϕVx √2Vxcos ωt+ϕVx−π/2.(43) Moreover, it is worth noting to note that the xand yaxes can be located in multiple positions due to the circular loci as shown Fig. 10. This figure also depicts a general case where the current plane 5iis not oriented with the voltage VOLUME 11, 2023 24597 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems FIGURE 9. Ymatrix transformations. plane 5v. With the aim of simplifying as much as possible the mathematical formulation, the xand yaxes are aligned with the semi-major and semi-minor axis of the projected current loci, projλi, in the voltage plane 5v, as shown in Fig. 10. Note that, in this way, the xand ycurrent components are related by the loci eccentricity in a similar manner than (34): Iy=Ixq1−ϵ2 i.(44) Therefore, the current formulated in the xyo coordinates can be expressed as   Ix Iy Io  =Vx   yxx −jyyx yxy −jyyy yxo −jyyo   ,(45) and   ix(t) iy(t) io(t) =  √2Ixcos ωt+ϕIx √2Iycos ωt+ϕIx−π/2 √2Iocos ωt+ϕIo ,(46) where yjk with j,k∈ {x,y,o}corresponds to the different terms of the admittance matrix Yxyo. Note that the currents ix(t) and iy(t) are in quadrature, and in addition, the current Iois not null in a general case meaning that the current and voltage loci are in different planes. According (20), (21), (22) the power terms P,Q,p(t)only depends on xand ycomponents since the voltage vois always null. Therefore these power terms can be computed in this particular case as follows: P=VxIx1+q1−ϵ2 icos ϕ, (47) Q=VxIx1+q1−ϵ2 isin ϕ, (48) p(t)=P+ ˜p(t),(49) ˜p(t)=VxIx1−q1−ϵ2 icos 2ωt+2ϕVx−ϕ,(50) where ϕ=ϕVx−ϕIx.(51) FIGURE 10. Locus of current and its projection into the plane of voltages for unbalanced load and balanced voltages. On the other hand, q(t)is a function of io(t), as stated in (24). For this particular case with balanced voltages, the instantaneous reactive power can be formulated as: q(t)=Vxpf1(t)+f2(t)+f3(t)+C1+C2+C3,(52) where f1(t), f2(t), f3(t) are oscillatory terms and C1,C2,C3are constant terms: f1(t)=4I2 ocos 2ωt+2ϕIo, f2(t)=2 (sin ϕ)(Ixϵi)2sin 2ωt+2ϕVx−ϕ, 24598 VOLUME 11, 2023 F. Casado-Machado et al.: RRF Transformation Assessment in Unbalanced Three-Phase Four-Wire Systems FIGURE 11. Active and reactive power for the S-type and T-type loads versus the imbalance factor k. f3(t)=−I2 x 21−q1−ϵ2 i2 cos 4ωt+4ϕVx−2ϕ, C1=2I2 o, C2=(2Ixsin ϕ)2q1−ϵ2 i, C3=Ix1−q1−ϵ2 i21 2+sin2(ϕ). Note that q(t)is periodic and non-negative. In order to evaluate numerically the previous theoretical formulation, two types of load imbalance, one that tends to a single-phase load and other that tends to a two-phase load, namely type S and type T respectively, are considered with the following admittance matrices: YS abc =1 (R+jX)  1−k0 0 0 1 −k0 0 0 1 +2k ,(53) and YT abc =1 (R+jX)  1−k0 0 0 1 +k/2 0 0 0 1 +k/2 ,(54) where k∈[0,1) is a sweep variable which control the imbalance degree. Note that for k=0, type S and type T loads perform like a perfectly balance load, while for k=1 are a single-phase and two-phase loads respectively as shown in Fig. 11. This numerical analysis has been done assuming a perfectly 50 Hz balanced three-phase voltage source with amplitude 224.4 V, R=120 and L=1.2 mH. Note that in both cases the total active and reactive power load remain constant but, depending on the imbalance degree, this load is shared unevenly between the phases. The next subsections are devoted to evaluate these two types of load imbalance in the different reference frames. FIGURE 12. Voltages in αβγ and in xyo for any k value for load imbalance type S. 1) S-TYPE LOAD IMBALANCE This subsection analyzes the current loci in the αβγ ,mno and xyo reference frames for the S type load with k= {0,0.5,1.0}. This analysis just refers to currents since the voltage is assumed balanced and constant in all the analyzed cases. In fact, the voltage locus and the voltage components in each reference frame are exactly the same that the ones shown in Fig. 12. Given this voltage, the mno axes are the same than the conventional αβγ ones. For this reason, the current loci and the corresponding current components in these axes for any unbalance factor kare exactly the same, as shown in Fig. 13. Regarding the xyo transformation, the reference frame axes are aligned with the semi-major and semi-minor axes of the projection of the current loci on the voltage plane, as previously stated, to take advantage of the straightforward computation of the power terms using (47)-(51). For this reason, the current components in the xand yaxes are delayed with respect to their counterparts in the mno and αβγ axes. If the load imbalance increases, k=0.5, the current loci is out of the voltage plane and, consequently the third current, i.e. iγand io, orthogonal to the voltage plane appears in all the transformations. Moreover, the imbalance increases the eccentricity of the current locus, which modifies the phase difference between the current components in the mno and αβγ reference frame. On the contrary, the current components in the xyo frame remain with a phase difference of π/2 radians due to a suitable axes alignment. Finally, in the case of a single-phase load, i.e. k=1, it is interesting to note that iy=0 in the xyo transformation being the current locus described just with ixand iowhich are in phase denoting a linear locus. The power terms related to these current components for the different unbalance degrees of the S-type load are shown in Fig. 14. As expected, the instantaneous active power remains constant for the balance case, k=0, but starts oscillating with the load imbalance, i.e. ϵi>0, according to (50). It is interesting to note that the total active power is constant for all the cases, which is consistent with the results shown in Fig. 11. Therefore, considering (44) and (47), the term Ix1+q1−ϵ2 iwhich is equal to Ix+Iy is constant irrespective of the imbalance factor, as shown in VOLUME 11, 2023 24599