Electric Power Systems Research 80 (2010) 788–798 Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr Tensor analysis of the instantaneous power in electrical networks A.J. Ustariz a,∗ , E.A. Cano a , H.E. Tacca b a b Department of Electrical, Electronic and Computer Engineering, Universidad Nacional de Colombia, Campus La Nubia. A.A. 127, Manizales, Colombia Department of Electronic Engineering, Universidad de Buenos Aires, Buenos Aires, Argentina a r t i c l e i n f o Article history: Received 8 September 2009 Received in revised form 12 December 2009 Accepted 14 December 2009 Available online 6 January 2010 Keywords: Instantaneous active power Instantaneous reactive power Vector analysis Tensor analysis Tensor product Instantaneous power tensor a b s t r a c t This paper proposes an alternative physical interpretation and calculation of instantaneous power, using the concept of tensor product as a mathematical tool. “Instantaneous power tensor” is the single expression defined using this theory of power, which involves the two components of instantaneous power (active and reactive), in order to geometrically interpret the behavior of electrical phenomena, analogous to studies of deformation in the mechanics of solids. Additionally, a comparison is made between the definition of instantaneous power in the literature (vector analysis) and the proposed definition (tensor analysis). © 2009 Elsevier B.V. All rights reserved. 1. Introduction Power components such as active power, reactive power and apparent power provide the necessary information to design, evaluate, monitor and compensate electrical systems. These powers are thus used to define system characteristics as power factor, installed power capacity, or demanded power. Nowadays, the components of power are well defined in linear single-phase systems or balanced three-phase linear systems under sinusoidal conditions. However, the increased usage of adjustable speed drives, arc furnaces and single-phase loads like computers and compact fluorescent lamps (CFLs), which are nonlinear loads, distort both the current and voltage waveforms [1]. In this case, the definition of these components is controversial allowing so many approaches, that while some authors restrict their analysis to the frequency domain or time, others formulate broader and more general theories by using the time-frequency domain [2–4]. Fig. 1 shows the theories that led to this division. In addition, since the beginning of the last century many theories have been proposed to solve the problem, based on some of the approaches already mentioned. These theories, besides choosing a specific approach (frequency, time or time-frequency), have used one of the following trends: (a) theories based on a periodic concept ∗ Corresponding author. Tel.: +57 68 879400x55725; fax: +57 68 879400x55725. E-mail addresses: ajustarizf@unal.edu.co, ajustarizf@hotmail.com (A.J. Ustariz), eacanopl@unal.edu.co (E.A. Cano), htacca@fi.uba.ar (H.E. Tacca). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.12.004 [5–9], which were developed for single-phase systems and have been implemented mainly for measuring the harmonic content and (b) theories based on an instantaneous concept and on a vector approach [10–16], which were developed for three-phase systems having as a main objective the design of control algorithms useful for active filters. This subject is still debated, and there is not yet a generalized power theory that can be assumed as a common base for physical interpretation of power phenomena, power quality assessment and non-active power compensation in multiphase power systems. Therefore, departing from the theories based on an instantaneous concept and on a vector approach, an alternative of instantaneous power calculation and physical interpretation is proposed using the concept of tensor product as a mathematical tool. The “instantaneous power tensor” is a unique compact expression defined using this concept which involves the two components of instantaneous power (active and reactive), in order to geometrically interpret the behavior of electrical phenomena, analogous to deformation formulations in the mechanics of solids. Also, the mathematical tool here proposed may be easily implemented in the time domain, offering an option for the power quality assessment and non-active power compensation in multiphase power systems. This paper is organized as follows. Section 2 describes the definition of instantaneous power using a vector analysis. Section 3 introduces the tensor analysis of the instantaneous power. In Section 4 practical applications the tensor analysis are presented. Finally, in Section 5 the most important conclusions are presented. A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 789 Fig. 1. Approaches to the definitions of power on a nonsinusoidal system. 2. Vector analysis of the instantaneous power 2.1. Mathematical foundations Nowadays a modern notation is used in three-phase systems (see Fig. 2(a)), where instantaneous voltages va , vb , vc and instantaneous currents ia , ib , ic are expressed as instantaneous space vectors: v =   va vb , vc i =   ia ib ic (1) These expressions allow instantaneous voltages and currents to be represented in an orthogonal system, as shown in Fig. 2(b). Based on this representation, two expressions define the components of instantaneous power. The first component is the instantaneous active power, expressed using the point or internal product as: p = v · i = va ia + vb ib + vc ic (2) The result is a scalar that involves direct products of each phase. Furthermore, the cross or external product is used to define the instantaneous reactive power, thus:  = v × i = q  qa qb qc  =  vb ic − vc ib vc ia − va ic va ib − vb ia  (3) The result of this operation is a vector that involves the cross product of each phase. These two expressions are valid for threephase systems with zero sequence components of voltage or current, satisfying the following orthogonal relation:    s = v · i =  p2 + q2 (4) where || · || denotes the Euclidian norm of the vector. 2.2. Physical interpretation Although, originally the physical interpretation of the components of instantaneous power was presented in the reference framework ␣-␤-0 in [17], the concept also applies to the definitions given in (2) and (3). Fig. 3 shows the interpretation given to the Fig. 3. Physical interpretation of the instantaneous active and reactive powers [17]. components of instantaneous power, where p represents the total instantaneous energy flow per unit time and q the energy exchange between the phases without energy transport. This interpretation presents as a main application the design of control algorithms for active filters. 3. Tensor analysis of the instantaneous power 3.1. Mathematical foundations The proposed method is based on the definition of an evolutionary expression of instantaneous power, called “instantaneous power tensor”, in order to geometrically interpret the dynamic behavior of electromagnetic phenomena, analogous to studies of deformation in the mechanics of solids. Therefore, the goal proposed is to obtain a single expression containing the two components of the instantaneous power described by Eqs. (2) and (3) using the dyadic or tensor product [18] the proposed expression is as follows: ℘ij = vi ⊗ ij = vi′ (5) where ℘ij is the instantaneous power tensor, v is the vector of phase voltages and i′ is the vector transpose of line current. On the other hand, a multiphase electrical network was defined as one n-phase and m-wire power system related to the dimension of vectorial space, where voltage vector and current vector were defined assuming that, R1 is the single-phase system, R2 is the two-phase system with two-wire or three-wire, R3 is the three-phase system with three-wire or four-wire and Rn is the multiphase system with n-phase and m-wire. Fig. 2. Three-phase power system: (a) electrical networks and (b) instantaneous space vectors. 790 A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 Therefore, by replacing (1) in (5) the instantaneous power tensor in R3 is expressed as: ⎡ va ia ⎤ ⎡ ⎤ ⎡ v a ia v a ib v a ic ℘ij = ⎣ vb ⎦ ⊗ ⎣ ib ⎦ = ⎣ vb ia vc ic ⎤ vb ib vb ic ⎦ vc ia vc ib vc ic (6) Moreover, in a system of n-phase and m-wire, where the instantaneous space vectors of voltage and current are expressed as: ⎤ ⎡ v1 ⎢ v2 ⎥ v = ⎢ . ⎥ , . ⎢ ⎥ ⎣ . ⎦ vn ⎡ ⎤ i1 ⎢ i2 ⎥ ⎢ ⎥ i = ⎢ ⎥ ⎢ .. ⎥ ⎣.⎦ (7) Fig. 4. Geometric representation of the tensor of instantaneous power in R3 . in 3.3. Relation to vector analysis Then, the instantaneous power tensor in Rn is equal to: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ v1 i1 v1 i2 · · · v1 in i1 ⎢ v2 ⎥ ⎢ i2 ⎥ ⎢ v2 i1 v2 i2 · · · v2 in ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎥ ℘ij = ⎢ . ⎥=⎢ . . . ⎥ ⎢ .. ⎥ ⊗ ⎢ ⎢ . . . . . . . ⎣ . ⎦ ⎣.⎦ ⎣ . . . ⎦ Applying a tensor invariant (the trace) and using the Einstein sum convention, the instantaneous active power is defined as: v1 vn in (8) vn i1 vn i2 · · · vn in when n = 1, it is the case of a single-phase system, and the instantaneous values of voltage and current are scalars. Therefore, according to the traditional concept of instantaneous power in single-phase systems the instantaneous power tensor in R1 is equal to: ℘ij = v1 i1 (9) Moreover, any tensor of second order can be unambiguously expressed as the sum of a symmetric and an antisymmetric tensor [19] and: ℘ij = 1 1 ℘ij + ℘ij′ + ℘ij − ℘ij′ 2 2 (10) ℘ij′ where is the transpose of the instantaneous power tensor. This expression can be written in compact form, as: ℘ij = sym ℘ij + ant ℘ij p = trace(sym ℘ij ) = sym ℘ij ıij = sym ℘ii (12) where the symbol ıij is the “Kronecker delta”. The results are the same as those obtained with the internal product shown in (2). Similarly, the instantaneous reactive power may be calculated as follows: qij = 2(ant ℘ij ) = ℘ij − ℘ij′ (13) Since every antisymmetric second order tensor is associated with a dual vector [21], the tensor components qij are related to components of the vector qk given in Eq. (3) by: qk = 1 ε q 2 kij ij (14) where εijk is the “Levi-Civita tensor”. Finally, the instantaneous apparent power is defined as: s = ||℘ij || =  max (℘ij′ ℘ij ) (15) Wherever ||℘ij || is the Euclidian norm of the instantaneous power tensor and max is the maximum eigenvalue. (11) This decomposition is independent of the selected coordinate system. 3.2. Physical interpretation The tensor approach proposed considers a volume element to analyze the physical interpretation of the instantaneous power. The interpretation is done from a geometric point of view, making analogy with the analysis of deformations in the mechanic of solids [20]. For convenience, the volume element is cubic (see Fig. 4) due to the implementation of the instantaneous power tensor in a rectangular coordinate system. Therefore, the tensor components will act on the cube provoking changes in the dimensions. These measurements of shape deviation are the tools to analyze and characterize the electrical phenomena. Fig. 5 shows the proposed interpretation, where, ℘ij represents the total energy volume per unit of time. The representation of the instantaneous power tensor in R3 is valid only in three-phase, Three-wire and four-wire systems. However, the mathematical tool developed to analyze the deformations can be extended to Rn . 3.4. Simulation results An illustrative example to analyze the physical interpretation of the instantaneous power tensor in three-phase power system is shown below. The nonlinear load of the circuit shown in Fig. 6 is characterized by an unbalanced three-phase alternating-current regulator with inductive load. The voltage source is sinusoidal, balanced, and with positive sequence and a frequency of 60 Hz. Fig. 5. Physical interpretation of the tensor of instantaneous power in R3 . A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 791 (a) The symmetric tensor shows a change in the volume and in the shape of the cube, meaning pure deformation (see Fig. 7d). (b) The antisymmetric tensor shows changes in the angle of rotation without changing the volume of the cube, keeping the original shape, in other words, pure rotation (see Fig. 7e). 4. Practical applications 4.1. Non-active power compensation From (5): Fig. 6. Unbalanced three-phase alternating-current regulator. Fig. 7a and b shows the voltage and current signals of each phase. Fig. 7c shows two superimposed waveforms depicting the instantaneous active power behavior; one of them calculated using (2) and the other using (12). Fig. 7c also shows the behavior of instantaneous reactive power calculated using (3) and (14). Fig. 7d and e shows the deformations of the symmetric and antisymmetric tensor regarding the null power condition. In summary, the deformations of the analyzed cube regarding the null power condition determined that: ℘ij′ ′ = (vi′ ) = iv ′ (16) This expression implies that the current vector can be obtained from the instantaneous power tensor, thus: i = ℘ij′ v · v v (17) and using (13) and (17), the current vector may be expressed by: i = ℘ij − qij v = ℘ij v + −qij v v · v v · v v · v (18) As result of this current vector decomposition, based on this new tensor formulation, two main compensation approaches are pos- Fig. 7. Simulation results of the alternating-current regulator. 792 A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 Fig. 8. Matlab-Simulink implementation compensating strategies. sible: the instantaneous-time compensation or the average-time compensation. First, in the instantaneous-time compensation – ITC the reference current is obtained, as: ℘ij′ − ℘ij −qij i v v = ref = v · v v · v (19) In the second approach, several strategies (constant active power method – CAP, unity power factor method – UPF and perfect harmonic cancellation method – PHC) might be considered for the extraction of the reference currents [22]. Thus, the reference current for the first compensating strategy – CAP results: i ref ¯ ij ) trace(℘ v = i − v · v (20) For the second compensation strategy – UPF is defined as:  i ref = i − ¯ ij ) trace(℘ (1/T )  T (v · v) v (21) Finally, for the third compensating strategy – PHC is:  i ref = i − ¯ ij ) trace(℘ (1/T )  + T + (vf · vf ) v+ f (22) where v+ is the voltage space vector with a single fundamental f ¯ ij is the average of instantapositive-sequence component and ℘ neous power tensor. The average of instantaneous power tensor is Fig. 9. Three-phase and six-phase waveforms. A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 793 Fig. 10. Simulation result instantaneous-time compensation – ITC. moment of the filter connection. For all cases, the information is organized as follows: then calculated as ⎡   ⎤ ⎢ T v1 i1 dt T v1 i2 dt · · · T v1 in dt ⎥ ⎥ ⎢   ⎥ ⎢ ⎥ ⎢ v2 i1 dt v2 i2 dt · · · v2 in dt ⎥ 1⎢ ⎥ ⎢ ¯ ij = ℘ T T T ⎥ T ⎢ . . . . ⎥ ⎢ . . .. . . . . ⎥ ⎢ ⎥ ⎢   ⎦ ⎣ vn i1 dt T vn i2 dt · · · T (23) vn in dt T In this paper, the different compensation strategies have been applied to a nonlinear six-phase system (see Fig. 8). In this example, a three-phase source with a nonsinusoidal and unbalanced voltage supplies a three-winding transformer Y, Y. The secondary windings supply a 12 pulse rectifier with an inductive load in the dc side. This system has been chosen to illustrate the definitions proposed in the previous sections as it is a practical case where six-phase waveforms can be identified (as shown in Fig. 9). Fig. 9 displays the next important waveforms: the voltage on the transformer primary side – vp and voltage on the transformer secondary side – vs . Simulation results for the different compensation strategies are presented in Figs. 10–13 and Table 1. The emphasized signal corresponds to the first phase, while t = 350 ms is the ip : current in the transformer primary side. is : current in the transformer secondary side. iref : reference current for the selected strategy. p: instantaneous active power in the transformer primary side. q: instantaneous reactive power in the transformer primary side. The terms related to power concepts in Table 1 are based on the new definitions of power proposed by the IEEE Working Group on Nonsinusoidal Situations collected in IEEE Standard 1459 [8]. Table 1 Summary of simulation results for the different compensation strategies. Factor iLoad ip ITC ip CAP ip UPF ip PHC Irms phase-a [A] If phase-a [A] THD phase-a [%] dPF phase-a P [kW] Se [kVA] PF 1.6673 1.6480 15.3491 0.9542 116.49 123.19 0.9456 1.5814 1.5790 5.5156 1.0000 116.37 116.54 0.9985 1.5823 1.5812 3.7307 1.0000 116.43 116.58 0.9987 1.5763 1.5753 3.5637 1.0000 116.45 116.45 1.0000 1.5744 1.5744 0.0000 1.0000 116.35 116.43 0.9993 794 A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 Fig. 11. Simulation result average-time compensation – CAP. The simulation results for the ITC strategy are shown in Fig. 10 and summarized in the third column of Table 1. Notice that the instantaneous reactive power is fully compensated while the instantaneous active power delivered by the source equals the instantaneous active power of load. Therefore, in terms of distortion it does not cancel all the harmonic of the source current. Simulation results for CAP strategy are shown in Fig. 11 and summarized in the fourth column of Table 1. Here, in contrast to the ITC strategy the instantaneous active power delivered by the source equals the constant active load power. In terms of distortion it does not cancel all the harmonic present in the source current, but the distortion is minor than the one found in the ITC strategy. Simulation results for UPF strategy are shown in Fig. 12 and summarized in the fifth column of Table 1. Here, the UPF strategy does not cancel all the harmonics in the source current but it maintains the same distortion of the source voltage. As it can be notice from Table 1, only the UPF strategy attains a unity value for the power factor. However, the source delivers oscillating components of active power. As it can be stated from Fig. 13, the PHC strategy cancels all the current harmonics in the transformer primary and secondary windings. However, in terms of power, PHC strategy does not fully compensate the instantaneous reactive power. Additionally, the source delivers oscillating components of active power. 4.2. Power quality assessment Traditionally, in stationary state the power quality assessment is based on individual indicators such as total harmonic distortion (THDmax ), load current unbalance factor (UF) and reactive factor (QF) [23]. Also, the maximum value between all these indices is chosen as a global indicator of power quality (GI), thus: ⎛  THDmax = max ⎝ UF = QF = I− f I+ f Qa f I2 h= / 1 h a I1 a ,  I2 h= / 1 h b I1 b ,  I2 h= / 1 h c I1 c ⎞ ⎠(24) (25) + Qb f + Qc f Pa + Pb + Pc GI = max(THDmax , UF, QF) (26) (27) where the Ih k is the current harmonic in the phase-k (k = a, b, c), I− f and I+ f are the fundamental negative and positive sequence components and Qk f is the fundamental reactive power of phase-k. Now, a single discriminating factor to power quality assessment is proposed based on the decomposition of instantaneous power tensor. This factor afterwards could be used to penalize A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 795 Fig. 12. Simulation result average-time compensation – UPF. or to compensate the customers at the point of common coupling (PCC), similarly as the power factor has been used. Establishing this factor implies to define the ideal power system as one circuit composed of a balanced and sinusoidal voltage source (reference source) supplying a resistive balanced load (reference load). Any situation producing nonconformity with respect to these ideal conditions supposes a quality loss. Moreover, this ideal circuit is the unique power system that presents a totally symmetric instantaneous power tensor. The evaluation of these nonconformities can be quantified through the definition of a new power quality index, named instantaneous deviation indicator of power quality IDIpq , stated by   n n  i=1 j=1 (℘ij − ideal ℘ij )2 IDIpq =  n n ideal 2 i=1 j=1 ℘ij (28) and next, the average deviation indicator of power quality follows. Defined as: DI pq = 1 T  IDIpq dt (29) T In the expression (28) the term ideal ℘ij corresponds to the instantaneous power tensor of the ideal power system (tensor totally symmetric). The ideal instantaneous power tensor is calculated as follows: ideal ℘ ij + = vf ⊗ if+ active ¯ ) trace(℘ +  ij v+ = vf ⊗ f + +   (1/T ) (vf · vf ) + + = Ge (vf ⊗ vf ) (30) T where v+ is the voltage vector with a single fundamental positivef sequence component and Ge is the equivalent conductance of the ideal power system. In this paper, the power quality assessment approach proposed based on the instantaneous power tensor theory has been applied to a three-phase four-wire system similar to the one presented in Fig. 14. In addition, the IDpq index is in contrasted against the GI index. For comparison proposes, several simulations done using both ideal and distorted voltage under different load current conditions. The RMS voltage values and the load parameters values of the three-phase four-wire system presented in Fig. 14 are shown in Table 2. The details of each load connected at the PCC are shown in Fig. 15. 796 A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 Fig. 13. Simulation result average-time compensation – PHC. Table 2 System parameters. Three-phase supply Va [Vrms ] Vb [Vrms ] Vc [Vrms ] Ideal 127∠0◦ 127∠−120◦ 127∠−240◦ Distorted Fundamental 3rd harmonic 127∠0◦ 127∠−120◦ 101∠−240◦ 3.21∠0◦ 3.21∠0◦ 3.07∠0◦ Balanced load Load #1 Load #2 Load #3 Ra = Rb = Rc [] La = Lb = Lc [mH] 2.00 – 2.00 5.00 2.00 – Unbalanced load Load #1 Load #2 Load #3 Ra Rb Rc La Lb Lc 2.00 3.87 4.84 – – – 2.00 3.87 4.84 5.00 6.32 7.36 2.00 3.87 4.84 – – – Filter of dc side Load #1 Load #2 Load #3 L [mH] C [␮F] – – – – 1.00 1000.00 Fig. 14. The studied example: electric power utility and customers. A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 797 Fig. 15. Load types: (a) resistive, (b) inductive and (c) nonlinear. Table 3 shows the sequence of connection and disconnection of the switches S-1, S-2 and S-3 for all cases. 4.2.1. Case A: ideal voltage, balanced load current Simulation results for case A are summarized in Table 4. Here, one may observe that the DI pq is equal to zero when the voltage remains is sinusoidal and the load is resistive and balanced. This result agrees with the proposed definition of the ideal circuit. When the load connected at the PCC is only resistive, inductive or nonlinear, the DI pq becomes similar to GI. However, when the load connected at the PCC, results from the combination of the three types of load, the obtained DI pq is dissimilar to GI. 4.2.2. Case B: ideal voltage, unbalanced load current The simulation results for case B are summarized in Table 5. The results and conclusions are similar to the ones related to case A. The difference relies on the fact that for case B the load current unbalance factor (UF) is non-null for all the time intervals. Also, it is observed an increased value for the DI pq and GI indexes. Table 3 Sequence of connection and disconnection of switches. Switch 0 ≤ t < 0.2 0.2 ≤ t < 0.4 0.4 ≤ t < 0.6 0.6 ≤ t < 0.8 S-1 S-2 S-3 On Off Off Off On Off Off Off On On On On Table 6 Summary of simulation results for case C. Factor 0 ≤ t < 0.2 0.2 ≤ t < 0.4 0.4 ≤ t < 0.6 0.6 ≤ t < 0.8 THDmax [%] UF [%] QF [%] GI [%] DI pq [%] 3.26 7.21 0.00 7.21 14.37 1.54 7.10 94.33 94.33 94.93 36.68 7.30 −6.51 36.68 37.47 15.79 7.18 14.89 14.87 25.84 Table 7 Summary of simulation results for case D. Factor 0 ≤ t < 0.2 0.2 ≤ t < 0.4 0.4 ≤ t < 0.6 0.6 ≤ t < 0.8 THDmax [%] UF [%] QF [%] GI [%] DI pq [%] 3.26 32.29 0.00 32.29 47.12 1.90 24.73 78.23 78.23 85.78 60.71 24.80 −4.58 60.71 58.30 27.62 26.01 14.35 27.62 45.38 4.2.3. Case C: distorted voltage, balanced load current Simulation results for case C are summarized in Table 6. Now, with the voltage source nonsinusoidal and unbalanced, the differences between the indexes GI and DI pq are increased. 4.2.4. Case D: distorted voltage, unbalanced load current The simulation results for the case D are summarized in Table 7. These results and conclusions are similar to the case C. From the tables above presented, the DI pq index seems suitable to be adopted as a global factor for power quality assessment. Table 4 Summary of simulation results for case A. Factor 0 ≤ t < 0.2 0.2 ≤ t < 0.4 0.4 ≤ t < 0.6 0.6 ≤ t < 0.8 THDmax [%] UF [%] QF [%] GI [%] DI pq [%] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 94.24 94.24 94.15 38.76 0.00 −5.10 38.76 34.8 17.70 0.00 15.41 17.70 22.70 Table 5 Summary of simulation results for case B. Factor 0 ≤ t < 0.2 0.2 ≤ t < 0.4 0.4 ≤ t < 0.6 0.6 ≤ t < 0.8 THDmax [%] UF [%] QF [%] GI [%] DI pq [%] 0.00 27.88 0.00 27.88 39.10 0.00 21.95 74.79 74.79 82.41 62.96 19.52 −3.21 62.96 53.43 30.02 21.35 13.90 30.02 38.84 5. Conclusions In this paper the instantaneous power tensor theory was introduced using the concept of tensor product, in order to study electrical phenomena in an analogous way as formerly done for deformations in the mechanic of solids. The deformation of the cube analyzed regarding the null power condition led to determine that: (a) the symmetric tensor shows change in the volume and in the shape of the cube and (b) the antisymmetric tensor shows changes in the angle of rotation without changing the volume of the cube, keeping the original shape. The mathematical tool introduced proposes a way for calculating the instantaneous power components (active and reactive) in multiphase power systems. This new formulation also allows different compensation strategies to be defined in n-phase systems, based on the symmetric and antisymmetric tensor decomposition. 798 A.J. Ustariz et al. / Electric Power Systems Research 80 (2010) 788–798 A single DI pq index has been proposed as the basis of the power quality assessment based on the instantaneous power tensor theory. The average deviation indicator of power quality proved to be successful, as found by comparisons made between the ideal power system introduced and many other typical loads selected from a variety of test cases. Acknowledgement This work is supported by the Universidad Nacional de Colombia, Sede Manizales. References [1] W.G. Morsi, M.E. El-Hawary, Defining power components in nonsinusoidal unbalanced polyphase systems: the issues, IEEE Trans. Power Deliv. 22 (October (4)) (2007) 2428–2438. [2] C.I. Budeanu, Reactive and fictitious powers, Publication No. 2 of the Rumanian National Inst. Bucharest, Romania, 1927. 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