# A new approach for three-phase loads compensation based on the instantaneous reactive power theory

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2 66 P. Salmerón et al. / Electric Power Systems Research 78 (8) representation, which simplifies the obtention of compensation currents. Finally, the different compensation strategies have been applied to a practical case, which has allowed the proposed approach to be verified. The paper is organized as follows. In Section, the p q theory is summarized and a new formulation, without using mapping matrices, is presented. Section presents the way of obtaining the compensation currents in the p q theory reference frame according to the strategy denominated constant power. In Section 4, the procedure to obtain the compensation currents with the compensation objective named unity power factor is established. In Section 5, a similar development with the compensation objective of obtaining balanced and sinusoidal source currents is presented. The two last control strategies have not been developed up now in the p q original theory frame. Finally, Section 6 presents the simulation results corresponding to the three compensation strategies applied to a practical case and Section 7 shows the results obtained by applying the three strategies to an experimental laboratory prototype.. Original p q theory The instantaneous reactive power theory was formulated at the beginning of the eighties [6], and this is the formulation with the largest diffusion along these years. For that reason, it is the most used as control strategy in the APF. The p q theory was developed for three-phase three-wire systems with balanced and sinusoidal source voltages. The compensation objective assumed was the obtention of a constant instantaneous source power. The theory is based on a translation from the phase reference system (1) to the system, Fig. 1. The transformation matrix associated is as follows: e e α e β = u 1 u u (1) i i α i β = From Eqs. (1) and (), it can be deduced that i 1 i i () i N = i 1 + i + i = i () The different power terms are defined as follows: p p q e = e α e β e β e α i i α i β = [T ] i i α i β (4) where p is the zero sequence (real) instantaneous power, p the real instantaneous power and q is the imaginary instantaneous power. Considering the [T] inverse matrix, the calculation of the current components from the different power terms is possible. The expression is shown in the next equation: i i α = 1 e p e e e e α e e β p (5) i β e e β e e α q where e = e α + e β Although original p q theory is developed from the mapping matrices (4) and (5), an alternative develop is possible. This new model does not use any mapping matrices. Therefore, it makes easier the theory treatment and its application to the active filters control. In fact, the vectorial frame allows to obtain, in a simple way, any compensation strategy, while the mapping matrices have been always limited to the constant power strategy. Actually, the vectorial frame is a conceptual approach. So, three new voltage vector will be defined, e : e, e and e βα, Fig.. Fig. 1. referente system. Fig.. New voltage vectors in coordinates system.

3 The voltage and current space vectors are defined as follows: e e = e α ; e βα = e β ; v = e = ; e β e α i i = (6) i α i β where e βα is the orthogonal voltage vector and e is the zero sequence voltage vector. It is always verified that e = [ e e α e β ] t = e + e (7) P. Salmerón et al. / Electric Power Systems Research 78 (8) and e βα e = (8) In the reference frame, the current vector i may be expressed as the sum of its projections over the vectors e, e and e βα, that is i = p (t) e + q (t) e βα + p (t) e e e e βα e βα e e = p (t) e + q (t) e βα + p (t) e (9) e e e e e e where p = e i is the instantaneous real power in α β components, q = e βα i the instantaneous imaginary power and p (t) is the zero sequence instantaneous real power. They are identical to those power terms defined in Eq. (4). The fact that the orthogonal voltage vector norm and the voltage vector without zero-sequence component norm are the same has been considered in Eq. (9). Since now the compensation currents will be obtained from several strategies within the model presented in the Eq. (9), without using mapping matrices.. Constant power compensation The strategy assumed by the original or p q theory since its origin has been the obtention of a constant instantaneous power in the source side with the only restriction of getting a null average instantaneous power exchanged by the compensator, P c. Along the paper, lower case represents instantaneous values, upper case average values, subindex L load requirement, subindex C compensator supply and subindex S source supply. In fact, total power required by the load can be expressed as follows: p L (t) = p L (t) + p L (t) = P L + p L (t) + P L + p L (t)(1) where uppercase are referred to the average values and the terms with the character over it are referred to the power oscillatory component. To calculate the compensator current, and according to Fig., it is verified that p C (t) = p L (t) p S (t) = p L (t) P L (11) Fig.. Power system compensated by an active power line conditioner. where P L is the total active power incoming to the load. p S (t)=p L after compensation. Taking into account Eq. (1), and the independence of coordinates, the Eq. (11) can be expressed in the next way: p C (t) = p L (t) P L = p L (t) (1) p C (t) = p L (t) P L = p L (t) (1) These equations, effectively, means that the average value p C (t) = P C =, where p L (t) represents the ac or oscillatory part of p L (t). On the other hand, the instantaneous imaginary power exchanged by the compensator must be the same as the instantaneous imaginary power required by the load: q C (t) = q L (t) (14) Therefore, from Eq. (5), the compensation current is i C i Cα = 1 e p L (t) e e e e α e e β p L (t) (15) i Cβ e e β e e α q L (t) This strategy achieves a constant instantaneous power supplied by the source. Nevertheless, it does not eliminate the neutral current. After compensation, the source supplies a constant instantaneous power with a value identical to the load active power with a neutral current not null. The results presented in Eq. (15) have been got by mean of the mapping matrices introduced by Akagi et al. [6]. In the new model of the p q theory, without the use of mapping matrices, the constant power compensation strategy can be developed. Besides, the elimination of the neutral current is possible [9]. The corresponding procedure is presented now. After compensation, the source current i S (t) is the next: i S (t) = K e (16) Taking into account that e e =, from Eq. (6) to (7), the source total instantaneous real power must be as follows: p S (t) = e i S = K e e = K e e = P L (17)

4 68 P. Salmerón et al. / Electric Power Systems Research 78 (8) Imposing that p S (t)=p L, the proportionality factor can be obtained and it gets the next value: Thus, knowing that p C (t) = P C = the proportionality constant is as follows: K = P L e (18) G e = P L E (4) Therefore, from Eqs. (9) and (18), the compensation currents may be expressed like a vector as follows: i C (t) = i L (t) i S (t) = p (t) e e P L e = p (t) P L e e + p L(t) e e + p L(t) e e + q L e e βα e + q L e e βα (19) where p (t) represents the oscillatory part of the instantaneous real power in the α β plane. From Eq. (19), such component of the source current is as follows: i c = p L(t) e e () i cα = p L(t) P L e i cβ = p L(t) P L e e α q L e e β, 4. Unity power factor compensation e β + q L e e α (1) The unity power factor compensation is a strategy widely used along the time. The target is to obtain source currents with the same distortion and symmetry condition as source voltages. It means that the source current is collinear to the supply voltage. In this situation, the source supplies the load active power, but the instantaneous real power is not constant after compensation [7]. When the source voltages are sinusoidal and balanced, constant power and unity power factor compensations results are the same. In fact, for sinusoidal and balanced source voltages, if compensation currents cancel reactive, distortion and unbalanced current components, source currents will be sinusoidal and balanced in the same way as voltage. And for that reason, instantaneous power supplied by the source will be constant. However, when the supply voltage is unbalanced and non-sinusoidal the situation won t be the same. Since now, a control strategy will be obtained. It is based on the p q theory. This new strategy allows unity power factor in the source side to be obtained after compensation: i s = G e e () Taking into account Eqs. (11) and () and to satisfy that the active power exchanged by the compensator is null: p C (t) = p L (t) p S (t) = p L (t) G e e (t) () since p L (t) = P L and where E is the square root of voltage vector RMS value: E = 1 e dt (5) T T Finally, after compensation: i S = P e L (6) i Sα i Sβ E e α e β From Eq. (11) and considering the reference axes, compensation power can be expressed as follows: p C (t) = p L (t) p S (t) = p L (t) P L E e (7) p C (t) = p L (t) p S (t) = p L (t) P L E e (8) Besides, instantaneous imaginary power verifies: q C (t) = q L (t) (9) So, i C = i L i S = p L(t) e e P L E e + p (t) e e P L E e + q L(t) e e βα () In this way, the explicit expressions of the compensation currents in the α β coordinates are as follows: ( ) p (t) i C = e P L E e, i Cα = i Cβ = ( p L (t) e ( p L (t) e P L E P L E ) ) e α q L(t) e e β, e β + q L(t) e e α (1) This control algorithm does not permit to eliminate the neutral current, since: i S = i L i C = p (t) e e p (t) e e + P L E e = P L E e () This expression will be different of zero if zero-sequence voltage component exists.

5 P. Salmerón et al. / Electric Power Systems Research 78 (8) Sinusoidal and balanced source current compensation For last years, both compensation objectives presented in this paper have not been enough to solve the new problems associated to the voltage and current distortion. It is necessary to define a new compensation objective. It is the sinusoidal source current compensation strategy. The aim is to obtain a sinusoidal and balanced source current, in phase with voltage [11 1], when the voltage is sinusoidal and balanced or in phase with voltage positive sequence fundamental component in any other case. It must be fulfilled in any supply voltage conditions and load kind. If voltages are balanced and sinusoidal, currents proposed will be proportional to the voltages (as in previous strategy). The proportionality constant must have the value established in the Eqs. (4) and (5), too. If voltages are balanced non-sinusoidal, the reference source current must be proportional to its fundamental component. The proportionality constant is fixed to satisfy the restriction of having null active power exchanged by the compensator. The ideal current expression is as follows [1,1]: i S = P L () i Sα i Sβ E 1 e α1 e β1 e α1 and e β1 represent the fundamental components of the α, β voltage, respectively, and E1 represents the square root of the voltage e 1 fundamental component RMS value. To analyze the case of unbalanced non-sinusoidal voltage, it is necessary to consider that the vector e from Fig. may be divided in the positive sequence fundamental component e + 1, negative sequence fundamental component e 1, and harmonic components e N1. Current vector can be expressed, in this way, as follows: i L = p L(t) e = p L(t) e + P L e e + P L e e p L(t) e e + q L(t) e e βα e 1 + P L(t) e n e n e + q L(t) e e βα (4) So, in the case of unbalanced non-sinusoidal voltage, the source current expression becomes as shown in the following equation: i S = P L (5) i Sα i Sβ E 1 + e + α1 e + β1 E + 1 represents the square root of the voltage positive sequence fundamental component RMS value. In (5), i Si is proportional to the voltage positive sequence fundamental component. The proportionality constant value is calculated imposing the restriction of null active power exchanged by the compensator. Thus, the compensation current equations to obtain balanced and sinusoidal supply currents are calculated, from Eq. (5),as follows: i C = i L i S = + P L(t) e ( p L (t) e e n + P L n ) P L E 1 + e e p L(t) e e 1 e + q L(t) e e βα (6) In this case, it is necessary to impose restrictions not only to the instantaneous real powers p (t) and p (t), but besides to the instantaneous imaginary power q (t). This new restriction is necessary to get sinusoidal and balanced source currents in the case of non-sinusoidal or unbalanced voltages. In fact: q C = e βα i C = P L E 1 + e βα e q L(t) e e = P L E 1 + e βα e q L(t) (7) since in general e βα e + 1 In mapping matrices format, the control strategy may be expressed as i C i Cα = 1 e e e e e α e e β i Cβ e e β e e α p L (t) p L (t) P L E + (e α e + α1 + e βe + β1 ) 1 q L (t) P L (e α e + β1 e βe + α1 ) E + 1 (8) Moreover, this strategy achieves to eliminate the neutral current with a null active power exchanged by the compensator. 6. Simulation results In this paper, the different control strategies proposed within instantaneous reactive power theory have been applied to a threephase four-wire system similar to the presented in Fig.. It is a three-phase four-wire system composed of an unbalanced and non-sinusoidal voltage source that supplies a non-linear unbalanced load. The compensator injects the non-useful current required by the load. The three strategies presented in previous sections have been applied to the simulation platform shown in Fig. 4. This figure presents a Matlab-Simulink implementation diagram. On the one hand, the source is composed of three single-phase sources and a serial resistor in star configuration. On the other hand, the load is composed of two back to back SCRs and a serial resistor,

6 61 P. Salmerón et al. / Electric Power Systems Research 78 (8) Fig. 4. Matlab-Simulink implementation diagram. whose value is different in each phase, in star connection. Thus, a non-linear load is obtained. The main objective is the control strategies analysis to calculate the reference current. So, for the purposes of the simulation analysis, a relatively simple model has been considered for the compensator. Therefore, the converter is modeled by means of three controlled current sources. The compensation current is obtained through a set of calculation blocks, and they are directly injected in the system by the compensator. The block identified as control block is a mask whose content implements the different specified control algorithm. All the algorithms used in this paper are summarized in Fig. 5, where u R is the voltage used to calculate the reference current by each compensation strategy. The method used to calculate the voltage fundamental component, necessary in the last implemented strategy, is based on the use of low band filters, multipliers and adders, as shown in Fig. 6. In fact, any voltage or current signal, in coordinates, may be expressed in the next way: e α = E α1 cos(ωt + ϕ 1 ) + E α cos(ωt + ϕ ) + (9) The product of the voltage component e α (t) and the waveform sin ωt has a constant term: 1/E α1 cos ϕ 1 being E α1 the Fig. 6. Method to get voltage fundamental component. voltage fundamental component amplitude and ϕ 1 is the voltage fundamental component phase. On the other hand, the product composed by the same voltage component e α (t) and the waveform cos ωt has a constant term: 1/E α1 sin ϕ 1. The product of both constant terms per and per sin ωt and cos ωt, respectively, compose the phase 1 voltage fundamental component waveform, as shown in Fig. 6. The angular speed used to generate the sine and the cosine functions corresponds to the nominal pulsation of the voltage waveform fundamental component. It is because the objective is to extract the voltage fundamental component waveform, whose frequency is the nominal one. Fig. 5. Control block algorithms diagram.

8 61 P. Salmerón et al. / Electric Power Systems Research 78 (8) Fig. 8. Three-phase source current (A) before/after compensation with unbalanced and sinusoidal source voltages obtained in simulation assays. Fig. 9. Three-phase source current (A) before/after compensation with balanced and non-sinusoidal source voltages obtained in simulation assays. power supplied by the source, the power factor is not the unity and the source current is not balanced and sinusoidal. And so on, applying the other two compensation strategies. 7. Experimental results In this work, the APF control has been implemented by means of a specific digital signal processor (DSP) board developed by dspace. It includes a real-time processor and the necessary In/Out interfaces that allow the control operation to be carried out. In particular, DS11 peer to peer connection (PPC) controller board is equipped with a Power PC processor for fast floating point calculation at 4 MHz. This hardware allows to program via Simulink. In this way, all the control circuit components are configured graphically within the Simulink environment. The RTI translates the Simulink model to C language, it generates the real-time executable program, and it downloads it in the controller board. To check the approaches proposed in this paper, they were applied to the three-phase four-wire unbalanced ac-regulator compensation. Fig. 1 shows a general scheme of compensated system.

9 P. Salmerón et al. / Electric Power Systems Research 78 (8) Fig. 1. A general scheme of the compensated system. The voltage source is constituted by three autotransformer. They allow the voltage to be unbalanced before applying it to the system. The load is composed of three regulators with a serial inductive load in each phase connected in star. The compensator is constituted by a Semikron power inverter model SKM5GB1, with a three-phase IGBT bridge and two capacitors of F in dc side. In each phase, the connection transformers voltage ratio is /46 V and the inverter output inductance is Ls = 17 mh. The use of the coupling transformers allows working with a low dc voltage. The control block inputs are nine: the load voltages and currents, and the compensation currents. Besides, it is necessary another input to control the capacitor dc voltage. Additional inputs are used to check the APF compensation performance. The voltage and current sensors used are AD/DC LEM LA-5 NP and AC/DC LEM LV 5-6 models, respectively. The software running in the real-time processor carries out the control. It calculates the reference source current as it was presented in Sections 4. The difference between the real compensation current and the calculated before is the input to the PWM module. Its output are the power circuit IGBTs trigger signals. In fact, in an experimental develop the current control used is very important. In our experimental system, a hysteresis band PWM control has been implemented. The converter IGBTs commutation introduces unavoidable high order harmonics. To Fig. 11. Control system used in the experimental set-up. reduce its influence, the PWM control is carried out by mean of a specific circuit instead of by software, Fig. 11. The RMS voltage values and the load parameters values are shown in Table 1. The voltage applied is the net voltage unbalanced using an autotransformer. The three phases root mean square (RMS) values are indicated in Table 1 first row. These values point out a great level of unbalanced applied to the system. Table 1 System parameters Phase 1 Phase Phase RMS voltage Resistor ( ) Inductance (mh) SCRs angle 9 9 9

10 614 P. Salmerón et al. / Electric Power Systems Research 78 (8) Fig. 1. Phase 1 voltage (V) and current (A) obtained in the laboratory prototype. (a) Before compensation, (b) after compensation applying constant power strategy, (c) after compensation applying unity power factor strategy and (d) after compensation applying sinusoidal source current compensation. Fig. 1. Three-phase waveforms obtained in the laboratory prototype. (a) Voltage applied, (b) source current before compensation, (c) source current after compensation applying constant power strategy, (d) source current after compensation applying unity power factor strategy and (e) source current after compensation applying sinusoidal source current compensation.

11 P. Salmerón et al. / Electric Power Systems Research 78 (8) Besides, the feedback method used to control self-supporting dc bus is very important. It supposes, in general, a modification of compensation currents. Summarizing, the procedure used consists of controlling the capacitor voltage to a reference value. To achieve it, a PI control may be chosen for the error between the reference value and the capacitor voltage value at the end of each period. The result is the value of the compensator lost power. This value modifies the power terms which appear in the compensation currents expressions corresponding to each strategy. The modification is in the next way: referring to the first strategy, Eqs. () and (1), the power lost is subtracted to the p L term in the expressions of i cα and i cβ ; referring to the second strategy, Eq. (1), the power lost is added to the P L term in the expressions of i cα and i cβ ; referring to the third strategy, Eq. (8), the power lost is added to the load power P L. Fig. 1 presents the phase one source voltage and current before and after compensation. Fig. 1(a) graph shows the situation before compensation, i.e., phase one voltage and load current. Fig. 1 graphs (b), (c) and (d) show the results got by all the three control strategies: constant power, unity power factor and sinusoidal source current, respectively. The phase difference between voltage and current is eliminated by applying the three strategies. In fact, the first graph shows a phase difference between both waveforms. It is not in the other three graphs. Fig. 1 presents the three-phase waveforms corresponding to: voltage applied (graph 1 (a)), load current or source current before compensation (graph 1 (b)), and source current after compensation using each control strategy (graphs 1 (c e)): constant power, unity power factor and sinusoidal source current, respectively. Fig. 1(a) shows the voltage unbalanced. Fig. 1(b) presents an unbalanced and distorted three-phase load current due to the load non-linearity and to the source voltage unbalance. The source current distortion after compensation using any of the three control strategy is much lower than the presented by the load current. The unity power factor strategy does not get balanced source current and it does not eliminate the neutral current. The sinusoidal source current strategy obtains balanced and sinusoidal source current after compensation, Fig. 1(e). Even so, a ripple in the waveforms is unavoidable due to the threshold band imposed by the PWM control. The source current corresponding to the constant power strategy presents a considerably distortion and the corresponding to the unity power factor strategy shows the same unbalance as the voltage applied to the system. The voltage RMS and total distortion demand (TDD) values and the source current RMS and TDD values before and after compensation are presented in Table. The TDD is calculated as follows: h TDD = + h + +h n RMS (4) where h i represent the order i harmonic RMS value and RMS represents the waveform RMS value. Besides, Table presents the three-phase RMS value corresponding to voltage and currents. It is defined as follows [16]: V1 V e = + V + V, I e = I 1 + I + I + I 4 (41) where V i represents the phase voltage corresponding to the phase i and I i represents the phase i current RMS value. Phase 4 corresponds to the neutral current. This table shows as the three control strategies decrease the current distortion considerably. Except the values corresponding to the phase two, the distortion values are very similar applying the three control strategies. However, that distortion represents high order harmonics in the case of sinusoidal current compensation and low and high order harmonics in the other two, Fig. 1. It is corroborated by the values corresponding to the phase two. The values achieved applying sinusoidal current is lower than the achieved in the other two phases. In an ideal case, they would be null, but in this case the PWM block has a great influence on the results, and therefore the compensation current does not track its reference exactly. On the other hand, the source current after compensation applying the control strategy derived from constant power compensation presents a distortion appreciatively bigger than the corresponding to the sinusoidal source current strategy as can be corroborated by Fig. 1. The unity power factor compensation, does not obtain balanced source currents. Respect to the three-phase RMS values, they are very similar in the case of applying constant power and unity power factor compensation strategies and a little bigger in the case of sinusoidal source current. It is necessary to obtain balanced and sinusoidal source current if voltage is not balanced and sinusoidal. The column Power factor in Table shows that the three strategies improve the power factor value after compensation. Table Experimental results, voltage values (V) and currents values (A) RMS TDD Phase 1 Phase Phase Neutral V e or I e Power factor Phase 1 Phase Phase Voltage Source current Before compensation Constant power Unity power factor Sinusoidal source current

12 616 P. Salmerón et al. / Electric Power Systems Research 78 (8) Table Source voltage and current harmonic content (% over the fundamental component) before and after compensation Besides, the corresponding to the unity power factor strategy is the highest of them. Table presents the main harmonic order corresponding to the source current three phases before and after compensation. 8. Conclusions This paper has developed an exhaustive analysis of the p q instantaneous reactive power theory based on a new formulation without using mapping matrices. The calculation of the reference compensation currents is based on the current vector projections over three new voltage vectors defined in coordinates system. The three main compensation objectives (constant power, unity power factor and sinusoidal source current) have been studied to generate three control strategies based on the p q instantaneous reactive power theory, one corresponding to each compensation objective. These control strategies have been implemented in a simulation platform to study the results obtained in three cases of voltage applied: sinusoidal and balanced, sinusoidal but unbalanced and balanced but nonsinusoidal. Thus, the analysis shows that, with few modifications in the control strategy developed by the authors of the original theory, any compensation objective imposed to the system may be achieved. So, the original p q theory can be used as the base of active power filters control algorithm, to get any compensation objective and with any voltage condition in the PCC. The strategies have been assayed by simulations in the Matlab-Simulink environment and tested in an experimental laboratory prototype. References [1] A. Ghosh, G. Ledwich, Power Quality Enhancement Using Custom Power Devices, Kluver Academic Publishers, Boston,. [] M. Aredes, K. Heumann, Three-phase four-wire shunt active filter control strategies, IEEE Trans. Power Electron. 1 () (1997) [] S.J. Chiang, W.J. Ai, F.J. Lin, Parallel operation of capacity-limited threephase four-wire active power filters, IEE Proc.-Electr. Power Appl. 149 (5) () 9 6. [4] G.W. Chang, T.C. Shee, A novel reference compensation current strategy for shunt active power filter control, IEEE Trans. Power Deliv. 19 (4) (4) [5] L.H. Tey, P.L. So, Y.C. Chu, Improvement of power quality using adaptive shunt active filter, IEEE Trans. Power Deliv. () (5) [6] H. Akagi, Y. Kanazawa, A. Nabae, Instantaneous reactive power compensators comprising switching devices without energy storage components, IEEE Trans. Ind. Appl. IA- () (1984) [7] A. Cavallini, G.C. Montanari, Compensation strategies for shunt activefilter control, IEEE Trans. Power Electron. 9 (6) (1994)

13 P. Salmerón et al. / Electric Power Systems Research 78 (8) [8] A. Horn, L.A. Pittorino, J.H.R. Enslin, Evaluation of active power filter control algorithms under non-sinusoidal and unbalanced conditions, in: Proceedings of the Seventh International Conference on Harmonics and Quality of Power, 1996, pp [9] H. Akagi, S. Ogasawara, H. Kim, The theory of instantaneous power in three-phase four-wire systems: a comprehensive approach, Conf. Rec. IEEE Ind. Appl. Conf. 1 (1999) [1] H. Kim, H. Akagi, The instantaneous power theory on the rotating p q r reference frames, in: IEEE Proceeding on Power Electronic and Drive Systems, [11] P. Salmerón Revuelta, R.S. Herrera, Application of the instantaneous power theories in load compensation with active power filters, in: Proceeding on CD of European Conference on Power Electronic, EPE, Toulouse, France,. [1] F.Z. Peng, L.M. Tolbert, Compensation of non-active current in power systems definitions from compensation standpoint, in: Power Enginnering Society Summer Meeting,, vol.,, pp [1] P. Salmeron, J.C. Montaño, J.R. Vázquez, J. Prieto, A. Pérez, Compensation in non-sinusoidal, unbalanced three-phase four-wire systems with active power line conditioner, IEEE Trans. Power Deliv. 19 (4) (4). [14] A. Gosh, A. Joshi, A new approach to load balancing and power factor correction in power distribution system, Power Deliv., IEEE Trans. 15 (1) () [15] M. Depenbrock, V. Staudt, H. Wrede, A theoretical investigation of original and modified instantaneous power theory applied to four-wire systems, IEEE Trans. Ind. Appl. 9 (4) () [16] A.E. Emanuel, Summary of IEEE Standard 1459: definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced and unbalanced conditions, IEEE Trans. Ind. Appl. 4 () (4)

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