2.1 Balanced three-phase systems

Balanced three-phase sinusoidal systems are composed of three sinusoidal voltage sources connected to a load, whose electrical phase shift is 120° from each other with equal amplitudes. Three-phase sources and loads can be connected in delta or wye connection (also known as star) (^{Beig, 2016}).

This paper uses a star-star connection to develop the deductions. Figure 1 shows the star-star connection, describing the three-phase source on the left and the load on the right; line currents and phase voltages are also denoted. It also illustrates the phase voltages, measured at terminals a, b, and c, referencing voltages to the neutral terminal n. Line currents are flowing through lines a, b, and c. The voltages of a balanced three-phase system are defined in the time-domain through equations (1), (2) and (3) (^{Beig, 2016}).

Figure 1 Wye-wye three-phase four-wire system. 

Similarly, line currents are defined in (4), (5) and (6).

Where Vk and Ik (k = a, b, c) are the RMS values associated to the sinusoidal waveforms of voltage and current, Vk=Vkejϕk and Ik=Ikejβk are the phasors associated with the respective phases and ω=2πf is the fundamental angular frequency of the system, defining frequency f in Hz. Moreover, ϕk and βk are the initial phase angle of each voltage and current phasor, respectively.

A balanced three-phase system has three voltage phasors and three current phasors with identical magnitudes, which means that Va=Vb=Vc and Ia=Ib=Ic; it also has a shift angle equal to 120° among phasors, which is equivalent to ϕa=ϕb=ϕc and βa=βb=βc (^{Tleis, 2019}).

2.2 Unbalanced three-phase systems

A three-phase system is unbalanced when the magnitudes of the phasors are not equal or the phase between any pair of phasors differs by 120°. Fortescue's symmetric component decomposition can be applied to represent an unbalanced three-phase system as the sum of three balanced three-phase systems, which are called symmetric components of positive, negative, and zero sequence (Fortescue, 1918). Symmetric component sequences of an unbalanced three-phase system Va, Vb and Vc can be calculated using (7), where α is defined as 1∠120°, which offsets a signal by 120 degrees. Figure 2 represents the decomposition of an unbalanced three-phase system into its symmetrical components, where Va0, Va- and Va+ are the zero, positive and negative sequences, respectively. The same definition applies for currents.

Figure 2 Decomposition of an unbalanced three-phase system into symmetrical components. 

Symmetric components constitute a useful method for analyzing power in unbalanced three-phase four-wire systems (IEEE Std. 1459, 2010). The equations below use symmetric components for defining power in three-phase systems. For the following definitions V+, V- and V0 are the RMS values of positive, negative and zero sequence components of voltage; I+, I- and I0 are the RMS values of positive, negative and zero sequence components of current and, θ+, θ- and θ0 are their respective phase angles between voltage and current phasors by each sequence.

Active power (P), also called real power, is the average value of the instantaneous power during the interval of a period. In terms of the symmetric components, active power is defined in (8) and (9), where Pk k=+,-,0 represents the contributions of each symmetric component to the net active power (P).

Reactive power (Q) corresponds to the part of the power that cannot become useful work and that has a bidirectional flow between a source and a load. In terms of the symmetric components, reactive power is defined in (10) and (11), where Qk k=+,-,0 represents the contributions of each symmetric component to the net reactive power (Q).

Apparent power (S) is defined in ^{Emanuel (2004)}, as the maximum power transmitted to the load (or delivered by a source) while keeping the same line losses and the same load (or source) voltage and current. For unbalanced loads two apparent power definitions are prevalent: arithmetic apparent power and vector apparent power (Emanuel, 1999). The arithmetic apparent power is defined in (12), where Sa=VaIa, Sb=VbIb and Sc=VcIc. The vector apparent power is defined in (13).

The power factor (PF) is a number that indicates how much active power (P) is transmitted out of the maximum possible power (S) that will cause the same power loss in the supplying equipment and lines (^{Emanuel, 1998}). A power factor is defined for both arithmetic and vector apparent powers; the arithmetic power factor is defined as PFA=PSA and the vector power factor is defined as PFV=PSV.

Both power factors PFA and PFV are not accurate for nonsinusoidal waves or when the load is unbalanced (^{Emanuel, 2004}). Hence a new definition for apparent power known as effective apparent power was suggested by F. Buchholz (^{Frank, 1992}). The effective apparent power correctly reflects the power loss in the neutral current path as well as the effect of unbalance, which are not reflected properly by the arithmetic and vector apparent powers (Emanuel, 2004).

The concept of effective apparent power assumes a virtual balanced circuit that has the same line power losses as the actual unbalanced circuit. This equivalence leads to the definition of an effective line current Ie and an effective line-to-neutral voltage Ve.

The effective current of a four-wire system is defined in (14). In this equation ρ is defined as the ratio ρ=rnr, where rn is the neutral wire resistance and r is the line resistance. In case that the value of the ratio ρ is not known, it is recommended to use ρ=1.0 (IEEE Std. 1459, 2010).

The effective voltage is found by representing the load by three equivalent equal resistances RY, connected in star and three equal resistances RΔ connected in delta. Using the ratio ξ=3RYRΔ, the power equality between the actual load and the optimized system yields the expression for effective voltage, shown in (15) (^{Emanuel, 2004}). If the value of the ratio ξ is not known, it is recommended to use ξ =1.0 (IEEE Std. 1459, 2010).

The expression for effective apparent power Se, in terms of the effective current and the effective voltage is shown in (16). The effective apparent power factor PFe is defined as PFe=PSe.

2.3 Nonsinusoidal unbalanced three-phase systems

The effective apparent power of nonsinusoidal and unbalanced three-phase systems is calculated using equation (16). Expressions for calculating the effective voltage and the effective current for these systems are shown in equations (17) to (22), where the subscript 1 means fundamental (60Hz or 50Hz) and the subscript H the inclusion of all the harmonics (^{Emanuel, 2004}). In equations (18) and (19) the subscripts ab, bc and ca represent the phase-to-phase voltages of the system.

The Total Harmonic Distortion (THD) is a measure that quantifies the nonsinusoidal property of a waveform (^{Asadi & Eguchi, 2020}). The equivalent THD of the effective voltage and current are defined in equations (23) and (24), respectively.

The current distortion power, voltage distortion power, and harmonic apparent power are defined in equations (25), (26) and (27).

3.1 Interface 1: three-phase sinusoidal unbalanced systems

Figure 3 shows interface 1, which makes power calculations of three-phase sinusoidal systems. The upper left part of the interface shows the time-domain plot of the three-phase system voltages, as well as the decomposition in symmetric components. The upper right part of the interface shows the same plots but for currents. The lower left part of the interface shows a bar plot with the values of active, reactive, apparent arithmetic, apparent vector, and apparent effective powers, while the lower right part shows the numeric values of these powers and their respective power factors. The interface has two set of sliders, which allow to set up the RMS values and their phases values of line-to-neutral voltages and line-to-line currents of the three-phase system.

To show the operation of this interface, the three-phase system of Figure 4 was used; this example was taken from ^{Alexander & Sadiku (2012)}. The line currents flowing through the three-phase load are presented in (28), (29) and (30). Figure 5 and Figure 6 show the interface sliders fixed to the corresponding values of voltage and current of the example system analyzed.

Figure 4 Example circuit. 

Figure 5 Voltage sliders. 

Figure 6 Current sliders. 

Figure 3 shows the power analysis of the example system. Note that the phasor diagram of the voltages corresponds to the positive sequence of the symmetrical components, while the negative and zero sequences are null, which happens because the voltage supply of the system is balanced. In the time-domain plot of the currents it can be appreciated that these are not balanced, while their amplitudes are not equal, and the shift angles differ from 120°; moreover, the decomposition in symmetrical components show a significant contribution of the positive and negative sequences.

The values of active and reactive power calculated are the same as those obtained in (^{Alexander & Sadiku, 2012}), and the relation S_A≤S_V≤S_e is fulfilled as mentioned in the IEEE Std. 1459.

3.2 Interface 2: three-phase nonsinusoidal unbalanced systems

Figure 7 shows interface 2, which makes power calculations of three-phase nonsinusoidal systems. The upper left part of the interface shows the time-domain plot of the three-phase system voltages, while the upper right part shows the same kind of plot but for currents. The middle part of the interface shows the frequency spectrum of the voltage signals of the system, and the lower part shows the frequency spectrum of the current signals.

Figure 7 Interface 2: three-phase nonsinusoidal and unbalanced systems. 

The program requires to load a csv file with the harmonics data of voltages and currents of the system. The csv file must have the structure shown in tables 1 and 2, where the first column corresponds to the harmonic order and the other columns correspond to the RMS values and shift angles of the line-to-neutral voltages and line currents of the system.

To show the operation of this interface, the data of Table 1 and Table 2 were used. The voltage and current data of these tables correspond to the measures of a lightning circuit in the laboratory building of the East Sectional of the University of Antioquia. The data were measured with a Metrel MI2892 power quality analyzer.

Figure 7 shows the harmonic analysis of the example system. The time-domain plot of the voltages does not reflect a significant distortion caused by the presence of harmonics; on the contrary, current signals are affected in a considerable way by harmonics. The 5th harmonic RMS value of both voltage and current signals are the highest of all harmonics, which was expected since lightning loads usually have the higher impact in harmonic distortion in this harmonic order. Figure 8 presents the power analysis carried out by the analysis tool, which shows the calculated values of power parameters using the equations described in sections 2.2 and 2.3. It is necessary to emphasize that there is no reactive power in the previous analysis, since the shift angles between voltage and current signals were assumed to be zero for each harmonic; this is since the shift angles of the signals could not be measured with the equipment available.

Figure 8 Output data of interface 2. 

The calculation of power quantities using IEEE 1459 is a time-consuming activity; furthermore, the commercial power analyzers commonly used for such computations are expensive. The use of the proposed tool allows the calculation of power magnitudes when no power analyzer is available, only an oscilloscope or similar devices are required to obtain voltage and current waveforms which are the input of the developed tool. Figure 8 shows that the results are organized and can be used for further processing of the information for decision making.