## 1 Introduction  
In electrical power instrumentation, the apparent power measured by a power analyzer placed at the point of common coupling (PCC) of two subsystems of a three-phase power system (Figure 1) generally varies with the selected voltage reference point, a neutral point (N) of subsystem 1, a neutral point (n) of subsystem 2, or a grounding (earthing) point (G). This phenomenon is relevant when measuring, for example, the apparent power of one of the subsystems and the corresponding neutral point (N or n) is not accessible, such as may occur in a star-connected three-phase motor with only three terminals, or when the neutral points are far from the measurement point. The apparent power of any part of a power system is, in our opinion, a formal quantity that measures the combined effects of all the power phenomena in that part of the system. Hence, when a power analyzer registers different values of the apparent power at the PCC of two directly connected subsystems (Figure 1) based on the use of different voltage reference points, it suggests the existence of various power phenomena with effects derived from the voltage reference point. Therefore, if both subsystems in Figure 1 are star-connected with neutral points N and n, the power analyzer would register the apparent power of subsystem 1 if N is used as the voltage reference point, and that of subsystem 2 if n is used as the voltage reference point. If the grounding point G is used as the voltage reference, the apparent power registered by the power analyzer would not be that of either subsystem. Investigating the phenomenon associated with the above effect of the voltage reference point, it should be noted that, when the power system in Figure 1 is sinusoidal and balanced, the apparent powers measured at the PCC of the two subsystems are independent of the point of voltage reference (N, n, or G). This is because the reference points would have the same voltage. However, when subsystems are unbalanced and/or distorted, there would be voltage differences between the above three points [1; 2], resulting in differing apparent powers measured when using them as the voltage reference points. Since voltage differences between the different reference points are the neutral-point displacement voltages [3], the neutral displacement phenomenon is responsible for the different values of the measured apparent powers, according to the voltage reference point. The manifestation of over-voltages and sub-voltages is the main known effect of this phenomenon. The neutral-point displacement is a well-known phenomenon in power systems since the early 20 th century. [PERSON] and [PERSON] [4] explained in 1931 the neutral displacement phenomenon, which they referred to as neutral inversion. Additionally, [PERSON] [5] analyzed in 1936 the phenomenon caused by open circuits in power systems grounded through neutral earthing transformers and Petersen coils. Many works found in the technical literature deal with the problems caused by the phenomenon of neutral displacement in grounded systems [6; 7; 8; 9; 10; 11; 12]. Thus, several procedures of detection [13; 14] and correction [15; 16] of the neutral displacement phenomenon were developed over time. The effects caused by the neutral-point displacement voltages were also studied in power systems with compensating capacitor banks [17; 18; 19]. However, we did not find applications of the neutral displacement phenomenon to power measurement in power systems, perhaps because the power effects of this phenomenon are not so well known and, on many occasions, they are confused with the apparent power in the neutral conductor impedance. Indeed, all the above studies focused on measuring and reducing the neutral displacement voltages, but the powers associated with the neutral displacement phenomenon were not formulated until now. Figure 1: Voltage reference points for apparent power measurement.