# Relationship between [PERSON]'s Apparent Power and Instantaneous Power in Three-Phase Systems
## 5 Application Example  
A three-phase, four-wire, real-world, urban facility (Figure 2a) is analyzed in this section in regard to neutral conductor deterioration. This facility was supplied by a 400 kVA, 20 kV/395 V, Dyn11, three-phase transformer through three lines with a cross-sectional area of 16 mm\({}^{2}\) and a length of 12.5 m (resistance of 0.0148 \(\Omega\)) up to the PCC with the load in phases A and C, and a cross-sectional area of 10 mm\({}^{2}\) and a length of 111 m (resistance of 0.21 \(\Omega\)) in phase B. The neutral conductor resistance was 0.0148 \(\Omega\). The transformer short-circuit complex impedances were 0.0156 \(\angle\)74.57\({}^{\circ}\) (\(\Omega\)) at the fundamental-frequency (50 Hz). The loads were houses and shops of a little village. The installation was strongly unbalanced, as shown from the active and reactive power registered by our SIMPELEC analyzer (Figure 2b), and slightly distorted (Figure 2c). As the neutral conductor disrepair can be dangerous for loads, the effects of the neutral conductor deterioration were analyzed using an Excel simulation. The evolution of the line-to-neutral voltages and active power in the load phases as a function of the increase of the neutral conductor resistance are represented in Figure 3, respectively. When the neutral conductor deteriorated, the voltage and active power of the least loaded phase (phase C) increased, whereas a decrease in voltage was observed for the other two phases, engendering a decrease in active power. When the resistance of the neutral conductor increased by a deterioration of up to approximately 40 times its rated value, the voltages in phase C were greater than 240 V, thereby causing the phase C loads to be disconnected by the protection device or become out of order. The active power  
Figure 2: Real-world urban installation: (**a**) schematic representation; (**b**) active and reactive power; (**c**) main screen of the SIMPELEC nolsyr representing the waves and harmonics of line currents. in phase C became zero (Figure 3b) and the installation was more unbalanced. Therefore, phase B was the least loaded in these new conditions and the voltages and active power increased in this phase until the neutral conductor resistance reached 400 times its rated value (Figure 3). This engendered the disconnection of the phase B loads; only the phase A loads were supplied, reducing the voltages and active power in that phase according to the increase in neutral conductor resistance (Figure 3). The load and source apparent power were reduced during neutral conductor deterioration (Figure 4); the same phenomenon occurred in regard to the total active power in the system, even when the active power in a phase of the load individually increased. The load and source unbalanced power increased when the phase C loads were disconnected. However, the reduction in the apparent and unbalanced power was less important in the loads than in the sources because of the strong increase in the load neutral displacement power (Figure 4a) compared to the negligible effects of the neutral conductor deterioration at the source (Figure 4b). The minor effects engendered by the neutral conductor deterioration on the source are shown in Figure 5a in which a fast increase in the neutral-displacement power in the load is observed alongside a minor decrease in the source, even with remarkable neutral conductor deterioration. This result occurred in all of the power systems, independent of the power consumption and topology of the load or the value of the neutral resistance. Figure 4: Installation power: (**a**) load; (**b**) source. Figure 3: Voltages and powers in the load phases: (**a**) active power; (**b**) line-to-neutral voltages. Further, the fast increase in the load neutral-displacement power is shown in Figure 5a.