Services in Multiphase Systems

J. Brenner, Nuremberg

How Do We Optimise Electrical Power in Multiphase Systems?

The technical paper Power in Multiphase Systems by J. Brenner addresses the theoretical principles and practical applications of electrical power in multiphase power supply systems. The focus lies on different power components such as active power, reactive power, and apparent power, along with their calculation and significance for industrial applications. This work emphasizes the unique characteristics of power transmission in three-phase systems and provides valuable insights into the optimisation and analysis of modern electrical grids.

Range and Magnitude of Active Electrical Power in Multiphase Systems

Figure 1 shows the interface between the source and load of a multiphase system with n = m phase conductors plus 1 neutral conductor.

Figure 1: Interface of a multiphase system with neutral conductor

If the instantaneous values present are periodic time functions with the same period T, then the root mean square (RMS) values

and

and, depending on the direction of energy flow, the active electrical power [1]

can be measured at this interface. The relationship between these measured quantities is described by Schwarz’s inequality [2].

or by using equations (1.1), (1.2), and (1.3)

This results in the following regarding the range and magnitude of active electrical power in the multiphase system with neutral conductor

and

Figure 2 shows the interface between the source and load of a multiphase system with n = m phase conductors. If the instantaneous values present are periodic time functions with the same period T, then the root mean square (RMS) values

Figure 2: Interface of a multiphase electrical system without neutral conductor

and

and, depending on the direction of energy flow, the active electrical power [1]

can be measured. The relationship between these measured quantities is described by Schwarz’s inequality [2].

or by using equations (1.8), (1.9), and (1.10)

This results in the following regarding the range and magnitude of active electrical power in the polyphase system without neutral conductor

and

Apparent Electrical Power in Multiphase Systems

With the definition: The apparent power is the amount of the largest active power that can be achieved with the respective effective voltage and current values, the following follows from the magnitude inequality (1.7) for the apparent power in the polyphase system with neutral conductor

and from the magnitude inequality (1.14) for the apparent power in the polyphase system without neutral conductor

Using the root mean square (RMS) value equation [3]

the apparent power equation (2.2) is alternatively

Power Factor in Multiphase Electrical Systems

The apparent power equations (2.1) and (2.2) are used to calculate the range inequalities (1.6) and (1.13) in multiphase electrical systems.

or

With the definition: The power factor is the ratio of active electrical power to apparent electrical power

The inequality (3.2)

That is, based on this inequality, it is possible to define the cosine of an angle ϕ as the power factor.

The angle ϕ is uniquely determined when restricted to 0 ≤ ϕ ≤ π.

Active Electrical Power in Multiphase Systems

Using equation (3.5), the following expression for active electrical power follows from equation (3.3)

Positive P values (active power consumption) correspond to 0 ≤ ϕ < π/2, while negative P values (active power generation) correspond to π/2 < ϕ ≤ π. The value P = 0 corresponds to ϕ = π/2.

Reactive Electrical Power in Multiphase Systems

With the definition: The magnitudes of active power and reactive power are the two orthogonal components of apparent power. These three power quantities, all defined as positive values, can be represented as a right-angled triangle as shown in Figure 3.

Figure 3: Orthogonal Components of Apparent Electrical Power

Consequently

or

The following applies to the reactive electrical power

Using equation (4.1), this equation becomes

or

Apparent Electrical Power in Special Multiphase Systems

The apparent power in the single-phase two-wire system is calculated according to equation (2.1) with m = 1

or with _U1N= U and _I1= I

Der Beitrag „Leistungen in Mehrphasensystemen“ behandelt Wirkleistung, Scheinleistung und Blindleistung mit Beispielen.

The apparent power in the two-phase two-wire system is calculated according to equations (2.2) and (2.3) with m = 2

or with _U10 = _U20 _=U12 / 2 and _I1 = _I2 = I

The apparent power in the two-phase three-wire system is calculated according to equation (2.1) with m = 2

The apparent power in the three-phase three-wire system is calculated according to equations (2.2) and (2.3) with m = 3

The apparent power in the three-phase four-wire system is calculated according to equation (2.1) with m=3

The apparent power in the six-phase six-wire system is calculated according to equations (2.2) and (2.3) with m = 6

whereby

In the symmetrical three-phase three-wire system, U₁₀ = U₂₀ = U₃₀, U₁₂ = U₁₃ = U₂₃, and I₁ = I₂ = I₃. Consequently, the apparent power according to equation (6.4) in this system is

In the symmetrical three-phase four-wire system, U_₁N = U_₂N = U_₃N and I₁ = I₂ = I₃. Consequently, the apparent power according to equation (6.5) in this system is