Active and reactive power in electrical circuit with distributed and lumped parameters

It is to note that in educational and special scientific literature it does not exist some common and strongly valid approach to the concept of active and reactive power of the long line even in the case of steady-state sinusoidal condition, not to mention the non-sinusoidal and transient processes. The full (apparent) power can be decomposed in two orthogonal components and theirs distributions along the line can be obtained by complex amplitude method (CAM). However, it remains unclear what is implied by reactive power of the whole line without resorting to its simplified representation as a lumped RLC-circuit . Meanwhile, from the position of mathematical physics, the incertitude in interpretation of power of electrical system with distributed parameters can be easily eliminated by taking into consideration that the original equations of the linear circuit theory are of hyperbolic type and the energy integral (or the conservation law of full electromagnetic energy) is well known for this kind of equations.,,. The problem of reactive power for high-current lines has been repeatedly discussed in different aspects and interpretations, but some common and valid standard is not elaborated up to now. The transmission of electromagnetic power along the long line by means of conduction currents can me described by well-known telegrapher's equations , that represent the Kirchhoff’s laws for closed circuit generated by subcircuit with the length <math>dx</math>:

<math>L \frac{\partial i}{\partial t}+\frac{\partial u}{\partial x}+Ri=0 </math> , <math> C \frac{\partial u}{\partial t}+\frac{\partial i}{\partial x}+Gu=0 </math> , (1)

The parameters in these relations are the followings: ’’L’’ [H/m] is the inductance of the loop formed by direct and return lines; ’’R’’ [Ω/m] is the longitudinal resistance; ’’C’’ [F/m], ’’G’’ [S/m] are the transverse capacity and conductance of the wire insulation leakage.

Starting from (1) by some equivalent transformations one can obtain the following relationship (correct for any periodic solution) between the active and reactive powers of the source, of the receiver and of the long line ,,:

<math>\displaystyle P(0)-P(l) \, = \, \frac{1}{T} \int \limits _{0}^{T}\int \limits _{0}^{l}\left[R i^{2} + G u^{2} \right]\, dx \, dt </math> , (2)

<math>\displaystyle Q(0)-Q(l) \, = \, \frac{1}{\omega T} \int \limits _{0}^{T}\int \limits _{0}^{l}\left[L\left(\frac{\partial i}{\partial t} \right)^{2} -C\left(\frac{\partial u}{\partial t} \right)^{2} \right]\, dx \, dt </math> , (3)

where <math>P(x) \, = \, \frac{1}{T} \int \limits _{0}^{T} i(x,t)u(x,t)\, dt </math> , <math>Q(x) \, = \, - \frac{1}{\omega T} \int \limits _{0}^{T} i(x,t) \frac{du(x,t)}{dt}\, dt </math>.

If reactive power of the long line is equal to zero: <math>Q(0) = Q(l)</math>, i.e. the zero unbalance between the magnetic field power and electric field power takes place, then it is said that the line is balanced by reactive power. Generally, the reactive power evaluation by means of only one real number gives just an approximate qualitative representation of the electromagnetic power circulation in linear circuit. If this number is positive, i.e. <math>Q(0) > Q(l)</math>, then the power of magnetic field prevails over the power of electric field of the line. In this case, it is assumed that the line consumes the reactive power. In case of negative power values, i.e. <math>Q(0) < Q(l)</math>, the situation is straight opposite and it is assumed that the line generates the reactive power. The relations (2), (3) can be easily extended to multiwire lines by replacing the scalar values by corresponding vectors of currents and voltages as well as the matrixes of primary parameters (self and mutual capacitances and inductances).

In high-current circuits under the great electromagnetic power flux, the reactive power of the line is frequently reduced to the limit using the different "bucking out" (compensation) systems. Particularly, under the modeling of various types of reactive power compensation the formulas (2), (3) gain a self-dependent meaning.

See also

Electric power