Load flow calculation

In electrical power engineering, load flow calculation is a method of numerical analysis of power supply networks . In contrast to traditional circuit analysis, simplified representations such as the single-line diagram or per-unit system (pu) are used with regard to various forms of electrical power such as reactive power , active power and apparent power instead of current and voltage . The power grid is analyzed in the normal state (stable state). There are various software solutions for load flow calculations.

In addition to the load flow calculation, many software solutions have other methods of analysis, such as short circuit analysis and profitability analysis . Many programs use linear programming to find the optimal power flow, the state with the lowest cost per kilowatt produced.

The great importance of the load flow calculation lies in the planning of future expansions of energy supply networks as well as in the determination of the optimal operating status of existing systems. The basic information that is obtained is the voltage level and phase angle of each distribution rail or active and reactive power on each line.

Problem formulation

The aim of the load flow calculation is to obtain complete information (voltage / phase angle ) for each busbar with regard to load and generator active power. If this information is known, the active and reactive power flow in each branch as well as the generator output power can be determined analytically. Because of the non-linear nature of this problem, numerical methods are used to obtain solutions within acceptable tolerances.

The solution begins with identifying the known and unknown variables of the system. These variables depend on the type of distribution bus. A rail without a generator is called a load rail. Rails with at least one generator are generator rails. The exception is a randomly selected rail with a generator. Such rails are used as balance node ( English Slack bus hereinafter).

The solution to the problem assumes that the active power P _D and reactive power Q _{D are known} for each load rail. For this reason, load rails are called PQ rails. For generator rails it is assumed that the generated active power P _G and voltage | V | are known. For the balance node it is assumed that voltage | V | and phase angle Θ are known. For each load rail, the voltage and phase angle are unknown and must be calculated; the phase angle must be calculated for each generator rail; there are no unknown variables for the balance node. In a system with N rails and R generators there are unknowns. ${\displaystyle 2(N-1)-(R-1)}$

In order to solve for the unknowns, equations must be set up that do not use any further unknowns. The possible equations use the power equilibrium, which can be established for each rail with regard to active and reactive power. ${\displaystyle 2(N-1)-(R-1)}$ ${\displaystyle 2(N-1)-(R-1)}$

The equation of the power balance is:

${\displaystyle 0=-P_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos \theta _{ik}+B_{ik}\sin \theta _{ik})}$

this is the power fed into the rail i , the active component of the element in the busbars admittance matrix Y is _BUS corresponding to the i . Line and k . Column, is the imaginary part of the element in busbar admittance matrix Y _BUS corresponding to i . Line and k . Column and is the difference in phase angle between i . and k . Rail. ${\displaystyle P_{i}}$ ${\displaystyle G_{ik}}$ ${\displaystyle B_{ik}}$ ${\displaystyle \theta _{ik}}$

The reactive power equation is:

${\displaystyle 0=-Q_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin \theta _{ik}-B_{ik}\cos \theta _{ik})}$

where is the reactive power that is fed into rail i . ${\displaystyle Q_{i}}$

The equations contain the active and reactive power components for each load rail and the active power balance for each generator rail. Only the active power equilibrium is established for the generator rail, because it is assumed that the reactive power fed in is unknown. For the same reason, no equations are set up for the balance node.

Solution methods

There are different methods of solving nonlinear systems of equations such as Newton's method . This method starts with the estimation of all unknown variables (voltage and phase angle of the load rails, and phase angle of the generator rails). Then a Taylor series is set up, the result is a linear system of equations:

${\displaystyle {\begin{bmatrix}\Delta \theta \\\Delta |V|\end{bmatrix}}=-J^{-1}{\begin{bmatrix}\Delta P\\\Delta Q\end{bmatrix}}}$

Equations are set up for and : ${\displaystyle \Delta P}$ ${\displaystyle \Delta Q}$

${\displaystyle \Delta P_{i}=-P_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\cos \theta _{ik}+B_{ik}\sin \theta _{ik})}$

${\displaystyle \Delta Q_{i}=-Q_{i}+\sum _{k=1}^{N}|V_{i}||V_{k}|(G_{ik}\sin \theta _{ik}-B_{ik}\cos \theta _{ik})}$

and is a matrix of partial derivatives known as Jacobian matrix : ${\displaystyle J}$

${\displaystyle J={\begin{bmatrix}{\dfrac {\delta \Delta P}{\delta \theta }}&{\dfrac {\delta \Delta P}{\delta |V|}}\\{\dfrac {\delta \Delta Q}{\delta \theta }}&{\dfrac {\delta \Delta Q}{\delta |V|}}\end{bmatrix}}}$.

The linearized system of equations is solved by finding the closest estimate ( m + 1) of voltage level and phase angle based on:

${\displaystyle \theta ^{m+1}=\theta ^{m}+\Delta \theta \,}$

${\displaystyle |V|^{m+1}=|V|^{m}+\Delta |V|\,}$

The process is repeated until a stop condition occurs. The stop condition usually occurs when the solution of the equivalent equations lies within a certain tolerance.

A rough approach to solving the load flow problem is:

1. Preparation of an initial estimate for voltage and phase angle. It is common practice to set all phase angles to zero and all voltages to 1.0 per unit.
2. Solve the power balance equations using the latest phase angles and voltage values.
3. Linearization of the system around the last phase angle and voltage values.
4. Resolution after changing phase angle / voltage.
5. Update of phase angle / voltage.
6. Check for stop condition, if not fulfilled, repeat from step 2.

literature

Klaus Heuck, Klaus-Dieter Dettmann, Detlef Schulz: Electrical energy supply: generation, transmission and distribution of electrical energy for study and practice . 8th edition. Vieweg + Teubner, 2010, ISBN 978-3-8348-0736-6 .

Individual evidence

1. ↑ Ismail Kasikci: Compendium planning of electrical systems theory, regulations, practice, software application , 2nd, updated and exp. Edition 2014. Edition, Springer Berlin, Berlin 2014, ISBN 978-3-642-40969-1 .