Editing Power-flow study (section)

==Power-flow problem formulation==
The goal of a power-flow study is to obtain complete voltage angles and magnitude information for each bus in a power system for specified load and generator real power and voltage conditions.<ref>{{cite book|first1=J. |last1=Grainger |first2= W.|last2= Stevenson|title=Power System Analysis|publisher= McGraw–Hill|location= New York|year=1994| isbn= 0-07-061293-5}}</ref> Once this information is known, real and reactive power flow on each branch as well as generator reactive power output can be analytically determined. Due to the nonlinear nature of this problem, numerical methods are employed to obtain a solution that is within an acceptable tolerance.

The solution to the power-flow problem begins with identifying the known and unknown variables in the system. The known and unknown variables are dependent on the type of bus. A bus without any generators connected to it is called a Load Bus. With one exception, a bus with at least one generator connected to it is called a Generator Bus. The exception is one arbitrarily-selected bus that has a generator. This bus is referred to as the [[slack bus]].

In the power-flow problem, it is assumed that the real power <math>P_D</math> and reactive power <math>Q_D</math> at each Load Bus are known. For this reason, Load Buses are also known as PQ Buses. For Generator Buses, it is assumed that the real power generated <math>P_G</math> and the voltage magnitude <math>|V|</math> is known. For the Slack Bus, it is assumed that the voltage magnitude <math>|V|</math> and voltage phase <math>\theta</math> are known. Therefore, for each Load Bus, both the voltage magnitude and angle are unknown and must be solved for; for each Generator Bus, the voltage angle must be solved for; there are no variables that must be solved for the Slack Bus. In a system with <math>N</math> buses and <math>R</math> generators, there are then <math>2(N-1) - (R-1)</math> unknowns.

In order to solve for the <math>2(N-1) - (R-1)</math> unknowns, there must be <math>2(N-1) - (R-1)</math> equations that do not introduce any new unknown variables. The possible equations to use are power balance equations, which can be written for real and reactive power for each bus.
The real power balance equation is:

<math display=block>0 = -P_{i} + \sum_{k=1}^N |V_i||V_k|(G_{ik}\cos\theta_{ik}+B_{ik}\sin\theta_{ik})</math>

where <math>P_{i}</math> is the net active power injected at bus ''i'', <math>G_{ik}</math> is the real part of the element in the [[Ybus matrix|bus admittance matrix]] Y<sub>BUS</sub> corresponding to the <math>i_{th}</math> row and <math>k_{th}</math> column, <math>B_{ik}</math> is the imaginary part of the element in the Y<sub>BUS</sub> corresponding to the <math>i_{th}</math>  row and <math>k_{th}</math>  column and <math>\theta_{ik}</math> is the difference in voltage angle between the <math>i_{th}</math>  and <math>k_{th}</math>  buses (<math>\theta_{ik}=\theta_i-\theta_k</math>). The reactive power balance equation is:

<math display=block>0 = -Q_{i} + \sum_{k=1}^N |V_i||V_k|(G_{ik}\sin\theta_{ik}-B_{ik}\cos\theta_{ik})</math>

where <math>Q_i</math> is the net reactive power injected at bus ''i''.

Equations included are the real and reactive power balance equations for each Load Bus and the real power balance equation for each Generator Bus. Only the real power balance equation is written for a Generator Bus because the net reactive power injected is assumed to be unknown and therefore including the reactive power balance equation would result in an additional unknown variable. For similar reasons, there are no equations written for the Slack Bus.

In many transmission systems, the impedance of the power network lines is primarily inductive, i.e. the phase angles of the power lines impedance are usually relatively large and very close to 90 degrees. There is thus a strong coupling between real power and voltage angle, and between reactive power and voltage magnitude, while the coupling between real power and voltage magnitude, as well as reactive power and voltage angle, is weak. As a result, real power is usually transmitted from the bus with higher voltage angle to the bus with lower voltage angle, and reactive power is usually transmitted from the bus with higher voltage magnitude to the bus with lower voltage magnitude. However, this approximation does not hold when the phase angle of the power line impedance is relatively small.<ref>[http://www.eeh.ee.ethz.ch/uploads/tx_ethstudies/modelling_hs08_script_02.pdf Andersson, G: Lectures on Modelling and Analysis of Electric Power Systems] {{webarchive|url=https://web.archive.org/web/20170215042633/http://www.eeh.ee.ethz.ch/uploads/tx_ethstudies/modelling_hs08_script_02.pdf |date=2017-02-15 }}</ref>