- Power flow study
In

power engineering , the**power flow study**(also known as**load-flow study**) is an important tool involvingnumerical analysis applied to a power system. Unlike traditional circuit analysis, a power flow study usually uses simplified notation such as aone-line diagram andper-unit system , and focuses on various forms ofAC power (ie: reactive, real, and apparent) rather thanvoltage and current. It analyses the power systems in normal steady-state operation. There exist a number of software implementations of power flow studies.In addition to a power flow study itself, sometimes called the "base case", many software implementations perform other types of analysis, such as

short-circuit fault analysis and economic analysis. In particular, some programs uselinear programming to find the "optimal power flow", the conditions which give the lowest cost perkilowatt generated.The great importance of power flow or load-flow studies is in the planning the future expansion of power systems as well as in determining the best operation of existing systems. The principal information obtained from the power flow study is the magnitude and phase angle of the voltage at each bus and the real and reactive power flowing in each line.

**Power flow problem formulation**The goal of a power flow study is to obtain complete voltage angle and magnitude information for each bus in a power system for specified load and generator real power and voltage conditions [

*J. Grainger and W. Stevenson, "Power System Analysis", McGraw-Hill, New York, 1994, ISBN 0-07-061293-5*] . 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 "P

_{D}" and reactive power "Q_{D}" 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 "P_{G}" and the voltage magnitude |"V"| is known. For the Slack Bus, it is assumed that the voltage magnitude |"V"| and voltage "Θ" 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. In a system with "N" buses and "R" generators, there are then $2(N-1)\; -\; (R-1)$ unknowns.In order to solve for the $2(N-1)\; -\; (R-1)$ unknowns, there must be $2(N-1)\; -\; (R-1)$ 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:

$0\; =\; -P\_\{i\}\; +\; sum\_\{k=1\}^\{N\}|V\_\{i\}||V\_\{k\}|(G\_\{ik\}mbox\{cos\}\; heta\_\{ik\}+B\_\{ik\}mbox\{sin\}\; heta\_\{ik\})$

where $P\_\{i\}$ is the net power injected at bus "i", $G\_\{ik\}$ is the real part of the element in the Ybus corresponding to the "i"th row and "k"th column, $B\_\{ik\}$ is the imaginary part of the element in the Ybus corresponding to the "i"th row and "k"th column and $heta\_\{ik\}$ is the difference in voltage angle between the "i"th and "k"th buses. The reactive power balance equation is:

$0\; =\; -Q\_\{i\}\; +\; sum\_\{k=1\}^\{N\}|V\_\{i\}||V\_\{k\}|(G\_\{ik\}mbox\{sin\}\; heta\_\{ik\}-B\_\{ik\}mbox\{cos\}\; heta\_\{ik\})$

where $Q\_\{i\}$ 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 not assumed to be known 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.

**Newton-Raphson solution method**There are several different methods of solving the resulting nonlinear system of equations. The most popular is known as the Netwon-Raphson Method. This method begins with an initial guesses of all unknown variables (voltage magnitude and angles at Load Buses and voltage angles at Generator Buses). Next, a

Taylor Series with the higher order terms ignored of each of the power balance equations included in the system of equations is written.The result is a linear system of equations that can be expressed as:$egin\{bmatrix\}Delta\; heta\; \backslash \; Delta\; |V|end\{bmatrix\}\; =\; -J^\{-1\}\; egin\{bmatrix\}Delta\; P\; \backslash \; Delta\; Q\; end\{bmatrix\}$

where $Delta\; P$ and $Delta\; Q$ are called the mismatch equations:

$Delta\; P\_\{i\}\; =\; -P\_\{i\}\; +\; sum\_\{k=1\}^\{N\}|V\_\{i\}||V\_\{k\}|(G\_\{ik\}mbox\{cos\}\; heta\_\{ik\}+B\_\{ik\}mbox\{sin\}\; heta\_\{ik\})$

$Delta\; Q\_\{i\}\; =\; -Q\_\{i\}\; +\; sum\_\{k=1\}^\{N\}|V\_\{i\}||V\_\{k\}|(G\_\{ik\}mbox\{sin\}\; heta\_\{ik\}-B\_\{ik\}mbox\{cos\}\; heta\_\{ik\})$

and $J$ is a matrix of partial derivatives known as a

Jacobian :$J=egin\{bmatrix\}\; dfrac\{delta\; Delta\; P\}\{delta\; heta\}\; dfrac\{delta\; Delta\; P\}\{delta\; |V\; \backslash \; dfrac\{delta\; Delta\; Q\}\{delta\; heta\}\; dfrac\{delta\; Delta\; Q\}\{delta\; |Vend\{bmatrix\}$.

The linearized system of equations is solved to determine the next guess ("m" + 1) of voltage magnitude and angles based on:

$heta^\{m+1\}\; =\; heta^\{m\}Delta\; heta$

$|V|^\{m+1\}\; =\; heta^\{m\}Delta\; |V|$

The process continues until a stopping condition is met. A common stopping condition is to terminate if the norm of the mismatch equations are below a specified tolerance.

A rough outline of solution of the power flow problem is:

# Make an initial guess of all unknown voltage magnitudes and angles. It is common to use a "flat start" in which all voltage angles are set to zero and all voltage magnitudes are set to 1.0 p.u.

# Solve the power balance equations using the most recent voltage angle and magnitude values.

# Linearize the system around the most recent voltage angle and magnitude values

# Solve for the change in voltage angle and magnitude

# Update the voltage magnitude and angles

# Check the stopping conditions, if met then terminate, else go to step 2.**Power flow methods***

Newton–Raphson method

*Gauss–Seidel method **References**

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Severn Tidal Power Feasibility Study**— is the name of a UK Government project looking at the possibility of using the huge tidal range in the Severn Estuary and Bristol Channel to generate electricity. The tidal range in the Severn Estuary is the second highest in the world and can… … Wikipedia**power**— is the concept which is at the heart of the subject of social stratification . It is therefore not surprising that we have seen so many disputes concerning its meaning (including disputes about what particular sociologists meant when they used… … Dictionary of sociology**Electric power transmission**— Electric transmission redirects here. For vehicle transmissions, see diesel electric transmission. 400 kV high tension transmission lines near Madrid Electric power transmission or high voltage electric transmission is the bulk transfer of… … Wikipedia**Timeline of steam power**— See Steam engine, Steam power during the Industrial Revolution. Steam power developed slowly over a period of several hundred years, progressing through expensive and fairly limited devices in the early 1600s, to useful pumps for mining in 1700,… … Wikipedia**Intermittent power source**— [ Erie Shores Wind Farm monthly output over a two year period] An intermittent power source is a source of electric power generation that may be uncontrollably variable or more intermittent than conventional power sources, and therefore non… … Wikipedia**Naval Nuclear Power School**— Infobox University name = Naval Nuclear Power School motto = Knowledge, Integrity, Excellence established = 1955 type = Military Technical School head label = Commanding Officer head = CAPT Thomas Bailey, USN city = Goose Creek state = South… … Wikipedia**thermoelectric power generator**— Introduction any of a class of solid state devices (solid state device) that either convert heat directly into electricity or transform electrical energy into thermal power for heating or cooling. Such devices are based on thermoelectric… … Universalium**Nuclear power plant**— This article is about electricity generation from nuclear power. For the general topic of nuclear power, see Nuclear power. A nuclear power station. The nuclear reactor is contained inside the cylindrical containment buildings to the right left… … Wikipedia**Sidoarjo mud flow**— The Sidoarjo mud flow or Lapindo mud, also informally abbreviated as Lusi , a contraction of Lumpur Sidoarjo ( lumpur is the Indonesian word for mud), is a since May 2006 ongoing eruption of gas and mud in the subdistrict of Porong, Sidoarjo in… … Wikipedia**Marine current power**— is a form of marine energy obtained from harnessing of the kinetic energy of marine currents, such as the Gulf stream. Although not widely used at present, marine current power has an important potential for future electricity generation. Marine… … Wikipedia