Power flow study

In power engineering, the power flow study (also known as load-flow study) is an important tool involving numerical analysis applied to a power system. Unlike traditional circuit analysis, a power flow study usually uses simplified notation such as a one-line diagram and per-unit system, and focuses on various forms of AC power (ie: reactive, real, and apparent) rather than voltage 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 use linear programming to find the "optimal power flow", the conditions which give the lowest cost per kilowatt 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:

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:

.

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