subject: Power Flow Study [print this page] Power flow problem formulation 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. 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 PD and reactive power QD 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 PG and the voltage magnitude |V| is known. For the Slack Bus, it is assumed that the voltage magnitude |V| and voltage phase 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 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:
where Pi is the net power injected at bus i, Gik is the real part of the element in the Ybus corresponding to the ith row and kth column, Bik is the imaginary part of the element in the Ybus corresponding to the ith row and kth column and ik is the difference in voltage angle between the ith and kth buses. The reactive power balance equation is:
where Qi 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 Newton-Raphson Method. This method begins with initial guesses of all unknown variables (voltage magnitude and angles at Load Buses and voltage angles at Generator Buses). Next, a Taylor Series is written, with the higher order terms ignored, for each of the power balance equations included in the system of equations . The result is a linear system of equations that can be expressed as:
where P and Q are called the mismatch equations:
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:
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
Newtonaphson method
Fast-Decoupled-Load-Flow method
Gausseidel method
References
^ J. Grainger and W. Stevenson, Power System Analysis, McGraw-Hill, New York, 1994, ISBN 0-07-061293-5
