Power flow study
Encyclopedia
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. 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: voltages, voltage angles, real power and reactive power). It analyzes the power systems in normal steady-state operation. There exists a number of software implementations of power flow studies.
In addition to a power flow study, 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 kilowatthour delivered.
Power flow or load-flow studies are important for planning 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.
Commercial power systems are usually too large to allow for hand solution of the power flow. Special purpose network analyzers
were built between 1929 and the early 1960s to provide laboratory models of power systems; large-scale digital computers replaced the analog methods.
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 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 the Slack Bus. In a system with N buses and R generators, there are then
unknowns.
In order to solve for the
unknowns, there must be 
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
is the net power injected at bus i, 
is the real part of the element in the bus admittance matrix
YBUS corresponding to the ith row and kth column,
is the imaginary part of the element in the YBUS corresponding to the ith row and kth column and 
is the difference in voltage angle between the ith and kth buses. The reactive power balance equation is:
where
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.
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
and 
are called the mismatch equations:
and
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 is below a specified tolerance.
A rough outline of solution of the power flow problem is:
Power engineering
Power engineering, also called power systems engineering, is a subfield of engineering that deals with the generation, transmission and distribution of electric power as well as the electrical devices connected to such systems including generators, motors and transformers...
, the power flow study (also known as load-flow study) is an important tool involving numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
applied to a power system. A power flow study usually uses simplified notation such as a one-line diagram
One-line diagram
In power engineering, a one-line diagram or single-line diagram is a simplified notation for representing a three-phase power system. The one-line diagram has its largest application in power flow studies. Electrical elements such as circuit breakers, transformers, capacitors, bus bars, and...
and per-unit system
Per-unit system
In the power transmission field of electrical engineering, a per-unit system is the expression of system quantities as fractions of a defined base unit quantity. Calculations are simplified because quantities expressed as per-unit are the same regardless of the voltage level...
, and focuses on various forms of AC power
AC power
Power in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy flow...
(ie: voltages, voltage angles, real power and reactive power). It analyzes the power systems in normal steady-state operation. There exists a number of software implementations of power flow studies.
In addition to a power flow study, sometimes called the base case, many software implementations perform other types of analysis, such as short-circuit fault analysis and economic
Energy economics
Energy economics is a broad scientific subject area which includes topics related to supply and use of energy in societies. Due to diversity of issues and methods applied and shared with a number of academic disciplines, energy economics does not present itself as a self contained academic...
analysis. In particular, some programs use linear programming
Linear programming
Linear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
to find the optimal power flow, the conditions which give the lowest cost per kilowatthour delivered.
Power flow or load-flow studies are important for planning 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.
Commercial power systems are usually too large to allow for hand solution of the power flow. Special purpose network analyzers
Network analyzer (electrical)
A network analyzer is an instrument that measures the network parameters of electrical networks. Today, network analyzers commonly measure s–parameters because reflection and transmission of electrical networks are easy to measure at high frequencies, but there are other network parameter...
were built between 1929 and the early 1960s to provide laboratory models of power systems; large-scale digital computers replaced the analog methods.
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 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 the Slack Bus. In a system with N buses and R generators, there are then
unknowns.In order to solve for the
unknowns, there must be 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
is the net power injected at bus i, is the real part of the element in the bus admittance matrixYbus matrix
In power engineering, Y Matrix or Ybus is an n x n symmetric matrix describing a power system with n buses. It represents the nodal admittance of the buses in a power system. In realistic systems which contain thousands of buses, the Y matrix is quite sparse. Each bus in a real power system is...
YBUS corresponding to the ith row and kth column,
is the imaginary part of the element in the YBUS corresponding to the ith row and kth column and is the difference in voltage angle between the ith and kth buses. The reactive power balance equation is:where
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 SeriesTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
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
and are called the mismatch equations:and
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 is 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
- Fast-Decoupled-Load-Flow method
- Gauss–Seidel method