Load flow - step by step
Load flow (power flow) analysis is a basic analysis for the study of power systems. It is used for normal, steady-state operation. It gives you the information what is happening in a system.
It is an answer to some fundamental questions, which power system engineer or electrical engineer can have:
- What are voltage levels in all power system nodes during operation?
- Are power system elements (transformers, generators, cables etc.) overloaded?
- What are the weakest points of network?
Load flow analysis is an important prerequisite for whatever you do in power systems, whether you do fault studies, stability studies, economic operation etc.
The load flow helps in continuous monitoring of the current state of the power system, so it is used on daily basis in load dispatch/power system control centers. It can also be a support during examining effectiveness of the alternative plans for future system expansion, when adding new generators or transmission lines is needed. You can refer to the description of different load flow cases for more information.
Load flow objective
The objective of load flow calculations is to determine the steady-state operating characteristics of the power system for a given load and generator real power and voltage conditions. Once we have this information, we can calculate easily real and reactive power flow in all branches together with power losses.
What are the inputs and outputs of load flow analysis?
Minimum input data for power flow analysis:
- Bus data (types of buses explained in bus types):
- For PV buses:
- Real power (generation and demand),
- Reactive power (demand),
- Voltage magnitude.
- For PQ buses:
- Real power (generation and demand),
- Reactive power (generation and demand).
- For slack bus:
- Voltage magnitude (usually 1 per unit),
- Voltage angle(specified to be zero),
- Real power (demand),
- Reactive power (demand).
- Line data:
- Transmission lines:
- Resistance,
- Reactance,
- Capacitance (can be negligible).
- Transformers:
- Winding resistances on low and high voltage side,
- Leakage reactance on low and high voltage side,
- Magnetization reactance,
- Iron loss admittance.
Power flow analysis provides following output data for each node/branch:
- Voltage magnitude,
- Voltage angle,
- Real and reactive power,
- Power losses.
Bus types
Depending, upon which two variables you specify, the buses (nodes) can be categorized into three categories:
- Slack bus (swing or reference bus),
- PQ bus (sometimes called as a load bus),
- PV bus.
Bus types | Quantities specified | Unknown variables |
Slack | \begin{equation}|U|, \delta, P_D, Q_D\end{equation} | \begin{equation}P_G, Q_G\end{equation} |
PQ | \begin{equation}P_G, Q_G, P_D, Q_D\end{equation} | \begin{equation}|U|, \delta\end{equation} |
PV | \begin{equation}|U|, P_G, P_D, Q_D\end{equation} | \begin{equation}\delta, Q_G\end{equation} |
Slack bus
There is only one slack bus in system under consideration. Slack bus always has a generator attached to it, with no exception. Normally this generator is biggest in the system. Its two main tasks is to:
- Serve as the reference for voltage angle,
- Balance generation, load and losses, because the power losses are not known until end of load flow calculation. Slack bus needs to supply complex losses.
The rest of buses swing with the reference to this particular bus.
Whatever is extra left that will come from this slack bus remaining, anything which we could not fulfill from the rest of the buses will come from it.
PQ bus
Load buses may contain generators with specified real and reactive power outputs.
PV bus
Have generator connected to them. The PV buses can have voltage control capabilities and uses a tap-adjustable transformer and and/or VAR compensator instead of generator.
Load flow problem
To put it simple, let’s look on the exemplary system:
- Has network of nodes (buses) and branches (lines or transformers),
- Has consumers (loads), which are withdrawing power at nodes,
- Has suppliers (generators), which are injecting power at nodes,
- Has a defined node voltages which determine branch flows,
- Needs to meet a requirement that branch flows out of a node must equal the net nodal injection (generation-demand) by Kirchhoff Current Law.
Each network node is described by four main variables:
- \(U_i\) voltage magnitude
- \(\delta_i\) voltage angle
- \(P_i\) real power
- \(Q_i\) reactive power
In other words, the network can be represented by phasors:
- Vector of complex bus voltages, \(Ve^{j\Theta}\) voltage magnitude
- Vector of complex bus power injections, \(S=P+jQ\) voltage angle
Kirchoff Current Law leads to nonlinear complex power balance equation:
\begin{equation} \mathbf g(x), \text{where} \ \mathbf x = \left[ \begin{array}{c} \mathbf V \\ \mathbf S \\ \end{array} \right] \end{equation}
Without getting much into multiple algebraic equations we get a main load flow equations (called static load flow equations):
\begin{equation} P_i = |U_i|\sum_{k=1}^n|U_k||Y_{ik}|cos(\Delta_{ik}+\delta_k-\delta_i) \label 1 \end{equation} \begin{equation} Q_i = -|U_i|\sum_{k=1}^n|U_k||Y_{ik}|sing(\Delta_{ik}+\delta_k-\delta_i) \label 2 \end{equation}where:
\(n\) - number of buses
\(i=1,2,3...n\)
The problem to solve is that you have two equations, which are nonlinear algebraic equations. Why non-linear ? Because voltages are squared and cosine and sine functions are themselves non-linear. We have four variables unknown (real power, reactive power, voltage magnitude and angle at each bus, so the number of unknown variables is 4n. The number of variables are double than the number of equations. In order to solve it we need to assume 2n variables based on a system knowledge. These non-linear algebraic equations can be solved by using iterative numerical techniques.
Load flow calculation – step by step
We can set following general steps in order to describe load flow calculation:
- Specify values of elements for network components
- Specify place, values and constraints for loads in the power system
- Define specifications and constraints for generators in the power system
- Establish a math model describing load flow in the power system
- Solve the model equations for the voltage profile of the power system
- Solve the model equations for the power flows and losses in the power system
- Verify if there are some constraint violations
Load flow assumptions, constraints and limitations
Before going into the description of numerical techniques, let’s focus on load flow constraints and limitations. The steady state operation of a three phase power system is characterized by balanced conditions. All electrical elements are three phase symmetric objects. That’s why we can assume that all voltages and currents in the system are balanced three phase quantities which leads to that the system is balanced. Because of that, we can consider only positive sequence network in load flow studies.
Assumptions for load flow calculation:
- System is in steady state (no transient changes)
- Three phases system is assumed to have balanced loading
- Per-unit system is used for simplification
There also some constraints in load flow problem for:
- Voltage magnitude, \( {|U_i|}_{min} \leq |U_i| \leq {|U_i|}_{max} \)
- Voltage angle, \( |\delta_i - \delta_k| \leq |\delta_i-{\delta_k}_{max}| \)
Physical limitations:
- Real generated power, \( {P_{Gi}}_{min} \leq P_{Gi} \leq {P_{Gi}}_{max} \)
- Reactive generated power, \( {Q_{Gi}}_{min} \leq Q_{Gi} \leq {Q_{Gi}}_{max} \)
Numerical solution of load flow problem
The number of nodes in real power systems is so high that the calculation are to complex to make it by hand. That’s why, we use numerical methods. Main iterative numerical methods to solve non-linear algebraic equations (load flow equations) (tutaj numer równań), which are:- Gauss-Seidel
- Newton-Raphson
- Fast-decoupled
Gauss-Seidel | Newton-Raphson | Fast-decoupled | ||
Complexity | Easy | Complex | Less complex (constant Jacobian, you do not have to inverse it) | |
Convergence | Linear | Quadratic - the fastest | Geometric | |
Sensitivity | Not available | Available | Available | |
System size | Problematic with large systems (the time of calculation increases linearly with system size) | Appropriate for any system size (the time of calculation does not depend on system size, 3-4 iterations usually needed) | Appropriate for any system size (the time of calculation does not depend on system size, 5-6 iterations usually needed) | |
Type of system | May have a convergence problem with ill-condition system | No problem with ill-condition system | No problem with ill-condition system | |
Accuracy | Good | The best | Average | |
Sensitivity | It is sometimes advisable to have one or two Gauss-Seidel iterations before Newton-Raphson, which may decrease the iteration to some extent and it helps with the “flat start” which is sometimes causing non-convergence for Newton-Raphson method. | |||
No. of iterations and time of solution for system of: | 14 bus | 56 iterations, 0,0288 s | 4 iterations, 0,0050 s | 6 iterations, 0,0053 s |
118 bus | 388 iterations, 2,738 s | 5 iterations 0,0287 s | 6 iterations, 0,0117 s | |
1228 bus | 224 iterations, 112,4 s | 5 iterations, 0,210 s | 12 iterations, 0,16 s | |
11856 bus | 173 iterations, 9112 s | 4 iterations, 3,15 s | 5 iterations, 5,174 s | |
Year of publication | 1967 | 1974 |
Load flow cases
The load flow cases are classified as design and contingency cases.
The design case is when all the equipment is working within standard conditions i.e. the voltage at all nodes is within ± 5%.
The contingency case load flow is conducted after a fault is cleared. There are different types of contingencies like:
- short circuits,
- loss of electrical generator, transformer, transmission line,
- sudden drop of large load.