Load flow, power flow - step by step

Posted On: December 30, 2022
Posted By: Adam Kierad - Electrical Software Engineer

Load Flow, Power Flow - Step by step

Load flow (power flow) analysis is a primary way to study power systems. It gives information about what is happening in a system and answers some fundamental questions like:

What are the voltage levels in power system nodes?
What are the current values flowing in power system branches?
Are power system elements (transformers, generators, cables, etc.) overloaded?
What are the weakest points of a network?

It is performed for the steady-state operation of a power system. In general, power flow solutions are needed for the system under the following conditions:

Various systems loading conditions (peak and off-peak).
With certain equipment outaged.
Addition of new generators.
Addition of new transmission lines or cables.
Interconnection with other systems.
Load growth studies.
Loss of line evaluation.

Power flow analysis is fundamental to the study of power systems forming the basis for other analyses. Load flow analysis is a prerequisite for whatever you do in power systems, whether you do fault studies, stability studies, etc. The load flow helps in continuous monitoring of the current state of the power system, so it is used in load dispatch/power system control centers. It can support examining the effectiveness of the alternative plans for future system expansion when adding new generators or transmission lines is needed.

In several papers and books, especially old ones, the power flow analysis is called load flow analysis.This notation should be avoided since, quoting Concordia and Tinney: “Load does not flow, but power flows.”

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 a generator's 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.

Try our free(no credit card needed) cloud application for load flow studies - app.electrisim.com. You can model, simulate and analyze your electrical system directly in a web browser

What are the inputs and outputs of load flow analysis?

Minimum input data for power flow analysis:

Bus data (types of buses explained -> Jump to Chapter Bus Types):

For PV buses:

Real power
Reactive power
Voltage magnitude

For PQ buses:

Real power
Reactive power

For slack bus:

Voltage magnitude (usually 1 per unit)
Voltage angle (specified to be zero)

Line data:

Transmission lines:

Resistance
Reactance
Capacitance

Transformers:

Winding resistances on low and high voltage sides
Leakage reactances on low and high voltage sides
Magnetization reactance
Iron loss admittance

Active power generation is normally specified according to economic-dispatching practice and the generator voltage magnitude is normally maintained at a specified level by the automatic voltage regulator acting on the machine excitation. Loads are normally specified by their constant active and reactive power requirement, assumed unaffected by the small variations of voltage and frequency expected during normal steady-state operation.

Power flow analysis provides the 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 (sometimes called swing or reference bus)
PQ bus (sometimes called load bus)
PV bus

Slack bus

There is only one slack bus in the system under consideration. The slack bus for the system is a single bus for which the voltage magnitude and angle are specified. The real and reactive power is unknown. The bus selected as the slack bus must have a source of both real and reactive power since the injected power at this bus must “swing” to take up the“slack” in the solution. The best choice for the slack bus (since, in most power systems, many buses have real and reactive power sources) requires experience with the particular system under study. The behavior of the solution is often influenced by the bus chosen.

PQ bus(load bus)

The load bus is defined as any bus of the system for which the real and reactive power is specified. Load buses may contain generators with specified real and reactive power outputs; however, it is often convenient to designate any bus with specified injected complex power as a load bus.

PV bus(Voltage Controlled Bus)

Any bus for which the voltage magnitude and the injected real power are specified is classified as a voltage-controlled (or P-V) bus. The injected reactive power is a variable (with specified upper and lower bounds) in the power flow analysis. (The PV bus must have a variable source of reactive power such as a generator.)

Load flow problem

The exemplary system:

The system has:

network of nodes (buses) and branches (lines or transformers)
consumers (loads), which are withdrawing power at nodes
suppliers (generators), which are injecting power at nodes
defined node voltages which determine branch flows, 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:

voltage magnitude
voltage angle
real power
reactive power

In other words, the network can be represented by two phasors:

Vector of complex bus voltages
Vector of complex bus power injections

Kirchhoff's Current Law leads to two nonlinear complex power balance equations. The problem is to solve these algebraic equations, which are non-linear because voltages are squared and cosine and sine functions are themselves non-linear. These non-linear algebraic equations can be solved by using iterative numerical techniques.

Load flow calculation - step by step

We can set the following general steps 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

Numerical solution of load flow problem

The number of nodes in real power systems is so high that the calculation is too complex to make by hand. That’s why we use numerical methods. Using digital computers to calculate load flow started in the middle of the 1950s. Since then, a variety of methods have been used in load flow calculation. Much effort has been devoted to the study of iterative methods for the solution of nonlinear equations. The development of these methods is mainly led by the basic requirements of load flow calculation and the performance of any method is measured with the following criteria:

The convergence properties (existence of solution)
The computing efficiency and memory requirements
The convenience and flexibility of the implementation

The performance of a method depends on the specific properties of the problem under consideration. In the case of the power flow problem, much theoretical work has been done and much experience has been accumulated over the years. The characteristics of the algorithms as applied to the solution of the power flow problem are very well known. Yet there is no guarantee that given a system the solution can be found (convergence). On the other hand, practical experience with these methods indicates that the non-convergence of the power flow problem is rather seldom.

The first practical digital solution methods for load flow were the Y matrix iterative methods. These were suitable because of the low storage requirements but had the disadvantage of converging slowly or not at all. Z matrix methods were developed which overcame the reliability problem but a sacrifice was made for storage and speed with large systems.

The main iterative numerical methods to solve non-linear algebraic equations (load flow equations) are:

Gauss-Seidel (G-S).
Newton-Raphson (N-R)
Newton-Raphson with Iwamoto multiplier (N-R-I)
Fast decoupled
Backward/forward sweep

All the above methods are implemented in the Electrisim App.

The Newton-Raphson method is very reliable in system solving, given good starting approximations. Heavily loaded systems with phase shift angle up to 90" can be solved. The Newton-Raphson method has been widely accepted because of its excellent convergence characteristics and its reliability. The method is not troubled by ill-conditioned systems and the location of the slack bus is not critical. Due to the quadratic convergence of bus voltages, high accuracy (near exact solution) is obtained in only a few iterations. This is important for the use of load flow in shortcircuit and stability studies. The method is readily extended to include tap-changing transformers, variable constraints on bus voltages, and reactive and optimal power scheduling.

The Newton-Raphson with Iwamoto multiplier (N-R-I) is slower method than the Newton-Raphson, but more robust.

The attractive characteristics of the Newton-Raphson method are moderated by complex computational burdens. At each iteration of the method, the Jacobian matrix needs to be formed and inverted. For large-scale systems (systems with thousands of buses), excessive storage requirements and computations jeopardize the practicality of the method. The introduction of sparsity techniques has mitigated this obstacle. Sparsity techniques are the procedures by which one takes advantage of the fact that the Jacobian matrix is highly sparse to minimize computational effort and storage requirements. The sparsity-coded Newton-Raphson method has been proven practical for large-scale systems. However, research has indicated that further improvements can be made to the method. Much effort has been concentrated on the so-called quasi-Newton methods. The basic idea behind these methods is the following: Is it possible to approximate the Jacobian matrix, which depends on the iterate under consideration, with a constant matrix that can be used in any iteration? If this is possible, then the inverse of the approximate Jacobian can be computed in the beginning once and for all, and be employed in all subsequent iterations. In this way, the most demanding computational task of forming and inverting the Jacobian matrix in every iteration is avoided. One should expect that this may deteriorate the convergence characteristics of Newton's method. However, overall, the quasi-Newton method may be more efficient than Newton's method. Many attempts have been made in this direction. One of the quasi-Newton methods, known as the Fast Decoupled Power Flow, has been proven to be very successful with substantial improvements over the Newton-Raphson method. The Fast Decoupled Load Flow is based on a transformation of the iterative equations of the Newton-Raphson method in such a way that they involve a constant, but approximate Jacobian matrix.

An inherent characteristic of any practical electric power transmission system operating in the steady state condition is the strong interdependence between active powers and bus voltage angles, and between reactive powers and voltage magnitudes. Correspondingly, the coupling between these ‘active power-angle' and ‘reactive power-voltage’ components of the problem is relatively weak.

Many contributions seek to improve the convergence characteristics of the Newton method and the P - Q decoupled method. Along with the development of artificial intelligent theory, the genetic algorithm, artificial neural network algorithm, and fuzzy algorithm have also been introduced to load flow analysis. However, until now these new models and new algorithms still cannot replace the Newton method and P - Q decoupled method. Because the scales of power systems continue to expand and the requirements for online calculation become more and more urgent, parallel computing algorithms are also studied intensively now and may become an important research field.

The backward/forward sweep method is especially suited for radial and weakly-meshed networks.

Different types of load flow

DC power flow

In several applications, it is expedient to seek a quick but approximate solution to the power flow problem. This objective is accomplished with the so-called DC power flow (or DC Network Model). The DC power flow results from a simplification and linearization of the rather complex and nonlinear power flow equations. The simplification results in a model which describes the flow of real power only. The flow of reactive power is ignored. This model has been extensively used in the past and has been useful for planning studies where an approximate evaluation of the real power flow is needed.

Continuation Power Flow

The continuation power flow problem is essentially solving a sequence of power flow problems where are you modifying a given parameter such as the loading in the network. For example you might increase the loading until you see a voltage collapse in the system.

Optimal power flow (OPF)

The OPF procedure consists of determining the optimal steady-state operation of a power system, which simultaneously minimizes the value of a chosen objective function and satisfies certain physical and operating constraints. Today OPF has been playing a very important role in power system operation and planning: different classes of OPF problems, tailored towards special-purpose applications are defined by selecting a different function to be minimized, different sets of controls, and different sets of constraints.

Several options are available for controlling the circuit flows and bus voltages such as transformer taps, status of capacitor banks, generating unit reactive, or real power output, etc. The objective is to select the options (controls) in such a way that the resulting operating conditions meet certain performance criteria, for example, minimum operating cost, minimum losses, etc.

The basic objective of the OPF is to find the values of the system state variables and parameters that minimize some cost functions of the power system. The types of cost functions are system dependent and can vary widely from application to application and are not necessarily strictly measured in terms of money. Examples of engineering optimizations can range from minimizing:

active power losses
particulate output (emissions)
system energy, or
fuel costs of generation

to name a few possibilities. The basic formulation of the OPF can be represented as minimizing a defined cost function subject to any physical or operational constraints of the system.

The OPF problem is set up on the following basis:

The operating generating units are predetermined
The power outputs of hydro units are predetermined by reservoir dispatching
The structure of the transmission network is predetermined, which means the network reconfiguration problem is not considered in OPF The variables of the OPF problem consist of a set of dependent variables and a set of control variables. The dependent variables include node voltage magnitudes and phase angles, as well as the MVAr output of generators performing node voltage control. The control variables might include a real and reactive power output of generators, voltage settings of voltage control nodes, transformer tap positions, phase shifter angles, operating capacities of shunt capacitors, reactors, etc.

The constraints of OPF include:

Power flow equations
Upper and lower bounds on the generator's active power outputs
Upper and lower bounds on the generator reactive power outputs
Capacity constraints on shut capacitors and reactors
Upper and lower bounds on the transformer or phase shifter tap positions
Branch transfer capacity limits
Node voltage limits

Except for constraint (1), the other constraints are all inequality constraints. Among these constraints, (1) and (6) are functional-type constraints. When the rectangular coordinates format is used to describe node voltages, constraint (7) is also of the functional type. The others are constraints on variables. The objective function of an OPF problem may take many different forms according to the different applications.

Asymmetric/Three-phase Load Flow

For most purposes in the steady-state analysis of power systems, the system unbalance can be ignored and the single-phase analysis described in previous sections is adequate. However, in practice, it is uneconomical to balance the load completely or to achieve perfectly balanced transmission system impedance, as a result of untransposed high-voltage lines and lines sharing the same right-of-way for considerable lengths. A realistic assessment of the unbalanced operation of an interconnected system, including the influence of any significant load unbalance, requires the use of three-phase load flow algorithms. The object of the three-phase load flow is to find the state of the three-phase power system under the specified conditions of load, generation and system configuration.

Balanced AC power flow - Temperature-Dependent Power Flow

The description of the algorithm which takes into account a thermal inertia of transmission lines can be found here Github-Pandapower-Temperature-Dependent Power Flow..

Criteria for evaluation

The power flow cases are generally classified as design cases, contingency cases, and extreme contingency cases. The definition of the individual case and the acceptable performance under the given operating case has to be considered.

Base case - A base case is a design requirement case with all the equipment operating within the normal ratings. This is applicable for peak and off-peak load conditions. The system voltage at all the buses will be within ± 5% (or other value specified in a grid code). The base case criteria are applicable for all the planning studies of the bulk power system.

Contingency case - A contingency case is a power flow case with one component outage, followed by fault clearing. The fault may be any one of the following:

Loss of one component without a fault
A permanent three-phase fault on any bus section, any one generator, transmission line, or transformer cleared in normal fault clearing time
Simultaneous phase-to-ground faults on different phases of each of two double circuits installed on a double circuit tower, cleared in normal fault clearing time. Some utilities consider this a multiple contingency case.
A permanent three-phase to-ground fault on any bus section, any generator, transmission line, or transformer with delayed fault clearing.

Contingency cases must have all lines loaded within short-term emergency ratings and all other equipment loaded with long-term emergency ratings. Allowable system voltages are within a range of 0.95 per unit to 1.05 per unit. It is expected that within minutes all line and cable loading can be reduced to within the long-term emergency ratings by adjustment of phase shifting transformers and/or re-dispatch of generation. Sometimes, a contingency analysis is performed using the entire system. Then, the following types of cases are found in the results:

Acceptable cases - These are power flow cases, without any overloaded branches or undervoltage or overvoltage buses.

Cases with overloaded lines - If there are overloaded lines or transformers, then the line overloading can be brought to the normal ratings using transformer tap changing or other control actions

Cases with overvoltage or undervoltage - If there are overvoltage or undervoltage buses, then the bus voltages can be brought to the normal values using transformer tap changing or other control actions.

Cases with overloads lines and voltage-deviated buses - Actions required as above.

Not converged cases - The power flow solution is not converged for the given contingency case.

Islanded cases - During islanded operation, the system parts into two or more sections, and each section may tend to have overvoltage or undervoltage problems depending on the amount of generation available in each section.

The not converged and the islanding cases are not acceptable. All the cases require careful analysis to avoid any loss in the system's performance.

Multiple contingency cases - Sometimes more than one fault occurs in a power system due to a common cause (for example a lightning strike) or for other reasons. Though the power systems are not designed for multiple contingencies, the power system planners need to know the effect and remedial approaches for such events. Some of the multiple contingencies are:

Loss of an entire generating plant
Sudden dropping off of a very large load
Loss of all lines from a generating station or substation
Loss of transmission lines on a common right of way
Three-phase fault on a bus section, generator, or transmission line with delayed clearing

The effect of multiple contingencies might be line overloads, unacceptable bus voltages, islanding, or any other emergency condition. Therefore, planning studies are always needed in this direction to understand the system's behavior.

Control variables

These consist of all quantities that can be independently manipulated or by existing control loops to satisfy system objectives. For the power flow problem, these are:

Voltage magnitude at certain buses. For example, generation buses, buses connected to regulating transformers, or buses with synchronous condensers, etc.
Real power generation at generation buses. (These controls are not independent since at all times the total generation must equal the system load plus losses)
Tap the settings of transformers
Phase shift of phase shifting transformers
Switch the status of a capacitor and/or reactor banks (open/close)
Etc.