Power-flow study

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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 (i.e.: voltages, voltage angles, real power and reactive power). It analyzes the power systems in normal steady-state operation. A number of software implementations of power-flow studies exist.

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, stability studies (transient & steady-state), unit commitment and economic dispatch.[1] In particular, some programs use linear programming to find the optimal power flow, the conditions which give the lowest cost per kilowatt hour 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.


An AC power-flow model is a model used in electrical engineering to analyze power grids. It provides a nonlinear system which describes the energy flow through each transmission line. Due to nonlinearity, in many cases the analysis of large network via AC power-flow model is not feasible, and a linear (but less accurate) DC power-flow model is used instead. Both of those models are very crude approximations to reality.

Systems analysis[edit]


Prior to purchasing heavy or high-voltage electrical equipment (transformers, breakers, cables, etc.), it is customary for plant owners, engineering firms, design firms, power utilities, etc. to model their electrical system in a digital computer software for the purpose of equipment sizing and performing various "what if" scenarios. These electrical models range from 120V all the way to 765 kV at the transmission level. An electrical software is typically used for performing various analysis from steady state to transient behavior of the electrical system.

Network topology builder[edit]

Electrical models can be created by many available methods:

  • Dragging and connecting blocks from symbol library to create a logical electrical single-line diagram
  • Importing data from other databases and programs like Microsoft Excel, Access, PSS/E, etc.
  • Utilizing templates for substations, protection, distribution, switching stations, data center tiers, etc. Templates are groups of pre-built and connected symbols that represent standard electrical power system configurations

Model Verification and validation[edit]

The electrical system one-line diagram must be updated to accurately illustrate the power system. One-line diagram is an important maintenance document in any plant. When any change or addition is made to a power system, the one-line diagram should be updated immediately to show that change. All staff concerned with the maintenance and operation of the electrical system should have access to the latest revised copies on a regular basis. These diagrams should be reviewed and updated periodically. ETAP software provides an integrated facility to create, maintain, track and revise your power system one-line diagrams.

Load flow[edit]

A load flow study is especially valuable for a system with multiple load centers, such as a refinery complex. The power flow study is an analysis of the system’s capability to adequately supply the connected load. The total system losses, as well as individual line losses, also are tabulated. Transformer tap positions are selected to ensure the correct voltage at critical locations such as motor control centers. Performing a load flow study on an existing system provides insight and recommendations as to the system operation and optimization of control settings to obtain maximum capacity while minimizing the operating costs. The results of such an analysis are in terms of active power, reactive power, magnitude and phase angle.

Short circuit[edit]

As plant expansion and modification occurs, loads may be moved and larger ones added, leading to increased levels of available short-circuit currents. In addition, the power grid supplying the plant may have increased the available fault capacity due to the enlargement of its own system. The possibility of increasing the amount of short-circuit current available into a fault by these changes is the major reason for a periodic system study. If the short-circuit capacity of the system exceeds the capacity of the protective device, a dangerous situation exists for both plant personnel and system equipment. Under fault conditions, the protective devices would attempt to interrupt the fault current, which could cause a catastrophic failure. Therefore, the need and importance of determining the short-circuit capabilities of a system cannot be stressed enough.

Motor starting[edit]

There are many considerations to starting a motor other than effectively connecting it to the line voltage. Nuisance tripping and excessive running currents, as well as dimming of lights, are signs that a power system is not performing properly. The power system should be able to supply inrush to any motor on the system while supplying normal service for the rest of the system. If the system does not have sufficient capacity, there will be excessively low voltage drops and insufficient capacity for motor starting.

Protective devices[edit]

Protective device coordination is the ability of the closest upstream protective device to detect and clear the system fault without the operation of another protective device further upstream in the power system. As backup protection, if the closest device fails to operate for some reason, the next set of protective devices should be coordinated so they will operate before extensive damage results. Many cases of unexplained outages or heavily damaged equipment are accepted without question or knowledge that the real reason may be improper coordination of protective devices. A poorly coordinated system will result in nuisance outages during a fault condition. Damage to equipment are more likely under mis-coordinated protective devices hence, resulting in unplanned downtime and equipment repair or replacement.

Arc flash[edit]

Arc Flash studies are required and compliance with this standard is mandatory per OSHA. Companies will be cited and fined for not complying with these standards. A facility must provide, and be able to demonstrate, a safety program with defined responsibilities the following:

  • Calculations for the degree of arc flash hazard
  • Correct personal protective equipment (PPE) for workers
  • Training for workers on the hazards of arc flash
  • Appropriate tools for safe working
  • Warning labels on equipment

Dynamic stability[edit]

System stability study is essential when adding, upgrading or evaluating existing generators within the facility. This study will evaluate overcurrent relay settings and or modifications to the protection scheme associated with the generators and utility. This time-based analysis will determine relay settings that will allow the generator out-of-step protection and overcurrent protections to operate for a disturbance prior to loss of system stability and damage to equipment's. In general, load shedding (LS) can be defined as the amount of load that must nearly instantly be removed from a power system to keep the remaining portion of the system operational. This load reduction is in response to a system disturbance that results in an unbalanced condition of the amount of system load exceeding the available electric generation. Common disturbances that can cause this condition to occur include faults, loss of generation, switching errors, lightning strikes, etc. Consequences of Improper Load Shed:

  • Shedding too much load (Loss of Critical Process)
  • Total loss of production
  • Unsafe operating condition and environmental concerns
  • Costly outages and equipment damage

Harmonic analysis[edit]

In general, harmonic analysis is commonly used to predict distortion levels caused by the addition of a new harmonic-producing loads or capacitor banks. Consequences of excessive harmonic distortion include:

  • Control / Computer system interference
  • Overheating in rotating machinery
  • Overheating / failure of capacitors
  • Costly outages and equipment damage

Power-flow problem formulation[edit]

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.[2] 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 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}\cos\theta_{ik}+B_{ik}\sin\theta_{ik})

where P_{i} is the net power injected at bus i, G_{ik} is the real part of the element in the bus admittance matrix YBUS corresponding to the ith row and kth column, B_{ik} is the imaginary part of the element in the YBUS corresponding to the ith row and kth column and \theta_{ik} is the difference in voltage angle between the ith and kth buses (\theta_{ik}=\delta_i-\delta_k). The reactive power balance equation is:

0 = -Q_{i} + \sum_{k=1}^N |V_i||V_k|(G_{ik}\sin\theta_{ik}-B_{ik}\cos\theta_{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.

In many transmission systems, the voltage angles \theta_{ik} are usually relatively small. There is thus a strong coupling between real power and voltage angle, and between reactive power and voltage magnitude, while the coupling between real power and voltage magnitude, as well as reactive power and voltage angle, is weak. As a result, real power is usually transmitted from the bus with higher voltage angle to the bus with lower voltage angle, and reactive power is usually transmitted from the bus with higher voltage magnitude to the bus with lower voltage magnitude. However, this approximation does not hold when the voltage angle is very large.[3]

Newton–Raphson solution method[edit]

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:

\begin{bmatrix}\Delta \theta \\ \Delta |V|\end{bmatrix} = -J^{-1} \begin{bmatrix}\Delta P \\ \Delta Q \end{bmatrix}

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}\cos\theta_{ik}+B_{ik}\sin \theta_{ik})

\Delta Q_{i} =  -Q_{i} + \sum_{k=1}^N |V_i||V_k|(G_{ik}\sin\theta_{ik}-B_{ik}\cos\theta_{ik})

and J is a matrix of partial derivatives known as a Jacobian: J=\begin{bmatrix} \dfrac{\partial \Delta P}{\partial\theta} & \dfrac{\partial \Delta P}{\partial |V|} \\ \dfrac{\partial \Delta Q}{\partial \theta}& \dfrac{\partial \Delta Q}{\partial |V|}\end{bmatrix}.

The linearized system of equations is solved to determine the next guess (m + 1) of voltage magnitude and angles based on:

\theta^{m+1} = \theta^m + \Delta \theta\,
|V|^{m+1} = |V|^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 is below a specified tolerance.

A rough outline of solution of the power-flow problem is:

  1. 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.
  2. Solve the power balance equations using the most recent voltage angle and magnitude values.
  3. Linearize the system around the most recent voltage angle and magnitude values
  4. Solve for the change in voltage angle and magnitude
  5. Update the voltage magnitude and angles
  6. Check the stopping conditions, if met then terminate, else go to step 2.

Power-flow methods[edit]

  • Gauss–Seidel method: This is the earliest devised method. It shows slower rates of convergence compared to other iterative methods, but it uses very little memory and does not need to solve a matrix system.
  • Newton–Raphson method: Most current methods are based on this. The convergence rate is typically fast, but it may sometimes fail because of the inherent problems of fractality in the basins of attraction of the underlying iterative map.
  • Fast-decoupled-load-flow method: In the operations of a power system, it is important for personnel to have a high level of contingent information: they need to know what power-flow changes are likely to occur due to generator outages. The contingent information can also be used to anticipate future power disruptions in the power network. Fast decoupled load flow is used as a common method to retrieve contingent information conveniently.

-The Fast Decoupled Load Flow method is a variation on Newton-Raphson that exploits the approximate decoupling of active and reactive flows in well-behaved power networks, and additionally fixes the value of the Jacobian during the iteration in order to avoid costly matrix decompositions. Also referred to as "fixed-slope, decoupled NR".

-Within the algorithm, the Jacobian matrix gets inverted only once, and there are 3 assumptions. Firstly, the conductance between the buses is zero. Secondly, the magnitude of the bus voltage is one per unit. Thirdly, the sine of phases between buses is zero, whereas the cosine of phases is 1. In real life, Fast Decoupled Load Flow can return the answer within seconds whereas the Newton Raphson method takes much longer.

  • Holomorphic embedding load flow method: A recently developed method based on advanced techniques of complex analysis. It is direct and guarantees the calculation of the correct (operative) branch, out of the multiple solutions present in the power flow equations.


  1. ^ Low, S. H. (2013). "Convex relaxation of optimal power flow: A tutorial". 2013 IREP Symposium Bulk Power System Dynamics and Control - IX Optimization, Security and Control of the Emerging Power Grid. pp. 1–06. doi:10.1109/IREP.2013.6629391. ISBN 978-1-4799-0199-9.  edit
  2. ^ Grainger, J.; Stevenson, W. (1994). Power System Analysis. New York: McGraw–Hill. ISBN 0-07-061293-5. 
  3. ^ Andersson, G: Lectures on Modelling and Analysis of Electric Power Systems