DigSILENT PF - 03 loadflow theory

1. Version: 25. Juli 2003
Loadflow Calculations
Basic Principles and Models
Training Course Documents

2. - 2 -
Table of Contents
1 Application areas for the loadflow calculations................................................................ 3
2 Calculation Methods........................................................................................................... 4
3 Models of Network Elements for Load Flow Calculation ................................................ 5
3.1 Synchronous Generator ......................................................................................................... 5
3.2 External Grid .......................................................................................................................... 7
3.3 General Load........................................................................................................................... 8
3.4 Motor ....................................................................................................................................... 8
3.5 Overhead/ Transmission Lines.............................................................................................. 9
3.6 Transformer .......................................................................................................................... 11
4 Loadflow calculation in the transmission system............................................................ 12
4.1 Power Transfer ..................................................................................................................... 12
4.2 Active Power Distribution.................................................................................................... 12
4.3 Loadflow Optimization......................................................................................................... 14
4.4 Reactive Power Control........................................................................................................ 14
5 Loadflow Calculations in Medium Voltage Systems (Distribution) ..................... 15
6 Loadflow Calculations in Low-voltage Systems .............................................................. 16
7 References......................................................................................................................... 17

3. - 3 -
1 Application areas for the loadflow calculations
Loadflow calculations are used to analyse the power system under steady state and unfaulted (short-circuit-free)
conditions. The load flow calculates the active and reactive power flows for all branches, and the voltages for all
nodes, in magnitude and phase angle.
The usual parameters given for a loadflow calculation are generator dispatch or set points (active and reactive
power, or active power and voltage), the active and reactive power of loads and the impedances of all branches.
If more complex active and reactive power control mechanisms (secondary control, regulated static compensa-
tors or FACTS) are to be simulated in the loadflow calculation, the setpoints and control characteristics of these
elements also have to be provided.
Regarding the areas of application of the loadflow calculation one should differentiate between simulations un-
der "normal" and "abnormal" conditions.
This distinction affects the modelling of the operating plant. Under normal operating conditions the generator
dispatches as well as the loads are known. It is sufficient for the calculation to represent these generator dis-
patches and to provide the active and reactive power of all loads. The resulting loadflow calculation should be a
system condition in which none of the branch or generator limitations are exceeded.
The second area of application, representing the abnormal conditions, requires a higher degree of accuracy from
the models. Here it can no longer be assumed that all system plant is operating within limits. The models must
be able to correctly simulate conditions that are deviating from the normal operating conditions. For example,
the reactive power limits of generators or the voltage-dependency of loads have to be modelled. Further, in
many applications, the power balance cannot be established with a single slack bus (or machine). Instead, a
more realistic representation of the active and reactive power control mechanisms have to be considered to de-
termine a correct sharing of the active and reactive generation.
The main areas of application for loadflow calculations are:
• Calculation of branch loading, system losses and voltage profiles for system planning and operation
(normal and abnormal conditions)
• Contingency analysis, network security assessment (abnormal conditions)
• Optimisation tasks (minimizing system losses, minimizing generation costs, open tie optimisation in
distributed networks, etc.; normal or abnormal condition)
• Verification of system conditions during the reliability calculations. Automatic determination of opti-
mal system re-supplying strategies. Optimisation of load shedding (abnormal conditions)
• Calculation of steady state initial conditions for stability simulations or short-circuit calculations using
the complete superposition method (usually normal conditions).

4. - 4 -
2 Calculation Methods
The most important calculation methods for the solution of loadflow problems are described in detail in the re-
ferred literature (e.g. /1/ /2/). It is important for the application that the implemented procedure determines the
solution fast and reliably. Since the loadflow problem (in contrast to the short-circuit calculation) represents a
non-linear problem, the solution can only be determined with an iterative process, which may, in some cases, not
lead to a solution.
The main difference between the many methods, which are used for loadflow calculations, lies in the specifica-
tion of the nodal equations. These can be set up using either the energy conservation law or Kirchhoff's law
(current equation).
Depending on the application, both nodal specification methods offer advantages, which explains why Power-
Factory supports both. Although both methods normally converge without any problems, experience has shown
that the applications of the two methods are best used as follows:
• Power equation: Large transmission networks with large angle deviations.
• Current equation: Systems, in which the voltage and/ or reactive power control is problematic, as well
as systems with power electronics.
To solve the system of equations, PowerFactory generally uses methods that are based on a Newton Raphson
iteration, independently of whether the nodal equations are specified as power or current equations.
Further, PowerFactory allows for the calculation of balanced and unbalanced load flows. In the classical load-
flow calculation system, the unbalances between phases are completely neglected. For the analysis of transmis-
sion networks this assumption is generally admissible. For distribution networks this assumption may be inap-
propriate depending on the characteristics of the system1
. All balanced system elements can be represented by a
single-phase equivalent circuit, which only represents the positive sequence component of a network (refer to
paper on “Short-circuit calculations”).
1
Distribution networks in America or Africa generally utilise single- or two-phase technologies. For such sys-
tems it is important that the full unbalanced "three-phase loadflow calculation" is used, which correctly models
all phases including (mutual) couplings. On the other hand, in some central european distribution systems the
balanced calculation perfectly applies.

5. - 5 -
3 Models of Network Elements for Load Flow Calculation
In this section the most important aspects of the Network Element Models for loadflow calculations will be de-
scribed. Complete descriptions of the models can be obtained from the PowerFactory manual /3/.
These elements can be classified according to their function in the system: Supply, Load and transmission or
Distribution Elements.
The most important power supply elements are:
• (synchronous-) generators
• external infeeds/ grids
Most types of loads are represented with the Load elements:
• general loads
• motors
Finally, the transmission or distribution network elements consist mainly of
• overhead lines
• transformers
3.1 Synchronous Generator
3.1.1 Equivalent Circuit
As shown in Figure 3-1, a synchronous machine can be modelled in steady state by an equivalent voltage source
with the synchronous reactance as internal impedance. For load flow calculations however, it is inconvenient to
specify the internal voltage and phase angle. The generated active and reactive power, or the generated active
power and the voltage at the terminals of the generator, are rather specified.
In Figure 3-1 a regulated synchronous machine is shown modelled for a loadflow calculation. In reality the
active power output of a machine is controlled by a turbine governor, which adjusts the turbine power output
according to the speed, electrical power output or some other prime mover signal (i.e. exhaust temperature in a
gas turbine, steam pressure in a steam turbine, etc.)
For the reactive power control, it is possible to distinguish between voltage control and power factor control.
Large machines in power stations, connected to a transmission system, are usually required to supply reactive
power support to the system. The reactive power of the machines, in these cases, is usually modified in order to
control the voltage at the terminals of the generator (or the high voltage side of the generator transformer) to a
preset value (primary voltage control). Such generators can be modelled by a "PV" control characteristic (power
and voltage control).
Generators of small power stations, which feed into a medium voltage system, are usually operated with a con-
stant power factor. The power factor should be as close as possible to unity, so that for a specific turbine power
the stator current of the machine is as small as possible and thus the generator runs at its optimum point. Such
generators can be represented as “PQ”-sources (active and reactive power is specified).
However, at least one machine in the system must ensure that the sum of all sources, loads and losses is equal to
zero. This power balance is usually formed by the "balance machine," often termed the “slack or swing ma-
chine” (SL - Slack). The necessity of a “balance machine” is based on the principle of the conservation of en-
ergy; this machine “takes up the slack” (or sometimes provides the slack). This, however, gives no indication
yet to which machine should be chosen as the balance machine. The balancing of active power (secondary con-
trol) in a real system is normally performed only by a few power stations, i.e. the secondary controlled power
stations or AGC units. For this reason, in many applications is useful to define a group of generators with sec-

6. - 6 -
ondary power control as the balance machine. Since this application is only used in transmission networks, it is
described in more detail in section 4.
U
U δ,
dx I
I δ,
00
, U
U δ
)cos(, ϕP
UP,
UU δ,
max
Q
minQ
Figure 3-1: Synchronous generator model for the loadflow calculation
3.1.2 Capability Curve of Synchronous Machines
The permissible operating range of synchronous machines is usually defined by the following limitations:
1. Maximum turbine power output
2. Maximum stator current
3. Maximum rotor (excitation) current
4. Steady state stability limit
5. Under-excitation limit
To describe the operating range of a synchronous machine on a PQ diagram, the output power is defined:
( ) ( )( )IUIU
jUIIUjQPS δδδδ −+−==+= sincos33
*
At rated voltage and current the output power can be represented by a circle in the PQ diagram. The radius of
this circle is equal to the specific output power of the machine:
rrr IUS 3=
Depending on the rotor induced voltage (L-L) and the load angle the output power of a synchronous generator
can be determined:
( )
( )( )rUU
d
r
UU
d
r
UU
X
U
Q
U
X
U
P
−−=
−=
δδ
δδ
00
00
cos
sin
Since the rotor induced voltage in steady state is proportional to the excitation current, a third condition can be
added to the PQ diagram, which represents the trajectory of maximum rotor induced voltage. This trajectory is
represented by a circle with the following centre co-ordinates:
d
r
X
U
Q
P
2
0
−=
=

7. - 7 -
and radius:
d
rr
X
UU
r 0
=
The two circles, maximum stator current and maximum excitation current cut each other at the nominal operat-
ing point of the machine.
Figure 3-2: Typical PQ diagram of a synchronous machine (in relation to rated power output)
For under excited operation it is more difficult to determine the operation limits. On the one hand the border of
static stability specifies the limitation, which depends, however, on both the parameters of the machine, as well
as the surrounding system. On the other hand, and usually more restrictive, the under-excited operation is lim-
ited due to heating of the end region of the armature by eddy-current losses.
Nevertheless, in an over-excited operating condition parasitic effects also exist, which result in the actual PQ
diagram deviating from the analytically derived diagram. Saturation effects also play a role, as the synchronous
reactance is then not constant, but depends on the operating point. For salient pole machines the influence of
magnetic anisotropy, which results in a difference between the d- and q-axis reactances, may not be neglected.
This results in the circle characteristics described above appearing as a spiral.
Owing to these parasitic effects it is customary to specify the operating limits by explicit parameters of maxi-
mum and minimum reactive power, which are determined from measurements and a detailed analysis of the
design.
3.2 External Grid
An external grid represents a reduced, overlying system. In principle it can be treated like a generator. It should
be noted, however, that for the modelling of high and medium voltage systems, the active and reactive power
balance usually takes place on the transmission level, which explains why an external grid is usually defined as
the “slack” (or "balance machine").
If several external grids are connected at the same time, e.g. in the case of several infeeds in a 132kV system,
one of the grids is usually defined as “slack,” and the other grids are treated like large generators with PV con-
trol.
In few applications would it be necessary to set several external grids to have slack characteristics. The alloca-
tion of the active power flow is then defined by the relative voltage angles (as specified) at the points of infeed.
For example, for the analysis of transfer capacities it may be permissible to assume that the external grid is so
strong that the voltage angle at the infeed points hardly changes due to the power flow in the network under
study.

8. - 8 -
3.3 General Load
All non-motor loads, as well as groups of non-motor loads or whole sub-systems, for example, medium voltage
systems as viewed from a 132kV system or a low-voltage system viewed from a medium voltage system, are
modelled as a "general load".
Under "normal conditions" it is permissible to represent such loads as constant PQ loads.
Under "abnormal conditions", for example during voltage collapse conditions the voltage-dependency of the
loads should be considered.
PowerFactory uses an exponential model to simulate the voltage-dependency:
kqukpu
U
U
Q
U
U
PjQP 







+







=+
0
0
0
0
The special cases "constant power", "constant current" and "constant impedance" can be described by the follow-
ing choice of the coefficients kpu and kqu:
1. kpu=kqu=0: Constant active and reactive power = Voltage independence
00 jQPjQP +=+
2. kpu=kqu=1: Constant active and reactive current = Linear voltage dependence
( )UjIIU
U
Q
jU
U
P
jQP QP 00
0
0
0
0
3 −=+=+
3. kpu=kqu=2: Constant Impedance (or Admittance) = Quadratic voltage dependence
( ) 2
00
2
2
0
02
2
0
0
3 UjBGU
U
Q
jU
U
P
jQP −=+=+
3.4 Motor
Rs Xs
XmU
Xr Rr/s
Figure 3-3: Equivalent circuit of an asynchronous machine in steady state
In comparison to synchronous machines, asynchronous machines do not possess an excitation winding. Voltage
or reactive power control is thus not possible2
. For an asynchronous machine it is therefore not possible to spec-
ify active and reactive power independently, or to control the terminal voltage.
Only for analysis under normal conditions is it sometimes permissible to model asynchronous machines as con-
stant PQ loads. The nominal power factor of the machine can then be used as power factor for the load.
Generally however, an asynchronous machine model as shown in Figure 3-3 should be used. From the equiva-
lent circuit it follows that, in steady state conditions, an asynchronous machine can be modelled with a slip-
dependent impedance. The active power output depends on a specific value of the slip. The resulting active
power flow through the reactance of the machine also influences the reactive power needs.
2
An exception is the slip controlled asynchronous machine, as used for wind generators or small hydroelectric
power plants. These machines allow reactive power control using a current converter, which is connected to the
rotor circuit.

9. - 9 -
Unfortunately the resistance and reactance values of machines are usually unknown. Normally only the rated
power, power factor, voltage, number of pole pairs and maybe the maximum torque are available. PowerFac-
tory allows the user to enter just these operating values and is then able to approximate the equivalent electrical
circuit parameters from this limited information.
3.5 Overhead/ Transmission Lines
3.5.1 Equivalent Circuit
R'l X'l
U1 U2B'l/2 B'l/2
Figure 3-4: Equivalent circuit for an overhead line and cable
The single-phase equivalent circuit for a cable or overhead line is displayed in Figure 3-4. The line parameters
are entered per length. The shunt capacitances are distributed evenly at either end of the line.
PowerFactory permits lines to be divided into routes and sections. Line routes are used if further elements are
connected within this route, e.g. loads connected directly to the line. Each individual line route is modelled
according to Figure 3-4.
Line sections are used for lines or cables, which consist of several portions, each represented by a different im-
pedance (for example, a cable connected to an overhead line). At the transition point from one portion to the
next no other plant is connected, which allows the individual portions to be lumped into one model as per Figure
3-4 for the whole line/ cable. The parameters of the equivalent circuit in this case result by switching the indi-
vidual impedances in series, and the individual capacitances in parallel.
3.5.2 Overhead line inductances
d12
r
L1
L2
L3
d23
d31
Figure 3-5: Overhead line
For the calculation of line inductances, both the field outside of the conductors, and the field inside the conduc-
tors (internal inductance) must be considered.
The influence of the field outside of the conductors can be calculated with the help of Ampere's circuital law:

10. - 10 -
∫ = IHds
In the case of an infinitely long conductor with circular cross section is this equivalent to:
IxH =π2
The variable x describes the radial distance from the centre of the conductor. From this the magnetic flux den-
sity along a circle around the conductor can be calculated:
I
x
HB
π
µ
µ
2
0
0 ==
Resulting from a current in L1, the surface between two conductors (arranged according to Figure 3-5) is in-
duced by the following flux:
1
0
12 12
1
0
12
12
ln
22
)( I
r
d
I
x
dx
dxxB
d
r
d
r
π
µ
π
µ
===Φ ∫ ∫
For the calculation of the total flux, the currents in conductor 2 and 3 must also be considered. In the case of a
completely symmetrical arrangement all partial fluxes of conductor 3 eliminate themselves, which explains why
the current in conductor 3 does not influence the flux between L1 and L2. The total flux therefore is equal to:
( )21
0
2
0
1
0
12
12
ln
212
ln
2
12
ln
2
II
r
d
I
d
r
I
r
d
−





=





+





=Φ
π
µ
π
µ
π
µ
The inductance related the field external to the conductors is therefore:






=
r
d
Lext
12
ln
2
0
π
µ
The influence of the magnetic field inside the conductors can be considered by the internal inductance. The in-
ternal inductance is frequency dependent. In case of a cylindrical conductor, the following low-frequency ap-
proximation is valid:
42
0
int
r
L
µ
π
µ
=
The total inductance of the symmetrical three-phase line is:






+=+=
r
d
LLL r
ext
12
ln
42
0
int
µ
π
µ
Sometimes, the internal inductance is expressed by the equivalent geometric mean radius (GMR):
4
r
reGMR
µ
−
=
The inductance can now be expressed by:






=
GMR
d
L
12
ln
2
0
π
µ
For the calculation of the other overhead line parameters (resistance, capacitance) reference is made to the litera-
ture (e.g./2/).

11. - 11 -
3.6 Transformer
rH xH
bmuH
xL rL
uLgm
(1+t):1
Figure 3-6: Equivalent circuit of a two-winding transformer
Figure 3-6 shows the transformer model in the positive phase-sequence system. The equivalent circuit is de-
scribed using the per unit system, which explains why no ideal transformer is used to model the voltage ratio
from the high voltage to the low voltage level.
The sum of the series impedances corresponds to the short-circuit impedance of the transformer. The series
impedances rH and rL are proportional to the copper losses. The magnetizing current is modelled with bm, the
iron losses are modelled with conductance gm. In PowerFactory the values of the resistances and reactances are
determined automatically from the measured short-circuit voltage, the copper losses, the no-load current and the
iron losses.
The ideal transformer on the high voltage side represents the tap changer, which can be adjusted both in magni-
tude and phase angle. By adjusting the real part of t the voltage or the reactive power can be controlled. By
altering the imaginary part of t the active power can be controlled (phase shifter).
For the tap changer the number of steps, the voltage per step, as well as the angle of t are defined in the trans-
former type parameters. The control mode, as well as the tap changer setpoints, can be individually specified for
each transformer element. During a loadflow simulation, the automatic adjustment of all transformer tap chang-
ers can globally be switched ON or OFF (in the loadflow command dialogue).

12. - 12 -
4 Loadflow calculation in the transmission system
4.1 Power Transfer
U1 U2
R XP1,Q1 P2,Q2
Figure 4-1: Power transmission over/ through an impedance
The fundamental problem of the transfer of power using alternating current is represented in Figure 4-1. It deals
with the transfer of active and reactive power through an impedance, which is defined by a resistance and a reac-
tance. The power on side 2 is calculated as:
( )2
2
*
12*
3
UUU
Z
jQPS −=+=
where:
jBG
XR
jXR
jXRZ
−=
+
+
=
−
= 22*
11
Describing the voltages in polar coordinates:
2
1
22
11
δ
δ
eUU
eUU
=
=
The power flow on side 2 is:
( )
( )2
2122112212
2
2122112212
cossin3
sincos3
BUUBUUGUQ
GUUBUUGUP
+−−=
−−=
δδ
δδ
Particularly interesting are the approximated power equations for the case of small voltage drops and small angle
deviations:
( )
( )21212
2
2
12
2
2122
2
2
3
3
UUBGUQ
BUUUGP
∆−−≈
−∆≈
δ
δ
In transmission networks the relationship of R/X (and thus also of G/B) for overhead lines and cables lies in a
range of between 0.1 and 0.2. This explains why, for transmission lines, the voltage angles across a line primar-
ily depends on the active power flow, and the voltage drop across a line depends primarily on the reactive power
flow.
4.2 Active Power Distribution
For the analysis of transmission networks, in particular where "abnormal condition" operations are regarded, it is
important to represent the active power sharing among generating units as close as possible to the real or actual
network.
It is therefore necessary to categorize the control processes that, for instance, follow the loss of a large power
station.
The following description is an extensive simplification that, in many cases, gives reasonable approximations of
the system behaviour using loadflow calculation capabilities. The actual system behaviour is more complex, in

13. - 13 -
particular whereby the primary and secondary frequency control overlap each other. The description serves the
purpose of a better categorization and allows a good approximation of the active power sharing at different
points in time.
4.2.1 Inertial Loadflow
Immediately following a disturbance, the missing power is delivered from the kinetic energy stored in the rotat-
ing mass of the turbines. This leads to a deceleration and thus to a decrease of the frequency. The contribution
of each individual generator towards the total additional power required is proportional to its Inertia.
These relations can be described mathematically as follows. For each generator the following swing equation
applies:
n
elt PP
J
ω
ω
−
=&
Assuming a completely synchronized system, in which the frequency of all generators is the same, the time de-
rivative of frequency, which is due to the total missing power, can be calculated as follows:
∑
∆−
=
in
tot
J
P
ω
ω&
Inserting the frequency derivative into the swing equation of each generator, its active power contribution to-
wards the total required power can be determined:
tot
i
i
i
P
J
J
P ∆=∆
∑
4.2.2 Primary Controlled Loadflow
After a short time (within a few seconds), the governors of units taking part of the primary frequency control
increase the turbine power and drive the frequency back to a value close to nominal frequency. Using a pure
droop control each governor increases the turbine power proportionally to the frequency deviation. The resulting
system frequency deviation (∆F) will then be associated to the system droop. The active power generation in the
system is divided among units according to the gain (K) of the individual primary controllers.
For each turbine, the deviation of the power output from the dispatch power (∆P) is given by:
FKP ∆−=∆
Adding the power contribution or deviations from dispatch power from all power stations (∆Ptot), assuming a
uniform frequency in the entire system, the frequency deviation remaining after the primary control operation
can be described as follows:
∑
∆
−=∆
i
tot
K
P
F
Inserting this frequency deviation into the equation of each individual turbine, its active power contribution in a
state of primary control is obtained:
tot
i
i
i
P
K
K
P ∆=∆
∑
4.2.3 Secondary Controlled Loadflow
The secondary controlled loadflow is of special importance, since it deals with the power system in an almost
stationary condition. Although a stationary condition scarcely ever exists in a power system, as the loads are
changing continuously, the secondary controlled condition (system condition after the operation of a secondary
frequency control or AGC) is still quite observable.
The goal of the secondary frequency control is to reduce the remaining frequency deviation that has resulted
from the first few minutes of primary control operation. In an interconnected power system the defined power
exchanges between different areas is generally also re-established by this secondary control.

14. - 14 -
Usually with properly set power exchanges between areas and enough secondary spinning reserve in all areas,
the frequency deviation after the operation of the secondary control should equal zero. In this case, the output
power of the primary controlled generators should equal the output power just before the disturbance. On the
contrary, when there is a deficit of secondary reserve in an area of the system, the frequency will not return to its
nominal value and the machines under primary frequency control will still deviate from their dispatch power.
The few power stations in each network zone that participate in the secondary control adapt their output power in
such a way that the active power unbalance is compensated. The missing power due to the disturbance is thus
distributed among the secondary controlled generators according to the participation factors, which are individu-
ally set for the secondary controller of each machine.
4.3 Loadflow Optimization
From an economical point of view, the power station dispatch resulting from the secondary control action is not
necessarily optimal. It just guarantees that a disturbance (e.g. a power station loss) will be compensated for. In
the longer term however, the most favourable generator economic dispatch has to be strived for. The required
output power has to be shared among the individual power stations in the most cost effective way.
In a liberalised power market, however, contractually agreements need to be adhered to. Generator dispatch then
obeys market rules and not the total system optimisation.
4.4 Reactive Power Control
Not only the active, but also the reactive power control must be considered in a loadflow analysis of a transmis-
sion network. The reactive power reserves of synchronous generators in transmission networks are used to con-
trol the voltages at specific nodes in the system and/ or to control the reactive power exchange with neighbouring
network zones.
Each generator is equipped with a voltage regulator that controls the generator voltage. This voltage regulator
has a voltage setpoint, which is set manually, or from the secondary reactive power control, so that the desired
voltages and reactive power flows are attained in the system.
If the voltage of a specific node is controlled from several generators, the contribution of reactive power from
each of the generators must be defined. This definition can be made in PowerFactory using the object "auto-
matic station control" where the reactive power participation factors can be specified for each generator.
During the secondary reactive power control, not only the generators play an important role, but also the trans-
former tap changers and switchable shunts or regulated static compensators. It is difficult to illustrate the inter-
action of these control processes in a loadflow calculation.
The optimal operation of all secondary voltage and reactive power controllers can be best determined using an
optimal loadflow analysis tool.
For the realistic consideration of secondary reactive power controllers in an exceptional loadflow calculation, a
method has been developed that calculates the time constants of the controllers in such a way that realistic final
states of the controllers result.

15. - 15 -
5 Loadflow Calculations in Medium Voltage Systems
(Distribution)
The R/X ratio of medium-voltage cables and overhead lines is significantly higher compared to those used in
high and extra-high voltage systems. This ratio normally ranges between 1 and 4. Considering the whole distri-
bution system, including the transformers in substations, the overall R/X ratio normally is still less than 1 how-
ever. In medium voltage systems the reactive power flow and voltage drop are therefore still closely related.
The de-coupling of the reactive power/ voltage drop and active power/ voltage angle relationship is however by
far not as pronounced as in high voltage systems.
For the analysis of transmission networks the active powerflow distribution and the related generator dispatch is
of prime concern. For distribution networks, this is not the case. Here it can be assumed that the main portion of
power is drawn from a high-voltage transmission system. Should power stations, which feed directly into the
distribution network exist, these normally run at maximum output power and do not perform primary frequency
control. It can be assumed that, following a disturbance, the missing power is compensated almost completely
via the connected transmission system.
The main focus for loadflow analysis in distribution networks is centred on the determination of system losses as
well as the assessment of branch utilisation and voltage profiles.
Regarding the modelling, the main challenge lies in a realistic representation of the loads. Chronological
changes are usually represented by load profiles, which describe the actual load at any time during a day. To
allow for weekday or seasonal variations, PowerFactory allows the option of defining two-dimensional load
profiles, where each column of the daily load profile represents the load for a specific weekday or season.
As soon as the load profiles are defined, it is possible to calculate a loadflow for any time of the day/ weekend/
season. By running sequential loadflows, any variable of interest can be plotted against the time of day. This
kind of loadflow analysis allows for a very accurate estimate of line loadings and/ or voltage levels at specific
points in the network.
In radially operated distribution systems the problem often arises that very little is known about the actual load-
ings of the loads connected at each mini substation. The only information sometimes available is the total power
flowing into a radial feeder. To be able to still estimate the voltage profile along the feeder a load scaling tool is
used. In the simplest case the distribution loads are scaled according to the nominal power ratings of the trans-
formers in the mini substations. Of course such scaling leads to inaccurate results. Better results are obtained by
using the average annual load, which could be determined from the total annual energy consumed and billed.

16. - 16 -
6 Loadflow Calculations in Low-voltage Systems
Normally the focus for loadflow calculations in low-voltage systems is to determine maximum branch currents
as well as maximum voltage drop.
In low-voltage systems the R/X ratio is considerably larger than 1. The voltage drop therefore depends mainly
on the active power flow. Reactive power flow in low-voltage systems are of lesser interest.
The modelling of the loads, as for distribution systems, represents a major challenge. In addition to the time
dependence, a stochastic component is introduced in the low-voltage systems, which is usually expressed in the
form of a coincidence factor. This considers, that for two connections it is very improbable that both are draw-
ing maximum load at the same time. With three connections this is even more unlikely etc. The maximum load
current thus depends on the number of connections supplied from a cable.
In PowerFactory a special low-voltage load model is implemented, which is defined by the number of connec-
tions supplied. In addition, the load is assigned to a load category, which defines for each category the maxi-
mum load of the connection, as well as the coincidence factor, for an infinite number of connections (thus the
relationship of average and maximum load).
Considering the stochastic independent component of low-voltage loads, the maximum load (dependant on the
number of connections) can be described by the following formula:
)1()()( maxmax SnngnS =
The function g(n) describes the maximum coincidence, dependant on the number of connections, n. If a Gaus-
sian (normal) distribution is assumed for the coincidence, the coincidence functions reads:
n
g
gng ∞
∞
−
+=
1
)(
This function depends only on the coincidence of an infinite number of connections.
0
0,2
0,4
0,6
0,8
1
1,2
0 10 20 30 40 50 60 70 80 90 100
Figure 6-1: Coincidence as a function of the number of connections (ginf=0.1)
For the calculation of maximum branch utilization as well as the maximum voltage drop, PowerFactory uses a
probabilistic loadflow calculation, which is able to calculate both maximum and average currents as well as the

17. - 17 -
average losses. The probabilistic loadflow calculation used by PowerFactory can be applied to any system to-
pology, thus also to meshed low-voltage systems.
Not only can these low-voltage loads be attached to network nodes, but also arbitrarily to line sections, which
thus reduces the clutter of the single line diagram.
7 References
/1/ B. Oswald. Berechnung stationärer und quasistationärer Betriebszustände in Elektroenergieversorgungs-
netzen. VDE-Verlag, 1992
/2/ J. J. Grainger und W. D. Stevenson. Power System Analysis Mc. Graw Hill, 1994
/3/ DIgSILENT GmbH. PowerFactory Manual V13, 2002