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Load Flow Analysis
Consider a four bus system.
The complex power in each bus is given by ‘S’.
Si = Pi
+ jQi = ViIi*
Consider bus 2
V2I2* = P2 + jQ2
I2 = (P2 – jQ2)/V2*
In terms of self and mutual admittances of the nodes
(P2 – jQ2)/V2* = Y21V1
+ Y22V2 + Y23V3 + Y24V4
Where:
V: stands for Voltage
Y: stands for admittance ie (1/Z)
Z: stands for impedance (the impedance could be
resistive(R,r); inductive(XL, jωL,2πfL); capacitive(XC,
1/(jωC),1/(j2πfC)))
Solving for V2 gives
V2 = 1/Y22[(P2 – jQ2)/V2*
- (Y21V1 + Y22V2 + Y23V3
+ Y24V4)] {A}
This is basic standard procedure.
From this analytical procedure, one can determine the
voltage at any given bus given the voltages at the other buses and the system
parameters as well as the system configuration and the loading at the various
buses.
Since the voltages at the various buses are not known
initially, we start with an educated guess and calculate these voltages
iteratively until an acceptable degree of accuracy is attained.
There are two main options used for Load flow analysis.
These are the Gauss Seidel Method and the Newton Raphson method. The Gauss
Seidel method uses the algebraic expressions derived above iteratively to
determine the voltages at the various buses. On the other hand, the Newton
Raphson iterative method uses error analysis applied to the Taylor series
expansion of the power system showing an expression for complex power about a
singularity ignoring higher order derivatives and solving for the associated error
and using the error to improve the guessed value or that result after each
iteration until an acceptable degree of accuracy is attained.
In the expression above;
The idea is to calculate the voltage using estimated values
of voltages at the other buses or the previously calculated voltages (being an
iterative method) in addition to real and reactive power values delivered to
the bus from generators or supplied to the load connected to the bus as well as
self and mutual admittances of the nodes.
The quantities are used to form the network equation with
the aim of finding the voltage.
Equation {A} gives a corrected value of V2 based
upon scheduled P2 and Q2
(where V2 is the voltage at bus 2 and P2
is the real net power at bus 2 and Q2 is the imaginary or reactive
net power at bus2; net power : the net power is the difference between the
generated power and the demand power at any given bus (P2=PG2
– PD2, Q2 = QG2 – QG2 [the unit of
P is in Watt (W)[x100 = 1]or kilowatt(kW)[x103=1000] or
Megawatt(MW)[x106 = 1,000,000]. The unit of Q is Vars[100
=1 ] or KiloVars[x103 = x1000] or MegaVars[x106 = x1,000,000].
[S] stands for complex power so S = P + jQ. The unit of ‘S’ is VoltAamperes
(VA)[x100 = x1] or KiloVoltAmperes(KVA)[x103=x1000] or
MegaVoltamperes(MVA)[x106=1,000,000]]))
Where the values
estimated originally are substituted for the voltage expressions; on the right
side of the equation.
These improved values of voltages are re-substituted in an
iterative way until a desired degree of accuracy is attained.
The Gauss – Seidel Method
In the Gauss-Seidel Method, as the corrected voltage is
found at each bus, it is used in calculating the corrected voltage at the next.
This calculation is carried out at each bus except the swing bus throughout the
network to complete the first iteration. The swing bus is fully specified hence
no calculation is carried out on it or required to be carried out. You could
however calculate the total power of the system at this bus. Ie. The total
power delivered to the system.
This method involves an iterative solving of linear
algebraic equations.
Note that it is possible for the voltages to converge upon
an erroneous solution if the original voltages are widely different from the
correct values.
To prevent this, original values of reasonable magnitude and
which do not differ too widely in phase are chosen.
Unwanted solutions can be detected by the experienced
engineer easily by inspection of the result since the voltage of the system do
not normally have a range in phase wider than 45o and the difference
between nearby buses is less than 10o and often very small.
For ‘N’ buses, the voltage at any bus k where Pk
and Qk are given is
Vk = (1/Ykk)([(Pk-jQk)/Vk*]-[ YknVn])
{AA}
Where n =/= k
The value of the voltages on the right side of the equation
are the most recently calculated values for the corresponding buses (or the estimated
voltage if no iteration has yet been made at that particular bus)
Acceleration Factors
These are multipliers designed to improve convergence. They
are some constant values that increases the amount of correction to bring the
voltage closer to the value it is approaching. Without this, convergence takes
a bit longer in a normal Gauss-Seidel method.
The acceleration factor for the real and imaginary values
may differ. A wrong choice would obviously have an opposite effect.
At a bus where voltage magnitude rather than reactive power
is specified, a value of reactive power is calculated first to help find the real
and imaginary components of the voltage for each iteration.
Pk – jQk = (YkkVk
+ YknVn*)
Where n =/= k
If we allow n=k
Pk
– jQk = Vk*
Y
knV
n
Qk
= -Im{Vk*
Y
knV
n}
{b}
Reactive Power Qk is evaluated by eqn {b} for the
best previous voltage values at the buses, and this value of Qk is
substituted in eqn {AA} to find a new value of Vk. The components of
the new Vk are the multiplied by the ratio of the specified constant
magnitude of Vk to the magnitude of the Vk found by eqn
{AA}. The result is the corrected complex voltage of the specified magnitude.
The reason n =/=
k is because that expression when n =/= k
has been taken out of the Σ sign in one case and expressed
separately. In the other case it is used to divide the quantity hence using the
Σ
sign you have to remove it.
Buses
Buses are also called Busbars. They are connection points in
the Electrical Power system. Node in electrical circuit analysis, are
abstractions of buses for analytical purposes. There are three main types of
buses in the Electrical Power System. These buses are analysed differently. The
buses are:
The Slack bus, also called the Reference Bus or the Swing
bus.
The Voltage controlled bus or the generator bus
The Load bus
The Voltage controlled Bus
Usually, the voltage controlled bus has a Generator
connected to it hence it is also called a Generator Bus. PGi and Vi
are specified.
PGi is constrained b/w limits
PGimin ≤ PGi ≤ PGimax
QGi is also set within limits
QGimin ≤ QGi ≤ QGimax
If QGi falls outside the limit, the limit is
taken and Vi becomes the unknown. The calculation is carried out
with this understanding and if QGi falls between the limits, Vi
is fixed again and the process continued.
The Reference Bus
The Reference Bus has to be a Generator Bus (with a
Generator connected to it) or a Tie Bus (for interconnectivity of major system
networks)
Load Bus
This is the most numerous Bus in the Electrical Power
System.
All P’s and Q’s are known. The unknowns are the voltage
magnitude ‘V’ and the angle ‘δ’ V/δo.
Newton Raphson Iterative Method
The Electrical Power System is expressed as a function of ‘δ’
& ‘V’. It is then expressed as a Taylor Series.
It is noted that there are differential quantities in the Taylor
Series expression. Second order values of the differential and higher order values are ignored,
because their effect is minimal.
Incremental change ‘ΔV’
& ‘Δδ’ are approximated to error quantities or these differentials.
An iterative method is set in motion such that the difference between the calculated complex
power and the specified Power is as a result of this error quantity. This error
is hence calculated and used to improve the assumed value of the voltages and angles.
The exercise is continued until an acceptable degree of accuracy or a given level of accuracy is established or
reached.
Consider a guessed solution x(0)
Let f(x(0)) + Δx(0) = 0
If we expand f(x) about x(0) we obtain
f(x(0)) + Δx(0)(df/dx)(0) +
½ (Δx(0))2(d2f/dx2)(0)
+ ……..+ = 0
Δx(0) is the error associated with the guess x(0)
If the error is small, the higher order terms can be neglected to
give
F(x(0)) + Δx(0)(df/dx)(0) ≈ 0
The error can be calculated as
Δx(0) = - f(x(0))/((df/dx)(0))
If we add this error to the original guess, we then have an
improved guess, such that
X(1) = x(0) + Δx(0) = x(0)
– f(x(0))/(df/dx)(0)
This procedure is repeated until the correct or specified
degree of accuracy is attained.
X(a+1) = x(a) + Δx(a) = x(a)
– f(x(a))/(df/dx)(a)
Consider ‘n’ dimentional Equations
x
(0)
=
Here only the first derivatives
are considered.
F1(x(0))
+(df1/dx1)(0)Δx1 +………………….+ ((df1/dxn)(0)) Δxn
≈
0
.
.
.
.
Fn(x(0))
+(dfn/dx1)(0)Δx1 +………………….+ ((dfn/dxn)(0)) Δxn
≈
0
(df1/dx1)(0)--
-------(df1/dxn)(0)
| |
| |
| |
(dfn/dx1)(0)--
-------(dfn/dxn)(0)
|
+
≈
Using compact matrix – vector notation
f(x(0)) + J(0) Δx(0) ≈ 0
hence
Δx(0) ≈ -[J(0)]-1f(x(0))
This error vector is added to the original guess and the
process repeated in an iterative way such that.
x(a+1) = x(a)- [J(a)]-1f(x(a))
where ‘a’ is the iteration counter ie counts the number of
iterations. (0) stands for guessed values.
J ≡ is an nXn matrix of partial derivatives (dfi/dxj)
called a Jacobian matrix.
In Power Systems Analysis,
δ2(0)
δ3(0)
.
.
|vn-1(0)|
|vn(0)|
|
x(0) =
Notice that the reference bus is not included.
This is because it is fully specified. This gives a (2n-2) –
dimensional state vector.
By expanding functions for real and imaginary power ‘fip’
and ‘fiq’ in Taylor Series around the initial guess, we have.
Pi ≈ fip(0) + (dfip/dδ2).Δδ2(0)
+………+ (dfip/d|vn|)(0).Δ|vn(0)|
For i = 2,…………,n
Qi ≈ fiq(0) + (dfiq/dδ2).Δδ2(0)
+………+ (dfiq/d|vn|)(0).Δ|vn(0)|
fip(0) & fiq(0) represent
real & imaginary (reactive) power leaving bus ’i’ or associated with bus
‘i’ if the bus voltage are set at guessed values.
The difference in power is also called Power Mismatch at a
given bus. This is the difference between the measured Power at any given Bus
and the calculated power at the same given bus using bus parameters like
voltages and admittance and the angle of the voltage ie the power factor. Note
that these voltage values and their angles could be guessed values or improved
values after each iteration, until an acceptable degree of accuracy is
attained. Ideally if the values used for the calculations are correct, there
should be no mismatch. This means that there should be no difference. Where
there is an error in the values used for the calculation, there will be a
mismatch and this is used for calculating the error associated with these parameters
and variables. These calculated errors are then used to improve the values of
these parameters and these variables used for the calculation subsequently.
This procedure is repeated iteratively until an acceptable degree of accuracy
is attained.
ΔPi(0) ≈ Pi – fip(0)
ΔQi(0) ≈ Qi
– fiq(0)
≈
(df2p/dδ2)(0)--
-----(df2p/d|vn|)(0)
| |
| |
| |
(dfnq/dδ2)(0)--
-----(dfnq/d|vn|)(0)
|
ΔU(0) ≈ J(0).Δx(0)
Procedure For Load Flow Analysis
Using Newton Raphson Method
Guess x(0)
Compute ΔU(0)
Pi
=
|Y
ik||V
i||V
k|cos(δk+δi+ϒ)≈ f
ip
Qi
= -
|Y
ik||V
i||V
k|sin(δk+δi+ϒ)≈ f
ip
For i = 1,2,………………n
Note that
Si =
Si = Pk – jQk = Vk* YikVk
For i = 1, 2, ………………n
Compute the Jacobian Matrix
Solve the voltage error vector ie
Δx(0) ≈ (J(0))-1.ΔU(0)
Add voltage errors to the initial guesses to obtain an
upgraded state vector.
The process is repeated.
NB.
G-S – Voltage error used as a convergence measure. Ie the
difference b/w subsequent iterative results.
N-R – Power Mismatch, real and reactive used for this
purpose.
1 
If the core center line is chosen as
the enclosed path, the magnetic field intensity is zero everywhere because of
infinite core permeability.
