A review on Load flow
studies
Presenter:
Ugyen Dorji
Master’s student
Kumamoto University, Japan
Course Supervisor:
Dr. Adel A. Elbaset
Minia University, Egypt.
Outline
 Introduction
 Methodology
 Classical methods
 Gauss-Seidal method
 Newton Raphson method
 Fast Decoupled method
 Other methods
 Fuzzy Logic application
 Genetic Algorithm application
 Particle swarm method (PS0)
Load/Power Flow studies
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Load-flow studies are performed to determine the steady-state
operation of an electric power system. It calculates the voltage drop
on each feeder, the voltage at each bus, and the power flow in all
branch and feeder circuits.
Determine if system voltages remain within specified limits under
various contingency conditions, and whether equipment such as
transformers and conductors are overloaded.
Load-flow studies are often used to identify the need for additional
generation, capacitive, or inductive VAR support, or the placement of
capacitors and/or reactors to maintain system voltages within
specified limits.
Losses in each branch and total system power losses are also
calculated.
Necessary for planning, economic scheduling, and control of an
existing system as well as planning its future expansion
Pulse of the system
Power Flow Equation
Note: Transmission lines are represented by
their equivalent pi models (impedance in
p.u.)
Fig. 1. A typical bus of the power system.
Applying KCL to this bus results in
(1)
(2)
The real and reactive power at bus i is
Substituting for Ii in (2) yields
Equation (5) is an algebraic non linear equation which must be solved by iterative
techniques
Gauss-Seidel method
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Equation (5) is solved for Vi solved iteratively
Where yij is the actual admittance in p.u.
Pisch and Qisch are the net real and reactive powers in p.u.
In writing the KCL, current entering bus I was assumed positive. Thus for:
Generator buses (where real and reactive powers are injected), Pisch and Qisch
have positive values.
Load buses (real and reactive powers flow away from the bus), Pisch and Qisch
have negative values.
Eqn.5 can be solved for Pi and Qi
The power flow equation is usually expressed in terms of the elements of
the bus admittance matrix, Ybus , shown by upper case letters, are Yij = -yij, and
the diagonal elements areYii = ∑ yij. Hence eqn. 6 can be written as
Iterative steps:
•Slack bus: both components of the voltage are specified. 2(n-1) equations to be
solved iteratively.
• Flat voltage start: initial voltage of 1.0+j0 for unknown voltages.
• PQ buses: Pisch and Qisch are known. with flat voltage start, Eqn. 9 is solved for
real and imaginary components of Voltage.
•PV buses: Pisch and [Vi] are known. Eqn. 11 is solved for Qik+1 which is then
substituted in Eqn. 9 to solve for Vik+1
However, since [Vi] is specified, only the imaginary part of Vik+1 is retained, and
its real part is selected in order to satisfy
• acceleration factor: the rate of convergence is increased by applying an
acceleration factor to the approx. solution obtained from each iteration.
•Iteration is continued until
Once a solution is converged, the net real and reactive powers at the slack bus are
computed from Eqns.10 & 11.
Line flows and Line losses
Considering Iij positive in the given direction,
Similarly, considering the line current Iji in the given direction,
The complex power Sij from bus i to j and Sji from bus j to i are
Newton Raphson Method
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Power flow equations formulated in polar form. For the system
in Fig.1, Eqn.2 can be written in terms of bus admittance matrix
as
Note: j also includes i
Expressing in polar form;
Substituting for Ii from Eqn.21 in Eqn. 4
Separating the real and imaginary parts,
Expanding Eqns. 23 & 24 in Taylor's series about the initial estimate neglecting
h.o.t. we get
The Jacobian matrix gives the linearized relationship between small changes in Δδi(k) and
voltage magnitude Δ[Vik] with the small changes in real and reactive power ΔPi(k) and
ΔQi(k)
The diagonal and the off-diagonal elements of J1 are:
Similarly we can find the diagonal and off-diagonal elements of J2,J3 and J4
The terms ΔPi(k) and ΔQi(k) are the difference between the scheduled and
calculated values, known as the power residuals.
Procedures:
1. For Load buses (P,Q specified), flat voltage start. For voltage controlled buses
(P,V specified),δ set equal to 0.
2. For Load buses, Pi(k) and Qi(k) are calculated from Eqns.23 & 24 and ΔPi(k) and
ΔQi(k) are calculated from Eqns. 29 & 30.
3. For voltage controlled buses, and Pi(k) and ΔPi(k) are calculated from Eqns. 23 &
29 respectively.
4. The elements of the Jacobian matrix are calculated.
5. The linear simultaneous equation 26 is solved directly by optimally ordered
triangle factorization and Gaussian elimination.
6. The new voltage magnitudes and phase angles are computed from (31) and (32).
7. The process is continued until the residuals ΔPi(k) and ΔQi(k) are less than the
specified accuracy i.e.
3. Fast Decoupled Method
• practical power transmission lines have high X/R ratio.
•Real power changes are less sensitive to voltage magnitude changes and are most
sensitive to changes in phase angle Δδ.
•Similarly, reactive power changes are less sensitive to changes in angle and are
mainly dependent on changes in voltage magnitude.
•Therefore the Jacobian matrix in Eqn.26 can be written as
The diagonal elements of J1 given by Eqn.27 is written as
Replacing the first term of the (37) with –Qi from (28)
Bii = sum of susceptances of all the elements incident to bus i.
In a typical power system, Bii » Qi therefore we may neglect Qi
Furthermore, [Vi]2 ≈ [Vi] . Ultimately
In equation (28) assuming θii-δi+δj ≈ θii, the off diagonal elements of J1 becomes
Assuming [Vj] ≈ 1 we get
Similarly we can simplify the diagonal and off-diagonal elements of J4 as
With these assumptions, equations (35) & (36) can be written in the
following form
B’ and B’’ are the imaginary part of the bus admittance matrix
Ybus. Since the elements of the matrix are constant, need to be
triangularized and inverted only once at the beginning of the
iteration.
Other Methods
Repetitive solution of a large set of linear equations in
LF- time consuming in simulations
 Large number of calculations on the Jacobian matrix.
 Jacobian of load flow equation tends to be singular under
heavy loading.
 Ill conditioned Jacobian matrix
 Doesn’t require the formation of the Jacobian matrix
 Insensitive to the initial settings of the solution variables
 Ability to find multiple load-flow solutions.
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Fuzzy Logic application
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Repetitive solution of a large set of linear equations in the load flow
problem is one of the most time consuming parts of power system
simulations.
Large number of calculations need on account of factorisation,
refactorization and computations of Jacobian matrix.
Fundamentally FL is implemented in a fast decoupled load flow (FDLF)
problem.
Mathematical analysis of FDLF
In eqn. 1, the state vector θ is updated but
state vector V is fixed. Eqn. 2 is used to
update the state vector V while state vector θ
is fixed.The whole calculation will terminate
only if the errors of both these equations are
within acceptable tolerances
Main idea of FLF Algorithm
FLF algorithm is based on FDLF equation but the repeated update of the state vector
performed via Fuzzy Logic Control instead of using the classical load flow approach.
The FLF algorithm is illustrated
schematically in Fig. 1.In this Figure the
power parameters ΔFP and ΔFQ are
calculated and introduced to the P-θ
FLCP-θ and Q-V FLCQ-V, respectively.
The FLCs generate the correction of the
state vector DX namely, the correction
of voltage angle Δθ for the P-θ cycle and
the correction of voltage magnitude ΔV
for the Q- V cycle.
Structure of the fuzzy load flow controller (FLFC)
•Calculate and per-unite the power parameters ΔFP and ΔFQ at each node of the
system.
•The above parameters are elected as crisp input signals. The maximum (or worst)
power parameter (ΔFPmax or ΔFQmax) determines the range of scale mapping that
transfers the input signals into corresponding universe of discourse, at every
iteration.
• The input signals are fuzzified into corresponding fuzzy signals (ΔFPfuz or ΔFQ fuz
with seven linguistic variables; large negative (LN), medium negative (MN), small
negative (SN), zero (ZR), small positive (SP), medium positive (MP), large positive
(LP).They are represented in triangular function.
The rule base involves seven rules tallying with seven linguistic
variables:
Rule 1: if ΔFfuz is LN then ΔXfuz is LN
Rule 2: if ΔFfuz is MN then ΔXfuz is MN
Rule 3: if ΔFfuz is SN then ΔXfuz is SN
Rule 4: if ΔFfuz is ZR then ΔXfuz is ZR
Rule 5: if ΔFfuz is SP then ΔXfuz is SP
Rule 6: if ΔFfuz is MP then ΔXfuz is MP
Rule 7: if ΔFfuz is LP then ΔXfuz is LP
These fuzzy rules are consistent to that of Eqn.3.
•The fuzzy signals Δffuz are sent to process logic, which generates the
fuzzy output signals Δxfuz based on the previous rule base and are
represented by seven linguistic variables similar to input fuzzy signals.
•The maximum corrective action Δxmax of state variables determines the
range of scale mapping that transfers the output signals into the
corresponding universe of discourse at every iteration.
where FI expresses the real or reactive power balance equation at
node-I with maximum real or reactive power mismatch of the
system, XI represents the voltage angle or magnitude at node-I.
• finally the defuzzifier will transform fuzzy output signals into crisp
values for every node of the network. The state vector is updated as
Index i depicts the number of iterations.
Case studies
GA applications
Load flow problem
where Gij and Bij are the (I,j)th
element of the admittance matrix. Ei,
and Fi are real and imaginary parts of
the voltage at node i.
If node i is a PQ node where the load demand is specified, then the mismatches
in active and reactive powers, ΔPi and ΔQi , respectively, are given by
Pisp and Qisp are the specified
active and reactive powers at
node i.
When node i is a PV node, the magnitude of the voltage, Visp and the active
power,Q;P, at i are specified. The mismatch in voltage magnitude at node i can
be defined as
The active power mismatch is as given in
Eqn.3
Objective function H is to be minimized.
Where Npq , Npv are the total numbers of PQ and PV nodes.
Components in genetic approach
1. Chromosomes: The real and imaginary parts of the
voltages of the nodes in the power system are encoded
using floating-point numbers and are set as elements in the
chromosomes.
2. Fitness function:
M is a constant for amplifying the fitness value.
3. Crossover operation: 2 point crossover method to bring
more diversity in the population of chromosomes.
4. Mutation operation: An element of a chromosome is
selected randomly. The voltage value of the element is
replaced by a value arbitrarily chosen within a range of
voltage values.
Initialize s chromosomes in the population
Fitness f(x) of each chromosome (fittest
chromosomes always retained)
Fig.1 GA Load flow
Algorithm flowchart
Selection of chromosomes (roulette
wheel method)
Crossover (Pc= crossover
rate/probability)
No
No of
Offspring
=s?
Yes
Mutation
Replace the current population with new population
End
No
Yes
Max number
of
generation
reached?
Constraint satisfaction technique for updating candidate nodal
voltages
(a) Satisfying the powers at a PQ node i by updating a PQ node d.
(b) updating the voltage at a PV node to satisfy its voltage and active power
requirements
Constraint satisfaction for PQ nodes
Let the real and imaginary voltages of node d be Eid and Fid. The power mismatches
ΔPi in eqn. 3 and ΔQi in eqn. 4 for node i are now set to zero. From eqns. 1-4, when d
‡ i, Eid and Fid can be calculated according to
When d = i, the power constraints at PQ node d itself are required to be met. The
constraint equations for calculating Eid and Fid of node d can be derived from eqns. 14 by the same procedure above and by setting the subscript i in eqns. 1-4 to d.
Constraint satisfaction for PV nodes
Let the real and imaginary voltages of the PV node d in the chromosome be Edd and
Fdd . The mismatches ΔPd in eqn. 3 and ΔVd in eqn. 5 for node d can now be set to
zero. From eqns. 1, 3, 5 and 6, the expressions for Edd and Fdd are:
Methods for enhancing the CGALF Algorithm
a) Dynamic population technique
• Diversity of the chromosomes increased by introducing new chromosomes in
the population to escape from local minimum points.
• % of existing weaker chromosomes replaced by randomly generated
chromosomes when the values of objective function H are identical for a
specified number of generations or iterations- subject to constraint
satisfaction.
b) Solution acceleration technique
• Faster convergence.
• Modify the constrained candidate solution process such that the revised
solutions in the chromosomes are closer to the candidate solution in the best
or fittest chromosome found so far.
Vk’=2Vk,best – Vk
c) Nodal voltage updating sequence
i.
Update the voltages of the PV nodes in the sequence of the node number
using eqns. 12-14.
ii. Then, the PQ node, which has the largest total mismatch, is updated first using
the constraint satisfaction methods.
(iii) Repeat step (ii) until all the PQ nodes are processed.
In step (i) above, the update operation attempts to meet the voltage
magnitude constraints and active power requirements of the PV nodes.
The strategy employed in step (ii) guarantees a reduction of the
mismatch at the node with the largest total mismatch. The strategy is
applied dynamically during the processing of the nodes as indicated in
step (iii).
Application examples
•Klos-kerner 11 node test system.
•Two loading condition considered.
Node 1: Slack node,voltage level=1.05pu. Nodes 5 and 9 are PV nodes with target
voltages of 1.05pu and 1.0375pu.
Hybrid Particle swarm optimization
application
1. Problem Formulation: The load flow equations, at any given bus(i) in the
system, are as follows:
The optimization problem is formulated as follows:
2. Hybrid Particle Swarm Optimization.
The PSO model consists of a number of particles moving around in the
search space, each representing a possible solution to a numerical
problem. Each particle has a position vector (xi) and a velocity vector
(vi)), the position (pbesti) is the best position encountered by the
particle (i) during its search and the position (gbest) is that of the best
particle in the swarm group.
In each iteration the velocity of each particle is updated according to its bestencountered position and the best position encountered among the group,
using the following equation:
The position of each particle is then updated in each iteration by adding the
velocity vector to the position vector.
Inertia weight ‘w’ control the impact of the previous history of velocities on the
current velocity-it regulates the trade-off between the global and local
exploration abilities of the swarm.
Suitable value for w usually provide balance between global and local
exploration abilities and consequently a reduction on the number of iterations
for optimal solution.
Ability of breeding, a powerful property of GA is used.
Numerical Examples
IEEE 14 bus system:
Thank you
References:
1. Power System Analysis, Hadi saadat, McGraw Hill International editions.
2. Fuzzy Logic application in load flow studies,J.G.Vlachogiannis,IEE,2001.
3. Development
of
constrained-Genetic
Algorithm
load
flow
method,
K.P.Wong,A.Li,M.Y.Law,IEE,1997.
4. Load flow solution using Hybrid Particle Swarm Optimization, Amgad A.El-Dib et.al,
IEEE,2004.