1. A review on Load flow
studies
Presenter:
Ugyen Dorji
Master’s student
Kumamoto University, Japan
Course Supervisor:
Dr. Adel A. Elbaset
Minia University, Egypt.
3. Load/Power Flow studies
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
4. Power Flow Equation
Note: Transmission lines are
represented by their equivalent pi
models (impedance in p.u.)
Applying KCL to this bus results
in
(1)
Fig. 1. A typical bus of the power system.
5. (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
6. Gauss-Seidel method
Equation (5) is solved for Vi solved iteratively
Where yij is the actual admittance in p.u.
Pi
sch and Qi
sch 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), Pi
sch and
Qi
sch have positive values.
Load buses (real and reactive powers flow away from the bus), Pi
sch and
Qi
sch have negative values.
7. 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 are Yii = ∑ yij.
Hence eqn. 6 can be written as
8. 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: Pi
sch and Qi
sch are known. with flat voltage start, Eqn. 9 is
solved for real and imaginary components of Voltage.
•PV buses: Pi
sch and [Vi] are known. Eqn. 11 is solved for Qi
k+1 which is
then substituted in Eqn. 9 to solve for Vi
k+1
9. However, since [Vi] is specified, only the imaginary part of Vi
k+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.
10. 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
11. Newton Raphson Method
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
Expressing in polar form;
Note: j also includes i
Substituting for Ii from Eqn.21 in Eqn. 4
12. Separating the real and imaginary parts,
Expanding Eqns. 23 & 24 in Taylor's series about the initial estimate
neglecting h.o.t. we get
13. The Jacobian matrix gives the linearized relationship between small changes in
Δδi
(k) and voltage magnitude Δ[Vi
k] 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.
14. 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.
15. 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
16. 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
17. 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
18. 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.
19. 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.
20. Fuzzy Logic application
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
21. 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.
22. 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.
23. 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.
24. •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.
•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.
25. • 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.
28. 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
Pi
sp and Qi
sp are the specified
active and reactive powers at
node i.
29. When node i is a PV node, the magnitude of the voltage, Vi
sp 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.
30. 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:
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.
M is a constant for amplifying the fitness value.
31. Initialize s chromosomes in the population
Fitness f(x) of each chromosome (fittest
chromosomes always retained)
Replace the current population with new population
Mutation
Crossover (Pc= crossover
rate/probability)
Selection of chromosomes (roulette
wheel method)
No of
Offspring
=s? Max
number of
generation
reached?
End
Fig.1 GA Load flow
Algorithm flowchart
No
Yes
Yes No
32. 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
33. 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. 1- 4 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:
34. 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.
35. (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.
36. Node 1: Slack node,voltage level=1.05pu. Nodes 5 and 9 are PV nodes
with target voltages of 1.05pu and 1.0375pu.
37.
38.
39.
40.
41. Hybrid Particle swarm optimization
application
1. Problem Formulation: The load flow equations, at any given bus(i) in
the system, are as follows:
42. 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.
43. In each iteration the velocity of each particle is updated according to its
best-encountered 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.
46. 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.
Editor's Notes
Given a clearly defined problem to be solved and a bit string representation for candidate solutions….
