A review on Load flow studies

Presenter: Ugyen Dorji

Master’s studentKumamoto University, Japan

Course Supervisor:Dr. Adel A. Elbaset

Minia University, Egypt.

OutlineIntroductionMethodology Classical methodsGauss-Seidal methodNewton Raphson methodFast Decoupled methodOther methods Fuzzy Logic application Genetic Algorithm application Particle swarm method (PS0)

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

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.

(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

Equation (5) is solved for Vi solved iteratively

Where yij is the actual admittance in p.u. Pi

sch 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), Pi

sch and Qisch have positive values.

Load buses (real and reactive powers flow away from the bus), Pi

sch 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 are Yii = ∑ 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: Pi

sch and Qisch 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 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

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

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 simulationsLarge number of calculations on the Jacobian

matrix. Jacobian of load flow equation tends to be

singular under heavy loading. Ill conditioned Jacobian matrixDoesn’t require the formation of the Jacobian

matrix Insensitive to the initial settings of the solution

variablesAbility to find multiple load-flow solutions.

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

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 LNRule 2: if ΔFfuz is MN then ΔXfuz is MNRule 3: if ΔFfuz is SN then ΔXfuz is SNRule 4: if ΔFfuz is ZR then ΔXfuz is ZRRule 5: if ΔFfuz is SP then ΔXfuz is SPRule 6: if ΔFfuz is MP then ΔXfuz is MPRule 7: if ΔFfuz is LP then ΔXfuz is LPThese fuzzy rules are consistent to that of Eqn.3.

•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.

• 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 applicationsLoad 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 Qi

sp are the specified active and reactive powers at node i.

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.

Components in genetic approach1. 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.

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 generatio

n reached?

End

Fig.1 GA Load flow Algorithm flowchart

NoYes

Yes No

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 requirementsConstraint 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. 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:

Methods for enhancing the CGALF Algorithma) 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 sequencei. 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 application1. 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 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.

Ability of breeding, a powerful property of GA is used.

Numerical ExamplesIEEE 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.