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 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 V i solved iteratively Where y ij is the actual admittance in p.u. P i sch and Q i 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), P i sch and Q i sch have positive values. Load buses (real and reactive powers flow away from the bus), P i sch and Q i sch have negative values.

Eqn.5 can be solved for P i and Q i The power flow equation is usually expressed in terms of the elements of the bus admittance matrix, Y bus, shown by upper case letters, are Y ij = -y ij, and the diagonal elements are Y ii = ∑ y ij. 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: P i sch and Q i sch are known. with flat voltage start, Eqn. 9 is solved for real and imaginary components of Voltage. PV buses: P i sch and [Vi] are known. Eqn. 11 is solved for Q i k+1 which is then substituted in Eqn. 9 to solve for V i k+1

However, since [Vi] is specified, only the imaginary part of V i 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.

Line flows and Line losses Considering I ij positive in the given direction, Similarly, considering the line current I ji in the given direction, The complex power S ij from bus i to j and S ji 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 I i 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 Δδ i (k) and voltage magnitude Δ [V i k ] with the small changes in real and reactive power Δ P i (k) and Δ Q i (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 Δ P i (k) and Δ Q i (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, P i (k) and Q i (k) are calculated from Eqns.23 & 24 and Δ P i (k) and Δ Q i (k) are calculated from Eqns. 29 & For voltage controlled buses, and P i (k) and Δ P i (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 Δ P i (k) and Δ Q i (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) B ii = sum of susceptances of all the elements incident to bus i. In a typical power system, B ii » Q i therefore we may neglect Q i

Furthermore, [Vi] 2 ≈ [Vi]. Ultimately In equation (28) assuming θ ii- δ i+ δ j ≈ θ ii, 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 Y bus. 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.

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 Δ F P and Δ F Q are calculated and introduced to the P-θ FLC P- θ and Q-V FLC Q-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 Δ V for the Q- V cycle.

Structure of the fuzzy load flow controller (FLFC) Calculate and per-unite the power parameters Δ F P and Δ F Q at each node of the system. The above parameters are elected as crisp input signals. The maximum (or worst) power parameter ( Δ F Pmax or Δ F Qmax ) 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 ( Δ F Pfuz or Δ F Q 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 Δ F fuz is LN then Δ X fuz is LN Rule 2: if Δ F fuz is MN then Δ X fuz is MN Rule 3: if Δ F fuz is SN then Δ X fuz is SN Rule 4: if Δ F fuz is ZR then Δ X fuz is ZR Rule 5: if Δ F fuz is SP then Δ X fuz is SP Rule 6: if Δ F fuz is MP then Δ X fuz is MP Rule 7: if Δ F fuz is LP then Δ X fuz is LP These fuzzy rules are consistent to that of Eqn.3.

The maximum corrective action Δ x max of state variables determines the range of scale mapping that transfers the output signals into the corresponding universe of discourse at every iteration. where F I expresses the real or reactive power balance equation at node-I with maximum real or reactive power mismatch of the system, X I represents the voltage angle or magnitude at node-I. The fuzzy signals Δ f fuz are sent to process logic, which generates the fuzzy output signals Δ x fuz 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 applications Load flow problem where G ij and B ij are the (I,j)th element of the admittance matrix. E i, and F i 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, Δ P i and Δ Q i, respectively, are given by P i sp and Q i sp are the specified active and reactive powers at node i.

When node i is a PV node, the magnitude of the voltage, V i 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 N pq, N pv 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: 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 generation reached? End Fig.1 GA Load flow Algorithm flowchart No 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 requirements Constraint satisfaction for PQ nodes Let the real and imaginary voltages of node d be E id and F id. The power mismatches Δ P i in eqn. 3 and Δ Q i in eqn. 4 for node i are now set to zero. From eqns. 1-4, when d ‡ i, E id and F id 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 E id and F id of node d can be derived from eqns 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 E dd and F dd. The mismatches Δ P d in eqn. 3 and Δ V d in eqn. 5 for node d can now be set to zero. From eqns. 1, 3, 5 and 6, the expressions for E dd and F dd 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. V k ’=2V k,best – V k c)Nodal voltage updating sequence i.Update the voltages of the PV nodes in the sequence of the node number using eqns 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 pu.

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 (x i ) and a velocity vector (v i )), the position (pbest i ) 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 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, Development of constrained-Genetic Algorithm load flow method, K.P.Wong,A.Li,M.Y.Law,IEE, Load flow solution using Hybrid Particle Swarm Optimization, Amgad A.El-Dib et.al, IEEE,2004.