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MATLAB Simulation Program for LOAD FLOW ANALYSIS of 37-Bus System (DECOUPLED POWER FLOW WITHOUT MAJOR APPROXIMATION)

Carmencita P. Cajiles, REE Master of Science in Electrical Engineering University of the Philippines – Diliman 2015 – 90451 1|P age

CONTENTS

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2

3 4 5

Title Page Table of Contents

1 2

INTRODUCTION

3

1.1 Overview 1.2 Program Interface

3 3

PROGRAM DESIGN

4

2.1 Program Design Process 2.2 MATLAB Programming

4 5

STANDARD DATA STRUCTURES POWER FLOW ALGORITHMS

6 7

4.1 Algorithm Discussions

7

IEEE 37-BUS TEST SYSTEM

13

5.1 Formulation of the Solution Equations Algorithm Iteration Schemes 5.2 Methodology 5.2.1 Flowchart 5.2.2 Solution Steps 5.3 Tabulation of Data Bus Output Data Branch Output Data 5.4 PowerWorld Simulator

13 15 16 17 17 19 31 31 32 33

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Appendix

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INTRODUCTION 1.1 Overview Load flow study is a numerical analysis of the flow of electric power in an interconnected system. A power flow study usually uses simplified notation such as a one-line diagram and per-unit system, and focuses on various aspects of AC power parameters, such as voltages, voltage angles, real power and reactive power. It analyzes the power systems in normal steady-state operation. (https://en.wikipedia.org/wiki/Power-flow_study)

1.2 Program Interface MATLAB Simulation Program for LOAD FLOW ANALYSIS of 37 – Bus System is a collection of routines written in m-files which implement the important functions to compute and analyze the applications of power flow. The program is made up of three main algorithms: Gauss-Seidel, Newton-Raphson and P-Q decoupling methods.

Simulation Program Gauss-Seidel Algorithm Newton-Raphson Algorithm

MATLAB Environment

P-Q Decoupling Algorithm

Visualized Computation

Output Solution

Figure 1.1 Diagram of MATLAB Simulation for Load Flow Analysis of 37-Bus System

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PROGRAM DESIGN 2.1 Program Design Process The power flow problem is the computation of voltage magnitude and phase angle at each bus in a power system under balanced three-phase steady state conditions. The designs of useful utilities translated into the MATLAB language are the primary and most useful way to visualize the computations. Subprograms do some specific part of the main task.

Input Data (Data Pre-processing)

Execute Algorithms

Output Solution

Figure 2.1 Flowchart for Program Design

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Step 1: Input Data (Data Pre-processing). Input the electrical grid data, with containing varieties of information: bus data, transmission line data and transformer data. To make the computation more significant, adopt one data structure to be used as standard. Use IEEE Common Data Format. Step 2: Execute Algorithms. The power flow algorithms are normally including Gauss-Seidel algorithm, Newton-Raphson algorithm, and P-Q Decoupling algorithm, and other subroutines necessary for power flow computation. Step 3: Output Solution. After the power flow evaluation, we can get the results: the final voltage magnitude and phase angle at each bus under balanced three-phase steady-state conditions. Additionally, as a by-product of this solution, real and reactive power flows in equipment such as transmission lines and transformers, as well as equipment losses, can also be computed.

2.2 MATLAB Programming The name MATLAB stands for MATrix LABoratory. MATLAB was written originally to provide easy access to matrix software developed by the LINPACK (linear system package) and EISPACK (Eigen system package) projects. MATLAB is a high performance language for technical computing. It integrates computation, visualization, and programming environment. Furthermore, MATLAB is a modern programming environment: it has sophisticated data structures, contains built-in editing and debugging tools, and supports object-oriented programming. These factors make MATLAB an excellent tool for teaching and research. MATLAB has powerful built-in routines that enable a very wide variety of computations. It also has easy to use graphics commands that make the visualization of results immediately available. Specific applications are collected in packages referred to as toolbox. There are toolboxes for signal processing, symbolic computation, control theory, simulation, optimization, and several other fields of applied science and engineering. (Introduction to MATLAB for Engineering Students, David Houcque, v.1.2 Aug. 2005)

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STANDARD DATA STRUCTURES The data files importing to MATLAB Simulation Program for LOAD FLOW ANALYSIS of 37-Bus System are in a specific structure using an IEEE Common Data Format. See (https://www.ee.washington.edu/research/pstca/formats/cdf.txt) for the partial description of the IEEE Common Data Format for the exchange of solved load flow data.

Figure 3.1 Common Data Format for 37-Bus System

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POWER FLOW ALGORITHMS The goal of power flow study is to obtain complete voltage angle and magnitude information for each bus in a power system for specified load and generator real power and voltage conditions. Once this information is known, real and reactive power flow on each branch as well as generator reactive power output can be analytically determined. Due to the nonlinear nature of this problem, numerical methods are employed to obtain a solution that is within an acceptable tolerance. The solution to the power flow problem begins with identifying the known and unknown variables in the system. The known and unknown variables are dependent on the type bus. A bus without any generators connected to is called a Load Bus. With one exception, a bus with at least one generator connected to it is called a Generator Bus. The exception is one arbitrarily-selected bus that has a generator. This bus is referred to as the slack bus. In the power flow problem, it is assumed that the real power PD and reactive power QD at each load Bus are known. For this reason, Load Buses are also known as PQ Buses. For Generator Buses, it is assumed that the real power generated PG and the voltage magnitude |𝑉| is known. For the Slack Bus, it is assumed that the voltage magnitude |𝑉| and voltage phase 𝜃 are known. Therefore, for each Load Bus, both the voltage magnitude and angle are unknown and must be solved for; for each Generator Bus, the voltage angle must be solved for; there are no variables that must be solved for the Slack Bus. In a system with 𝑁 buses and 𝑅 generators, there are then 2(𝑁 − 1) − (𝑅 − 1) unknowns. In order to solve for the 2(𝑁 − 1) − (𝑅 − 1) unknowns, there must be 2(𝑁 − 1) − (𝑅 − 1) equations that do not introduce any new unknown variables. The possible equations to use are power balance equations, which can be written for real and reactive power for each bus. The real power balance equation is: 𝑁

0 = −𝑃𝑖 + ∑ |𝑉𝑖 ||𝑉𝑘 |(𝐺𝑖𝑘 cos 𝜃𝑖𝑘 + 𝐵𝑖𝑘 sin 𝜃𝑖𝑘 ) 𝑘=1

where 𝑃𝑖 is the net power injected at bus 𝑖, 𝐺𝑖𝑘 is the real part of the element in the bus admittance matrix 𝑦𝑏𝑢𝑠 corresponding to the 𝑖th row and the 𝑘th column, 𝐵𝑖𝑘 is the imaginary part of the element in the 𝑦𝑏𝑢𝑠 corresponding to the 𝑖th row and 𝑘th column and 𝜃𝑖𝑘 is the difference in voltage angle between the 𝑖th and 𝑘th buses (𝜃𝑖𝑘 = 𝛿𝑖 − 𝛿𝑘 ). The reactive power balance equation is: 𝑁

0 = −𝑄𝑖 + ∑ |𝑉𝑖 ||𝑉𝑘 |(𝐺𝑖𝑘 sin 𝜃𝑖𝑘 − 𝐵𝑖𝑘 cos 𝜃𝑖𝑘 ) 𝑘=1

where 𝑄𝑖 is the net reactive power injected at bus 𝑖. 7|P age

Equations included are the real and reactive power balance equations for each Load Bus and the real power balance equation foe each Generator Bus. Only the real power balance equation is written for a Generator Bus because the net reactive power injected is assumed to be unknown and therefore including the reactive power balance equation would result in an additional unknown variable. For similar reasons, there are no equations written for the Slack Bus. In many transmission systems, the voltage angles 𝜃𝑖𝑘 are usually relatively small. There is thus a strong coupling between real power and voltage angle, and between reactive power and voltage magnitude, while the coupling between real power and voltage magnitude, as well as reactive power and voltage angle, is weak. As a result, real power is usually transmitted from the bus with higher voltage angle to the bus with lower voltage angle, and reactive power is usually transmitted form the bus with higher voltage magnitude to the bus with lower voltage magnitude. However, this approximation does not hold when the voltage angle is very large.

Newton-Raphson Solution Method This method begins with initial guesses of all unknown variables (voltage magnitude and angles at Load Buses and voltage angles at Generator Buses). Next, a Taylor Series is written, with the higher order terms ignored, for each of the power balanced equations included in the system of equations. The result is a linear system of equations that can be expressed as: [

∆𝑃 ∆𝜃 ] = −𝐽−1 [ ] ∆𝑄 ∆|𝑉|

where ∆𝑃 and ∆𝑄 are called the mismatch equations: 𝑁

∆𝑃𝑖 = −𝑃𝑖 + ∑ |𝑉𝑖 ||𝑉𝑘 |(𝐺𝑖𝑘 cos 𝜃𝑖𝑘 + 𝐵𝑖𝑘 sin 𝜃𝑖𝑘 ) 𝑘=1 𝑁

∆𝑄𝑖 = −𝑄𝑖 + ∑ |𝑉𝑖 ||𝑉𝑘 |(𝐺𝑖𝑘 sin 𝜃𝑖𝑘 − 𝐵𝑖𝑘 cos 𝜃𝑖𝑘 ) 𝑘=1

and 𝐽 is a matrix of partial derivatives known as

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The linearized system of equations is solved to determine the next guess (𝑚 + 1) of voltage magnitude and angles based on:

𝜃 𝑚+1 = 𝜃 𝑚 + ∆𝜃 |𝑉|𝑚+1 = |𝑉|𝑚 + ∆|𝑉|

The process continues until a stopping condition is met. A common stopping condition is to terminate if the norm of the mismatch equations is below a specified tolerance.

Fast-decoupled Load Flow Technique An important and useful property of power system is that the change in real power is primarily governed by the charges in the voltage angles, but not in voltage magnitudes. On the other hand, the charges in the reactive power are primarily influenced by the charges in voltage magnitudes, but not in the voltage angles. Note the following facts: (a) Under normal steady-state operation, the voltage magnitudes are all nearly equal to 1.0. (b) As the transmission lines are mostly reactive, the conductances are quite small as compared to susceptance (𝐺𝑖𝑗 ≪ 𝐵𝑖𝑗 ). (c) Under normal steady-state operation the angular differences among the bus voltages are quite small (𝜃𝑖 − 𝜃𝑗 ≈ 0 (𝑤𝑖𝑡ℎ𝑖𝑛 5° − 10°)). (d) The injected reactive power at any bus is always much less than the reactive power consumed by the elements connected to this bus when these elements are shorted to the ground (𝑄𝑖 ≪ 𝐵𝑖𝑖 𝑉𝑖 2 ).

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With these facts at hand, re-visit the equations for Jacobian elements in Newton-Raphson method.

𝐺𝑖𝑖 and 𝐺𝑖𝑗 are quite small and negligible and also cos(𝜃𝑖 − 𝜃𝑗 ) ≈ 1 and sin(𝜃𝑖 − 𝜃𝑗 ) ≈ 0, as [(𝜃𝑖 − 𝜃𝑗 ) ≈ 0]. Hence,

Similarly,

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Substituting equations,

∆𝑃 depends only on ∆𝜃 and ∆𝑄 depends only on ∆𝑉. Thus, there is a decoupling between ′∆𝑃 − ∆𝜃′ and ′∆𝑄 − ∆𝑉′ relations.

Similarly,

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Combining equations,

Matrix 𝐵′ is a constant matrix having a dimension of (𝑛 − 1) × (𝑛 − 1). Its elements are the negative of the imaginary part of the element (𝑖, 𝑘) of the 𝑦𝑏𝑢𝑠 matrix where 𝑖 = 2,3, … . . 𝑛 and 𝑘 = 2,3, … . . 𝑛. Combining equations,

Other Power Flow Methods 

Gauss-Seidel Method. This is the earliest devised method. It shows slower rates of convergence compared to other iterative methods, but is uses very little memory and does not need to solve a matrix system.

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IEEE 37-BUS TEST SYSTEM 5.1 Formulation of the Solution Equations A Newton iteration is performed by solving the system of equations

Pre-multiplying the ∆𝑃 equations by 𝑀 𝐻 −1 and adding the resulting equations to the ∆𝑄 equations,

For equivalent matrices,

Submatrices 𝑀 and 𝑁 of the Jacobian matrix 𝐽 are the same matrices that appear in Taylor series development of functions,

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Decoupled procedures without any major approximations:

Coupling effect taken into account,

Consider the 𝑣-th iteration of algorithm,

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Consider the computation of the temporary angle vector,

Adding the two successive angle corrections,

Algorithms 

Primal Algorithm Applying the properties discussed above, matrix 𝐵′ is the Jacobian submatrix 𝐻 computed at 𝑉 = 1 𝑝. 𝑢. , 𝜃 = 0°. As for matrix 𝐵", it is given by the equivalent matrix 𝐿𝑒𝑞 computed at flat start.

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

Dual Algorithm Here matrix 𝐵′ is given by the Jacobian submatrix 𝐿 computed at flat start. And as for matrix 𝐵′, it is given by the equivalent matrix 𝐻𝑒𝑞 computed at flat start.

Iteration Schemes The standard iteration scheme allows for repeated 𝑃𝜃 and 𝑄𝑉 iterations. That is to say, after one of the sub problems has converged, repeated iterations may be performed on the other sub problem, until convergence is reached for both sub problems.

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5.2 Methodology 5.2.1 Flowchart Open MATLAB Simulation Program for LOAD FLOW Analysis of 37-Bus System

Run Machine_Problem_37Bus_Test_System.m

DATA PRE-PROCESSING

Import LINE DATA ieee_cdf_37bus_input_data.txt

Initializing YBUS Matrix ybus_matrix.m

EXECUTE ALGORITHMS

Iterations Solving for the Temporary Angle Corrections

Solving for the Temporary Voltage Corrections

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Solving for the Voltage Corrections

Solving for the Angle Corrections

Solving for the Additional Angle Corrections

Solving for the Additional Voltage Corrections

Power Flow Solutions

Computation for the Real and Reactive Power of Slack Bus and PV Buses

Computation for the Line Power Flows, Losses and Total System Loss

Write and Import the Output Data into a Standard Text File ieee_cdf_37bus_output_data.txt

Initialize Bus Output Data

Initialize Branch Output Data

OUTPUT SOLUTION

END PROGRAM

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5.2.2

Solution Steps 1. Open MATLAB Simulation Program for LOAD FLOW Analysis of 37-Bus System.

2. Run Machine_Problem_37Bus_Test_System.m file.

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3. Data Pre-processing. 3.1. Input line data ieee_cdf_37bus_input_data.txt file.

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fid = fopen('ieee_cdf_37bus_input_data.txt'); % Importing the LINE DATA cdf_input=fgetl(fid); cdf_input=fgetl(fid); Bus_Data=[]; Bus_Data1 =[]; No_of_Buses=0; while ischar(cdf_input) cdf_input=fgetl(fid); % disp(['#' cdf_input '#'] ); if(strcmp(cdf_input(1:4),'-999')==1); break; end Line_String_Numeric=cdf_input(1:4); Line_Numeric=str2num(Line_String_Numeric); Bus_Data1=[Bus_Data1; [Line_Numeric] ]; index=15; Line_String_Numeric=cdf_input(index:end); Line_Numeric=str2num(Line_String_Numeric); No_of_Buses=No_of_Buses+1; Bus_Data=[Bus_Data; [No_of_Buses Line_Numeric] ]; end % Importing the LINE DATA cdf_input=fgetl(fid); Line_Data=[]; No_of_Lines=0; while ischar(cdf_input) cdf_input=fgetl(fid); % disp(['#' cdf_input '#'] ); if(strcmp(cdf_input(1:4),'-999')==1); break; end index=1; Line_String_Numeric=cdf_input(index:end); Line_Numeric=str2num(Line_String_Numeric); No_of_Lines=No_of_Lines+1; Line_Data=[Line_Data; [No_of_Lines Line_Numeric]]; end

3.2. Initializing the ybus matrix. Bus_Data(:,19) = Bus_Data1; Ybus = ybus_matrix(Line_Data); % Initializing the Ybus matrix BranchNo = Line_Data(:,1); BranchType = Line_Data(:,7); TxRatio = Line_Data(:,16); TSide = Line_Data(:,15); BranchNl = Line_Data(:,2); BranchNr = Line_Data(:,3); BranchR = Line_Data(:,8); BranchX = Line_Data(:,9); BranchB = Line_Data(:,10)/2;

% % % % %

From Bus Number To Bus Number Resistance of the Line, R Reactance of the Line, X Susceptance between 2 Buses, B

BusNReal = Bus_Data(:,19); BusNo = Bus_Data(:,1); BusType = Bus_Data(:,5); BusV = Bus_Data(:,6); BusAngle = Bus_Data(:,7)*3.1416/180; BusLoadMW = Bus_Data(:,8)/100;

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BusLoadMVAr = Bus_Data(:,9)/100; BusGenMW = Bus_Data(:,10)/100; BusGenMVAr = Bus_Data(:,11)/100; BusInjMVAr = Bus_Data(:,16)/100; tht = zeros(length(BusNo),1); V = ones(length(BusNo),1);

4. Execute Algorithms. 4.1. Iterations. 4.1.1. Solving for the temporary angle corrections. % Process 01: Solving for the Temporary Angle Corrections mismatchP = dP(Px,Ybus,tht,V,BusNo,BusType,BusNReal); H = jacobianH(Ybus,tht,V,BusNo,BusType,BusNReal); A = H; b = mismatchP; n = length(A); x = zeros(n,1); % 1.1 Forward Elimination for k = 1:n mult = A(k,k); for z = k:n A(k,z) = A(k,z)/mult; end b(k) = b(k)/mult; if k == n continue else for z = k+1:n mult = A(z,k); for j = k:n A(z,j) = A(z,j) - mult*A(k,j); end b(z) = b(z) - mult*b(k); end end end % 1.2 Backward Elimination x(n) = b(n)/A(n,n); for k = n-1:-1:1 sum = b(k); for j = k+1:n sum = sum - A(k,j)*x(j); end x(k) = sum/A(k,k); end dthtH = x; tht = cpcH_theta(tht,dthtH,BusNo,BusType);

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4.1.2. Solving for the voltage corrections. % Process 02: Solving for the Voltage Corrections mismatchQ = dQ(Qx,Ybus,tht,V,BusNo,BusType,BusNReal,BusInjMVAr); H = jacobianH(Ybus,tht,V,BusNo,BusType,BusNReal); L = jacobianL(Ybus,tht,V,BusNo,BusType,BusNReal); N = jacobianN(Ybus,tht,V,BusNo,BusType,BusNReal); M = jacobianM(Ybus,tht,V,BusNo,BusType,BusNReal); Leq = jacobianLeq(H,L,N,M); A = Leq; b = mismatchQ; n = length(A); x = zeros(n,1); % 2.1 Forward Elimination for k = 1:n mult = A(k,k); for z = k:n A(k,z) = A(k,z)/mult; end b(k) = b(k)/mult; if k == n continue else for z = k+1:n mult = A(z,k); for j = k:n A(z,j) = A(z,j) - mult*A(k,j); end b(z) = b(z) - mult*b(k); end end end % 2.2 Backward Elimination x(n) = b(n)/A(n,n); for k = n-1:-1:1 sum = b(k); for j = k+1:n sum = sum - A(k,j)*x(j); end x(k) = sum/A(k,k); end dVLeq = x; V = cpc_VLeq(V,dVLeq,BusNo,BusType);

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4.1.3. Solving for the additional angle corrections. % Process 03: Solving for the Additional Angle Corrections H = jacobianH(Ybus,tht,V,BusNo,BusType,BusNReal); N = jacobianN(Ybus,tht,V,BusNo,BusType,BusNReal); NV = N*dVLeq; A = H; b = NV; n = length(A); x = zeros(n,1); % 3.1 Forward Elimination for k = 1:n mult = A(k,k); for z = k:n A(k,z) = A(k,z)/mult; end b(k) = b(k)/mult; if k == n continue else for z = k+1:n mult = A(z,k); for j = k:n A(z,j) = A(z,j) - mult*A(k,j); end b(z) = b(z) - mult*b(k); end end end % 3.2 Backward Elimination x(n) = b(n)/A(n,n); for k = n-1:-1:1 sum = b(k); for j = k+1:n sum = sum - A(k,j)*x(j); end x(k) = sum/A(k,k); end dthtN = x*-1; tht = cpcN_theta(tht,dthtN,BusNo,BusType);

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4.1.4. Solving for the temporary voltage corrections. % Process 04: Solving for the Temporary Voltage Corrections mismatchQ = dQ(Qx,Ybus,tht,V,BusNo,BusType,BusNReal,BusInjMVAr); L = jacobianL(Ybus,tht,V,BusNo,BusType,BusNReal); A = L; b = mismatchQ; n = length(A); x = zeros(n,1); % 4.1 Forward Elimination for k = 1:n mult = A(k,k); for z = k:n A(k,z) = A(k,z)/mult; end b(k) = b(k)/mult; if k == n continue else for z = k+1:n mult = A(z,k); for j = k:n A(z,j) = A(z,j) - mult*A(k,j); end b(z) = b(z) - mult*b(k); end end end % 4.2 Backward Elimination x(n) = b(n)/A(n,n); for k = n-1:-1:1 sum = b(k); for j = k+1:n sum = sum - A(k,j)*x(j); end x(k) = sum/A(k,k); end dVL = x; V = cpc_VL(V,dVL,BusNo,BusType);

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4.1.5. Solving for the angle corrections. % Process 05: Solving for the Angle Corrections mismatchP = dP(Px,Ybus,tht,V,BusNo,BusType,BusNReal); H = jacobianH(Ybus,tht,V,BusNo,BusType,BusNReal); L = jacobianL(Ybus,tht,V,BusNo,BusType,BusNReal); N = jacobianN(Ybus,tht,V,BusNo,BusType,BusNReal); M = jacobianM(Ybus,tht,V,BusNo,BusType,BusNReal); Heq = jacobianHeq(H,L,N,M); A = Heq; b = mismatchP; n = length(A); x = zeros(n,1); % 5.1 Forward Elimination for k = 1:n mult = A(k,k); for z = k:n A(k,z) = A(k,z)/mult; end b(k) = b(k)/mult; if k == n continue else for z = k+1:n mult = A(z,k); for j = k:n A(z,j) = A(z,j) - mult*A(k,j); end b(z) = b(z) - mult*b(k); end end end % 5.2 Backward Elimination x(n) = b(n)/A(n,n); for k = n-1:-1:1 sum = b(k); for j = k+1:n sum = sum - A(k,j)*x(j); end x(k) = sum/A(k,k); end dthtHeq = x; tht = cpcHeq_theta(tht,dthtHeq,BusNo,BusType);

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4.1.6. Solving for the additional voltage corrections. % Process 06: Solving for the Additional Voltage Corrections L = jacobianL(Ybus,tht,V,BusNo,BusType,BusNReal); M = jacobianM(Ybus,tht,V,BusNo,BusType,BusNReal); MT = M*dthtHeq; A = L; b = MT; n = length(A); x = zeros(n,1); % 6.1 Forward Elimination for k = 1:n mult = A(k,k); for z = k:n A(k,z) = A(k,z)/mult; end b(k) = b(k)/mult; if k == n continue else for z = k+1:n mult = A(z,k); for j = k:n A(z,j) = A(z,j) - mult*A(k,j); end b(z) = b(z) - mult*b(k); end end end % 6.2 Backward Elimination x(n) = b(n)/A(n,n); for k = n-1:-1:1 sum = b(k); for j = k+1:n sum = sum - A(k,j)*x(j); end x(k) = sum/A(k,k); end dVK = x*-1; V = cpc_VK(V,dVK,BusNo,BusType); if (max(abs(dthtH))= 0 fprintf(fid,'+j%.3f',imag(LineFlows(mocha,9))); else fprintf(fid,'-j%.3f',abs(imag(LineFlows(mocha,9)))); end fprintf(fid,' %9.3f', real(LineFlows(mocha,10))); fprintf(fid,' %9.3f\r\n', imag(LineFlows(mocha,10))); end fprintf(fid,'\r\n Total System Losses: '); fprintf(fid,' %2.3f %2.3f',real(TotalSLoss),imag(TotalSLoss));

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6. Output solution. % OPENING THE FILE open('ieee_cdf_37bus_ouput_data')

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5.3 Tabulation of Data Power Flow Output Solution Using Decoupled Algorithm Without Major Approximation Bus No. Voltage Angle ---- Load -------- Generation ---Injected Magnitude Degree MW Mvar MW Mvar Mvar 1 1.024 -15.822 0.000 0.000 0.000 0.000 0.000 3 1.002 -20.102 12.300 5.000 0.000 0.000 0.000 5 1.004 -23.939 15.300 3.200 0.000 0.000 0.000 10 1.022 -20.129 16.800 2.500 0.000 0.000 0.000 12 1.020 -21.442 22.900 6.500 0.000 0.000 0.000 13 1.001 -21.965 21.200 6.800 0.000 0.000 4.800 14 1.003 -23.795 22.200 15.200 0.000 0.000 7.250 15 1.006 -22.437 58.200 36.300 0.000 0.000 12.760 16 1.003 -22.615 57.800 40.400 0.000 0.000 28.960 17 1.013 -22.179 32.800 12.900 0.000 0.000 15.890 18 0.997 -24.715 51.200 15.200 0.000 0.000 14.310 19 1.013 -21.811 18.300 5.000 0.000 0.000 0.000 20 1.011 -22.871 15.300 5.000 0.000 0.000 7.360 21 1.014 -23.078 37.200 13.400 0.000 0.000 0.000 24 1.003 -22.866 36.300 10.400 0.000 0.000 0.000 27 1.005 -22.568 19.500 7.800 0.000 0.000 0.000 28 1.030 -12.806 0.000 0.000 300.000 -1.154 0.000 29 1.025 -15.395 0.000 0.000 0.000 0.000 0.000 30 1.008 -19.013 23.400 6.200 0.000 0.000 0.000 31 1.030 -14.780 0.000 0.000 216.997 51.157 0.000 32 1.011 -18.498 0.000 0.000 0.000 0.000 0.000 33 1.016 -19.676 14.000 3.000 0.000 0.000 0.000 34 1.002 -24.087 44.620 0.000 0.000 0.000 0.000 35 1.025 -16.088 0.000 0.000 0.000 0.000 0.000 37 0.995 -24.749 33.400 9.600 0.000 0.000 0.000 38 1.025 -15.748 0.000 0.000 0.000 0.000 0.000 39 1.015 -18.077 0.000 0.000 0.000 0.000 0.000 40 1.003 -19.819 0.000 0.000 0.000 0.000 0.000 41 1.003 -20.158 0.000 0.000 0.000 0.000 0.000 44 1.020 -22.477 59.800 12.300 20.000 28.772 0.000 47 0.998 -20.473 0.000 0.000 0.000 0.000 0.000 48 1.020 -22.704 55.800 12.500 0.000 0.000 0.000 50 1.020 -20.816 14.100 2.000 42.090 1.675 0.000 53 1.002 -20.464 59.500 27.800 140.000 45.000 0.000 54 1.010 -22.338 12.430 5.730 106.080 6.358 0.000 55 0.990 -23.124 45.300 12.300 0.000 0.000 0.000 56 1.024 -15.918 14.000 3.700 0.000 0.000 0.000 Total

813.650

280.730

Table 1. Bus Output Data

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825.167

131.808

91.420

Transmission Line Flow and Loss Bus nl. 1 1 3 3 5 5 10 10 10 12 12 12 12 12 13 14 14 15 15 15 15 15 16 17 18 18 20 20 20 21 21 24 28 28 29 30 30 31 31 32 33 33 35 35 35 39 39 39 39 44 44 47 48

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Bus nr. 40 31 40 41 18 44 13 39 19 27 40 40 17 18 55 44 34 54 54 54 24 16 27 19 37 37 50 48 34 48 48 44 29 29 41 41 32 38 28 29 50 32 56 39 31 47 40 38 38 41 41 53 54

Circuit No. 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 3 2 1 1 1 1 1 2 1 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1

Power Flow nl to nr 1.154+j0.357 -1.154-j0.357 -0.137-j0.025 0.014.-j0.025 0.184+j0.017 -0.337-j0.049 0.455+j0.091 -0.933-j0.090 0.308-j0.026 0.296+j0.121 -0.707-j0.157 -0.707-j0.157 0.208+j0.052 0.676+0.077 0.236+0.056 -0.298-j0.071 0.076-j0.009 -0.434-j0.044 -0.434-j0.044 -0.438-j0.046 0.244-j0.044 0.480-j0.057 -0.099-j0.152 -0.121+0.082 0.166+0.022 0.168+0.024 -0.420+j0.129 -0.107-j0.120 0.373+j0.015 -0.186-j0.067 -0.186-j0.067 -0.120-j0.127 0.936+j0.116 0.936+j0.116 0.693+j0.083 0.468+j0.046 -0.702-j0.108 1.664+j0.315 -1.125-j0.090 -0.993-j0.036 0.149-j0.111 -0.289+0.081 -0.036+j0.015 0.510+j0.072 -0.474-j0.087 0.839+0.202 0.396+0.084 -0.837-j0.169 -0.836-j0.168 -0.571-j0.025 -0.597-j0.025 -0.146-j0.506 -0.060+j0.262

Power Flow nr to nl -1.153-j0.271 1.157+j0.188 0.137+j0.007 -0.014+j0.012 -0.183-j0.016 0.341+0.057 -0.448-j0.076 0.930+0.124 -0.304+0.033 -0.295-0.184 0.703+j0.179 0.703+j0.179 -0.207-j0.051 -0.664-j0.039 -0.235-j0.053 0.302+0.030 -0.076+0.008 0.436+0.023 0.436+j0.023 0.440+j0.026 -0.243+j0.023 -0.479+j0.038 0.100+0.106 0.121-j0.083 -0.166-j0.047 -0.168-j0.049 0.428-j0.115 0.107+j0.120 -0.370-j0.008 0.186+j0.60 0.187+j0.061 0.121+j0.128 -0.936-j0.073 -0.936-j0.073 -0.684-j0.061 -0.466-j0.049 0.703+j0.111 -1.662-j0.408 1.127-j0.243 1.003+j0.073 -0.148+j0.112 0.290-j0.075 0.036-j0.035 -0.507-j0.072 0.474+j0.098 -0.834-j0.179 -0.394-j0.093 0.825+j0.204 0.823+j0.204 0.566+j0.048 0.593+j0.050 0.147+j0.347 0.062-j0.336

--- Power Loss --MW Mvar -2.308 -0.628 -2.311 -0.546 0.274 -0.031 0.028 -0.037 0.367 0.033 -0.678 -0.106 0.903 0.168 1.863 0.214 0.613 -0.059 0.591 0.305 1.410 0.336 1.410 0.336 0.415 0.103 1.340 0.116 0.470 0.109 -0.601 -0.101 0.153 -0.017 -0.870 -0.067 -0.870 -0.067 -0.877 -0.072 0.487 -0.067 0.959 -0.095 -0.199 -0.258 -0.242 0.165 0.332 0.069 0.337 0.073 -0.847 0.244 -0.214 -0.240 0.743 0.023 -0.372 -0.127 -0.373 -0.128 -0.242 -0.256 -1.872 -0.189 -1.872 -0.189 1.377 0.144 0.934 0.095 -1.405 -0.219 3.325 0.723 -2.252 0.153 -1.995 -0.109 0.297 -0.223 0.579 -0.156 -0.071 0.050 1.017 0.144 0.948 0.185 1.673 0.381 0.790 0.177 1.662 0.373 1.659 0.372 1.137 0.073 1.190 0.075 -0.293 -0.854 -0.122 0.598

48 54 54 56

47 55 53 29

1 1 1 1

-0.980-j0.629 0.222+j0.071 -0.658+0.198 -0.176-j0.002

0.978+j0.685 -0.218-j0.070 0.658-j0.175 0.176-j0.011

Total System Loss:

1.958 0.440 1.316 -0.351

1.314 0.141 -0.374 0.009

11.211

2.084

Table 2. Branch Output Data

5.4 PowerWorld Simulator The 37-bus, 9-generator, 57-line/transformer power system of PowerWorld Simulator Case Example 6_13 (Power System Analysis and Design 5th Edition by Glover, Sarma, & Overbye)

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Taken into account the comparison of results. A close approximation of values with only three iterations in MATLAB Simulation Program for Load Flow Analysis of 37-Bus System being compared to PowerWorld Simulator Case Example 6_13 with a convergence tolerance of 0.1000 and maximum of iterations 50.

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APPENDIX References: Hadi Saadat, Power System Analysis (2nd Edition), New York:McGraw-Hill, 2002. J. Duncan Glover, Mulukutla S. Sarma, Thomas J. Overbye, Power System Analysis and Design (4th Edition), Canada: Thomson Learning, 2012. Monticelli, Garcia, and Saavedra, Fast Decoupled Load Flow: Hypothesis, Derivations, and Testing, IEEE Transactions on Power Systems, Vol. 5, No. 4, Nov. 1990. P.Srikanth, O. Rajendra, A.Yesuraj, M. Tilak, and K. Raja, Load Flow Analysis of IEEE 14 Bus System Using MATLAB, International Journal of Engineering Research & Technology (IJET), Vol 2, Isuue 5, May 2013. Online: Power Flow Study, https://en.wikipedia.org/wiki/Power-flow_study David Houcque, Introduction to MATLAB for Engineering Students, v.1.2, Aug. 2005. Mariesa L. Crow, Computational Methods for Electric Power Systems, Second Edition, 2009.

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menchcajiles_2015

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