Comparative Analysis of Load Flow Computational Methods Using MATLAB

Comparative Analysis of Load Flow Computational Methods Using MATLAB

Gaganpreet Kaur

Student, Power Engineering (M.Tech), GZS PTU Campus, Bathinda

S. K Bath

Associate Prof., Electrical Engineering Deptt.

GZS PTU Campus, Bathinda

B. S Sidhu

Associate Prof., Electronics Engineering Deptt.

GZS PTU Campus, Bathinda

Abstract- The power system analysis and design is generally done by Using Load flow analysis. The main information attained from this study includes the magnitudes and phase angles of load bus voltages, reactive powers at generator buses, real and reactive power flow on transmission lines. This information is essential for the continuous monitoring of the current state of the system, Planning, operation, economic scheduling and exchange of power between utilities.

Different methods are used for load flow analysis. The objective of this study is to develop MATLAB programs for load flow study to calculate bus voltages, their phase angles, real power loss and reactive power loss in the power system, computational time, and number of iterations, accuracy and memory for IEEE 9, IEEE 14, IEEE 25 bus systems. The different methods of load flow study are analysed and compared with each other. Every method has advantages and disadvantages in different conditions. So, comparison of these methods can be useful to select the best method for a typical network. As a result, some suggestions are proposed to apply the methods.

Keywords-Load flow, An Approximate method, Gauss Seidel method, Newton Raphson method, Fast decoupled method.

1. INTRODUCTION

   Load flow studies [9] are used to ensure that electrical power transfer from generators to consumers through the grid system is stable, reliable and economic. Load flow analysis is fundamental to the study of power systems. This analysis is at the heart of contingency analysis and the implementation of real-time monitoring systems. The study gives steady state solutions of the voltages at all the buses, for a particular load condition. Different steady state solutions can be obtained, for different operating conditions, to help in planning, design and operation of the power system. Thus the load flow problem consists of finding the power flows (real and reactive) and voltages of a network for given bus conditions. At each bus, there are four quantities of interest to be

   known for further analysis: the real and reactive power, the voltage magnitude and its phase angle.

2. LOAD FLOW ANALYSIS

   The complex power (Si) injected by the source into the ith bus of a power system is:

   (1)

   Where Pi is the real power and Qi is injected power into the ith bus.

   i=1, 2..n (2)

   i=1, 2..n (3)

   Where Vi is the voltage at the ith bus, Vk is the voltage at the kth bus, Yik is the mutual admittance

   between nodes i and k, ik is the angle of Yik, k is the angle of Vk and i is angle of Vi.

   Bus Classification: A bus is a node at which one or many lines, one or many loads and generators are connected. In a power system each node or bus is associated with 4 quantities, such as magnitude of voltage, phage angle of voltage, active or true power and reactive power in load flow problem two out of these 4 quantities are specified and remaining 2 are required to be determined through the solution of equation. Depending on the quantities that have been specified, the buses are classified into 3 categories: Load Buses: In these buses no generators are connected. At this type of bus, the net power Pi and Qi are specified whereas |Vi| and i are unspecified.

   Voltage Controlled Buses: These are the buses where generators are connected. At this type of bus, the net power PGi and |Vi| are specified whereas Qi and i are unspecified. Slack or Swing Buses: Usually this bus is numbered as 1.This bus is distinguished from other two types of buses by the fact that real and reactive powers at this bus are not specified. Instead, voltage magnitude and phase angle are specified.

3. APPROXIMATE (APPROX.) LOAD FLOW

   In this method [7] following assumptions and approximations are made in the load flow equations:

   • Line resistances being small are neglected i.e. PL, the active power loss of the system is zero. Thus and.

   • (i k) is small ( /6) so that sin (i- k ) (i- k

     ).This is justified from considerations of stability.

   • All buses other than the slack bus are PV buses

     i.e. voltage magnitudes at all the buses including the slack bus are specified.

     Equations then reduce to

     ; (4)

     i= 2, 3n

     i=2, 3n (5)

     Since |Vi|s are specified Eq. (4) represents a set of linear algebraic equations in is which are (n-1) in number as i is specified at the slack bus. The nth equation corresponding to slack bus (n=1) is redundant as the real power injected at this bus is now fully specified as

     carried out till the magnitudes of all bus voltages do not change by more than the tolerance value.

     Algorithm for GS method

     Step1. Prepare data for the given system as required.

     Step2. Formulate the bus admittance matrix YBUS. This is generally done by the rule of inspection.

     Step3. Assume initial voltages for all buses =2, 3, n. In practical power systems, the magnitude of the bus voltages is close to 1.0 p.u. Hence, the complex bus voltages at all (n-1) buses (except slack bus) are taken to be 1.0 .This is normally refersed as the flat start solution.

     Step4. Update the voltages. In any iteration, from

     (6) the voltages are given

     (7)

     Here note that when computation is carried out for bus-i, updated values are already available for buses 2, 3.(i-1)

     Equation (2) can be solved explicitly for 2, 3 n

     when

     in the current (r+1)st iteration. Hence these values are used. For buses (i+1)..n, values from previous, rth iteration are

     substituted in Eq. (5) yields Qis, in the reactive power bus

     injections.

4. GAUSS SEIDEL (GS) LOAD FLOW

The GS method [4] is an iterative algorithm for solving

used.

Step5. Continue iterations till

nonlinear algebraic equations. An initial solution vector is assumed, chosen from past experiences, statistical data or from practical considerations. At all subsequent iteration, the solution is updated till convergence is reached.

Case (a): Systems with PQ buses only:

Initially assume all buses to be PQ type buses, except the slack bus.

i = 2, 3n. (8)

Where, is the tolerance value. Generally, it is customary to use a value of 0.0001 p.u.

Step6. Compute slack bus power after voltages have converged Using.

(9)

This can be written as

So that,

Whereas i = 2, 3,..n (6) Equation (6) is an implicit equation since the unknown variable, appears on both sides of the equation. Hence, it needs to be solved by an iterative technique. In Gauss Seidel method, the value of the updated voltages is used in the computation of subsequent voltages in the same iteration, thus speeding up convergence. Iterations are

Step7. Compute all line flows.

Step8. The complex power loss in the line is given by Sik + Ski. The total loss in the system is calculated by summing the loss over all the lines.

Case (b): Systems with PV buses also present: At PV buses, the magnitude of voltage and not the reactive power is specified. Hence it is needed to first make an estimate of Qi to be used in (7). From (0) we have

Where, Im stands fo the imaginary part. At any (r+1)st

iteration, at the PV bus-i,

The steps for ith PV bus are as follows:

1. Compute Using (10).

2. Calculate Vi Using (7) with Qi =

   (10)

   The above equation can be expanded Using Taylors series to give

3. Since is specified at the PV bus, the magnitude of Vi obtained in step 2 has to be modified and set to the specified value . Therefore,

(11)

The voltage computation for PQ buses does not change. Case (c): Systems with PV buses with reactive power generation limits specified: In the previous algorithm if the Q limit at the voltage controlled bus is violated during any iteration, i.e. is computed Using (10) is either less than Qimin or greater than Qimax, it means that the voltage cannot be maintained at the specified value due to lack of reactive power support. This bus is then treated as a PQ bus in the (r+1)th iteration and the voltage is calculated with the value of Qi set as follows:

If Qi < Qi,min Then Qi = Qi,min. If Qi > Qi,max Then Qi = Qi,max.

(12)

If in the subsequent iteration, if Qi falls within the limits, then the bus can be switched back to PV status.

Acceleration of convergence

It is found that in GS method of load flow, the number of iterations increases with increase in the size of the system. The number of iterations required can be reduced if the correction in voltage at each bus is accelerated, by multiplying with a constant , called the acceleration factor. In the (r+1)th iteration we can let

+Higher order terms = 0.

Where i = 1, 2…… n (16)

Where, are the partial

derivatives of fi with respect to

respectively, evaluated at ). If the higher order terms are neglected, then (16) can be written in matrix form as

= 0 (17)

In vector form (17) can be written as

(18)

is known as the Jacobian matrix equation (18) can be written as

(19)

Approximate values of corrections can be obtained from equation (19).These being a set of linear algebraic equations can be solved efficiently by triangularisation and back substitution. Updated values of x are then

Or in general, form the (r+1)th iteration

Where is a real number. When =1, the value of

(13)

(20)

Iterations are continued till equation (14) is satisfied to any desired accuracy i.e.

is the computed value. If 1 < < 2, then the value computed is extrapolated. Generally is taken between 1.2 to 1.6, for GS load flow procedure.

V. NEWTON RAPHSON (NR) LOAD FLOW

Newton Raphson (NR) [7] method is used to solve a system of nonlinear algebraic equations of the form f(x) =0. Consider a set of n nonlinear algebraic equations given by

(14)

Let be the initial guess of unknown variables and be the respective corrections. Therefore,

Where i=1, 2..n.

NEWTON RAPHSON Algorithm

First, assume that all buses are PQ buses. At any PQ bus the load flow solution must satisfy the following non-linear algebraic equations

(21a)

(21b)

Where expressions for Pi and Qi are given in equations. For a trial set of variables the vector of residuals f O of equation (18) corresponds to

(22a)

(15)

(22b)

While the vector of corrections corresponds to Equation (18) for obtaining the approximate corrections vector can be written for the load flow case as =

ith bus mth bus mth bus

(23)

Whereas,

, ,

Here, certain assumptions, the entries of the [H] and [L] submatrices will become considerably simplified

for i for i=j

Matrices [H] and [L] are square matrices with dimension (nPQ + nPV) and nPQ respectively. Equations (24) and (25)

It is to be observed that the jacobian elements corresponding to the ith bus residuals and mth bus corrections are 2*2 matrix enclosed in the box in equation (21a) where i and m are both PQ buses. Since at the slack bus P1 and Q1 are unspecified and |Vi| and i are fixed. Consider now the presence of PV buses. If the ith bus is a PV bus, Qi is unspecified so that there is no equation corresponding to equation (21b). Therefore the jacobian elements are

ith bus mth bus mth bus

If the mth bus is also a PV bus |Vm| becomes fixed so that |Vm| = 0 and jacobian elements are

can now be written as

Fast decoupled load flow algorithm

Step1. Omitting from [B] the representation of those network elements that predominantly affect reactive power flows.

Step2. Neglecting from [B] the angle shifting effects of phase shifters

Step3. Dividing each of the equation (26) and (27) by |Vi| and setting |Vj| = 1 p.u in the equations.

Step4. Ignoring series resistance in calculating the elements of [B] which then becomes the dc approximation

ith

th

bus

mth

bus

mth

bus

power flow matrix.

With above modifications, the resultant simplified FDLF

If the i bus is a PQ bus while mth bus is a PV bus, then

equations become

elements are

ith bus mth bus mth bus

(28)

(29)

It is convenient for numerical solution to normalize the voltage corrections

As a consequence of which, the corresponding jacobian elements become

,

1. FAST DECOUPLED LOAD FLOW

   If the coefficient matrices are constant, the need to update the Jacobian at every iteration is eliminated. This has resulted in development of Fast Decoupled Load Flow (FDLF) [7].Memory requirement of Newton-Raphson is reduced by this method. The property of weak coupling between P- and Q-V variables gave the necessary motivation in developing the fast decoupled load flow method. In which P- and Q-V problems are solved separately. The elements are to be neglected are submatrices [N] and [J] [P] = [H][] (24)

   In Equation (28) and (29) both [B] and [B] are real, sparse and have the structures of [H] and [L] respectively. Since they haves contained only admittances. Equations

   (28) and (29) are solved alternatively always employing the most recent voltage value. Single iteration implies one solution for [] to update [] and then one solution for [|V|] to update [|V|] to be called 1- and 1-V iteration. Separate convergence tests are applied for the real and reactive power mismatches as follows: max [P] P ; max [Q] Q.

   Where P and Q are the tolerances.

2. RESULTS AND DISCUSSION

   Above discussed load flow methods have been implemented by using MATLAB on sample test systems of IEEE 9-Bus System, 14- Bus System and 25- Bus power System. Performance of these methods have been studied in terms of number of iterations taken for a given accuracy, computational time, convergence obtained, requirement of computer storage memory etc. As discussed in the following sections.

   Computational Time (in seconds)

   From Fig. 1 it is clear that the time per iteration in An Approximate, Gauss Seidel and Newton Raphson methods increases almost directly as the number of buses of the

   system while the elapsed time of the Fast Decoupled is less than the Newton Raphson method. But as accuracy increase from 0.01 to 0.000001 computational time of Newton Raphson method is quite less than other methods.

   0.6
   0.5
   0.4
   0.3
   0.2
   0.1
   0
   -0.1
   -0.2
   	APPROX.	GS	NR	FDLF
   9 BUS	0.019591	0.036654	0.071281	0.031335
   14 BUS	0.028978	0.042215	0.0689	0.068463
   25 BUS	0.041169	0.050388	0.095743	0.075743

   Fig. 1. Comparison of the Computational time obtained from the 3-Test Bus Systems

   Number of Iteration

   It is clear from Fig. 6 the Gauss Seidel method requires larger number of iterations to converge to given voltage mismatch tolerance, compare with other methods

   Approximate method is non-iterative and Fast decoupled methods takes more number of iterations to converge.

   Fig. 2. Comparison of number of iterations obtained from the 3-Test Bus Systems

   Memory (in Bytes)It is clear from Fig. 14 that Newton- Raphson requires more memory than Gauss Seidel, Approximate method, and Fast decoupled method.

   Fig. 3. Comparison of the memory requirement obtained from the 3-Test Bus Systems

   Convergence Characteristics

   The convergence characteristic are described by plotting the bar graphs of change in voltage magnitudes during successive iterations as a function of required tolerances as shown in Figs. 10, 11 and 12.

   It is clear from below bar graphs that as accuracy increases absolute voltage mismatch of Newton Raphson method decreases i.e. this method is best to achieve the convergence.

   Fig. 4. Comparison of the voltage mismatch obtained from the 3-Test Bus Systems

   Real Power Losses (in MW)

   As we seen from bar graphs, it is clear that real power loss obtained from Newton Raphson method are constant i.e.

   .

   does not vary as accuracy increases, where as in other methods losses increases as accuracy level increase.

   Fig. 5. Comparison of Real Power Losses as a function of accuracy obtained from 9 Bus System

   Fig. 6. Comparison of Real Power Losses as a function of accuracy obtained from 14 Bus System

   Fig. 7. Comparison of Real Power Losses as a function of accuracy obtained from 25 Bus System

   Reactive Power Losses (MVAR)

   As we seen from bar graphs, it is clear that reactive power loss obtained from Newton Raphson method are constant,

   where as in other methods losses increases as accuracy level increase.

   Fig. 8. Comparison of Reactive Power Losses as a function of accuracy obtained from 9 Bus System

   Fig. 9. Comparison of Reactive Power Losses as a function of accuracy obtained from 14 Bus System

   Fig. 10. Comparison of Reactive Power Losses as a function of accuracy obtained from 25 Bus System

3. CONCLUSION

   From above results it indicates that Newton Raphson method is more reliable because it converges faster and it takes least number of iterations when compared with the other methods, In general the Newton Raphson algorithm takes the least number of iteration to converge despite its

   longer computing time but as accuracy increases computing time of Newton Raphson is quite less than other methods. The number of iteration for the Gauss-Seidel increases directly as the number of the buses of the network, whereas the number of iterations for the Newton Raphson method remains practically constant, independent of the system size and approximate method is a non-iterative method. It

   is based on approximations However, since the convergence characteristics of the Fast decouple method is geometric compare to the quadratic convergence of the Newton Raphson, thus it has more number of iteration. Therefore because of high accuracies obtained in only a few iterations, the Newton Raphson method is important for use and more reliable than any of the methods.

4. REFERENCES

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APPENDIX

IEEE 9-BUS SYSTEM INPUT DATA

IEEE 14-BUS SYSTEM INPUT DATA

IEEE 25-BUS SYSTEM INPUT DATA