Transmission line is a vital part of the power system that connects two major points, the generation, and the distribution. For an efficient design, stable control, and steady operation of the power system, adequate knowledge of the transmission line parameters resistance, inductance, capacitance, and conductance is of great importance. These parameters are essential for transmission network expansion planning in which a new parallel line is needed to be installed due to increased load demand or the overhead line is replaced with an underground cable. This paper presents a method to optimally estimate the parameters using the inputoutput quantities i.e., voltages, currents, and power factor of the transmission line. The equivalent πnetwork model is used and the terminal data i.e., sendingend and receivingend quantities are assumed as available measured data. The parameter estimation problem is converted to an optimization problem by formulating an errorminimizing objective function. An improved particle swarm optimization (PSO) in terms of timevarying control parameters and chaosbased initialization is used to optimally estimate the line parameters. Two cases are considered for parameter estimation, the first case is when the line conductance is neglected and in the second case, the conductance is considered into account. The results obtained by the improved algorithm are compared with the standard version of the algorithm, firefly algorithm and artificial bee colony algorithm for 30 number of trials. It is concluded that the improved algorithm is tremendously sufficient in estimating the line parameters in both cases validated by low error values and statistical analysis, comparatively.
The major part of the power system consists of transmission lines which are the main medium of power flow between generation and distribution ends. The Loss of transmission lines means loss of power between two vital points which is not affordable at any cost. Long transmission lines are normally characterized by their line parameters such as series resistance, series reactance, shunt capacitance, and shunt conductance. The efficiency and reliability of the system are assured with continuous monitoring, protection, and control of the power system [
Therefore, accurate information of transmission line parameters and range of variations with boundary limits are of great importance to monitor the performance of the line and to design the protection schemes for fault conditions, these schemes can be fault locationbased or current differential protection [
The paper provides a technique to accurately estimate transmission line parameters with minimum possible error and assumes that the inputoutput data of voltages, currents and power factor is available from measurement units at two ends of the line. This method considers distributed nature of the line parameters and estimates the per phase line parameters using the equivalent πnetwork model of the long transmission line. The inputoutput modeling used in this paper is based on the determination of the transmission line model from inputoutput measured/available data which is also known as the blackbox approach [
The PMUs are employed in the power system to measure magnitudes along with phase angles of voltages and currents at different locations [
Recently, metaheuristic optimization algorithms have gained wide applications in solving complex, nonlinear engineering optimization problems [
In this paper long transmission line parameters estimation problem is formulated as an optimization problem and then solved using an improved particle swarm optimization (PSO) algorithm. The control parameters of the algorithm are made timevarying to achieve a dynamic behavior in achieving the global optimum and a chaosbased strategy is used to initialize the swarm of candidate solutions. The results obtained are then compared with the standard version of the algorithm, the firefly algorithm and the artificial bee colony algorithm in estimating the parameters of the transmission line model.
The paper is organized as follows, this Section is followed by Section 2 which presents the model of the transmission line and problem formulation, Section 3 outlines the optimization algorithms, Section 4 presents the simulation results and discussion whereas conclusions and references are provided at the end of the paper.
General equations representing long transmission line voltage and current are given in (1) and (2).
The characteristics impedance of the line [
For a lossless line, the characteristics impedance [
In case when the losses are neglected the above equation can be called as surge impedance or natural impedance equation of the line.
The equivalent pi network model of the transmission line [
The impedance and admittance of the line is represented by (9) and (10).
From [
In this paper, the data is used from [
The problem formulation uses the available data of voltages, currents, powers, power factors from [
Combining real and imaginary parts, the sending end voltage equation will be represented by
It should be noted that the sending and receiving end power factor values are available from PMU or SCADA measurements at both ends of the transmission line. An error minimization objective function is formulated using
The perunit values of line parameters
The characteristics impedance per unit length of the line will be,
The perunit impedance of the line along its length will be,
The perunit admittance of the line will be,
Normally line losses are much greater than the insulation resistance of the line and the value of line conductance is very small. If due to environmental pollution and weather conditions the value of actual insulation resistance is very small, then the loss is represented by the conductance
By combining real and imaginary parts, we get the complete sending end voltage and current equations representing the πmodel of the long transmission line model are given by
The above equations are used to estimate long transmission line parameters considering the shunt conductance of the line. From the equivalent circuit, the perunit lengths of the line parameters R, L, C, and G are derived using the below equations.
Suppose
Assigning
Eliminating
From (36), we get
The per unit length values of the line parameters
In this paper, an improved version of the particle swarm optimization algorithm, termed Chaos Initialized TimeVarying Particle Swarm Optimization (CITVPSO) is employed to estimate the parameters of the transmission line.
The PSO is the most widely used swarm intelligencebased algorithm for engineering optimization problems. The algorithm simulates the food search behavior of birds. An optimization problem is formulated and optimized in terms of parameters update. In solving an optimization problem using the PSO algorithm; the candidate solution is termed as a particle. A group (swarm) of particles is employed to explore the problem searchspace with the potential global solution. The PSO involves only two equations to be updated in each iteration, the velocity and position of the swarm of the particles expressed by
In
In this work, a variant of PSO is proposed. The proposed variant differs from the standard PSO SPSO in terms of swarm initialization and algorithm parameters. In SPSO the particles are initialized randomly following a normal distribution whereas in the used variant the particles are initialized using a onedimensional chaotic map and in the SPSO the algorithm parameters (
In
Chaos can be termed as a bounded nonlinear system with deterministic nature having stochastic properties and much sensitivity to initial conditions and parameters [
There is a limitation associated with the tent map that is due to the limitation of computer word length causing fractional parts of digits of floatingpoint numbers to be zero after a certain number of iterations. This makes the numbers to stuck at the fixed point 0 due to plunging at (0.2, 0.4, 0.6, 0.8) and some unstable points like (0, 0.5, 0.75) [
1: Begin
2: Initialize chaotic variables randomly
3: While (maximum iterations)
4: If the chaotic variable plunges
5: Provide a minor perturbation
6: Else
7: Update the variables by the Tent map equation
8: End
9: Next generation until maximum iterations
10: Scale the chaotic variables into the problem search space
11: End
The chaotic variables are generated in the range between 0 and 1 and then scaled into the problem search space using the relation expressed in
The flow diagram of the CITVPSO is shown in
In this Section two case studies of long transmission lines are discussed, one without considering the conductance while in the other case shunt conductance is taken into consideration for estimation of line parameters. To make a fair comparison all the algorithms are tested for the same swarm size and 30 trial runs in estimating parameters in both the cases. The swarm size or population size is set as 100 for all algorithms. For SPSO
A threephase 220 kV overhead transmission line having a 300 km length, and frequency of 60 Hz, is considered. The per phase, per meter actual πmodel line parameters, are taken, as given in [
The actual and estimated values of the parameters
Available per phase transmission line data  

Sending end voltage/phase  127 kV 
Sending end current/phase  416 A 
Sending end real power  150 MW 
Receiving end power factor  0.999 lagging 
Receiving end voltage/phase  102 kV 
Receiving end current/phase  440 A 
Receiving end real power  135 MW 
The actual long transmission line is represented by considering the effect of conductance in parallel, though the effect is very small but cannot be neglected. A πtype underground cable is considered to have a unity power factor, supplying a load of 100 MW per phase at receiving end with a voltage of 345 kV, the length of the line is 15mile (24.14 km). The cable data is given in [
Parameters  Actual values  CITVPSO  % age error 

26.4000  3.9080e−12  
0.3883  3.8061e−03  
1.3555  6.9599e−03 
Statistics  CITVPSO  SPSO  ABC  FA 

Best  1.7764e−14  1.9146  1.3759e−4  
Worst  1.2022e−13  37.7220  5.5134e−4  
Average  2.1179e−14  14.1384  3.1719e−4  
Standard Deviation  1.8705e−14  8.4963  1.2400e−4 
Available data of underground cable  

Sending end voltage (LN)  198.9 kV 
Sending end current  0.596 kA 
Sending end real power (3phase)  302 MW 
Receiving end power factor  Unity 
Receiving end voltage (LN)  199.2 kV 
Receiving end current  0.502 kA 
Receiving end real power (3phase)  300 MW 
Assuming the above data as available/measured data of underground cable, the long transmission line parameters are estimated by considering the shunt conductance of the line. The parameters are presented in
The parameters are estimated using the available data and the four optimization algorithms i.e., CITVPSO, SPSO, FA and ABC for 30 trial runs. It turned out that the CITVPSO has tremendous performance in estimating the parameter with very low objective values, consistent in all trial runs, as compared to the other three algorithms.
The actual and estimated parameters for the best run of the CITVPSO algorithm along with percentage error are shown in Table. The algorithm is capable of precisely estimating the parameter with a very low percentage error evident from
Parameters  Actual values  CITVPSO  %age error 


1.250000  0  

0.015650  3.2203e−01  

2.148591  3.8482–04  

4.379999  1.5614e−12 
The statistics for the trial runs are presented in
Statistics  CITVPSO  SPSO  ABC  FA 

Best  2.5535e−15  8.5277e−4  2.5275e−4  1.1489e−4 
Worst  8.1183e−12  2.0894  4.5894e−3  3.9517e−4 
Average  2.7308e−13  0.7711  1.6320e−3  2.3853e−4 
Standard deviation  1.4817e−12  0.5544  9.9240e−4  6.7610e−5 
The paper presented an optimal method to estimate long transmission line parameters using inputoutput quantities i.e., voltages, currents, and/or powerfactor measured at both ends of the transmission line. The measured data should be carefully recorded from measurement devices to avoid any error which will adversely affect the estimation process. An improved particle swarm optimization algorithm to avoid premature convergence and trapping in a local optimal is suggested. The control parameters of the PSO are made dynamic and the initialization is made chaotic to achieve better exploration and exploitation to support in finding the global solution. The performance of the algorithm is evaluated for two cases of parameter estimation: one case neglects the effects of conductance whereas in the other case the conductance is considered. The improved algorithm when compared with the standard version of the PSO algorithm, Firefly algorithm and Artificial bee colony algorithm, in the parameter estimation problem, turned out to be more effective and efficient indicated by the low percentage error values. The algorithm is tested for 30 trial runs and statistical analysis is performed for the trial runs. The statistical analysis revealed a superior performance of the improved algorithm over the standard PSO, firefly and artificial bee colony algorithms in terms of achieving low average and standard deviation values for the trial runs. The CITIVPSO achieved 1.7764e14, 1.2022e13, 2.1179e14 and 1.8705e14 best, worst, average and standard deviation values for CaseI respectively and 2.5535e15, 8.1183e12, 2.7308e13 and 1.4817e12 best, worst, average and standard deviation values for caseII respectively which is far better than the values achieved by the SPSO, FA and ABC algorithms. In this paper, the CITVPSO algorithm proved to be a good algorithm for the transmission line parameter estimation problem, comparatively. The method can be implemented to the real transmission line to evaluate the performance of the line, expansion of transmission line network in case of load growth, or when underground cable replaces the overhead lines and parameters of parallel lines or the cable are required to be determined. In future, other recent algorithms can be applied to this problem for comparison and any better performance.
Particle swarm optimization
Chaos initialized particle swarm optimization
Firefly algorithm
Artificial bee colony
Inertia constant
Sendingend voltage
Receivingend voltage γ Propagation constant
Sendingend current
Receivingend current
Receivingend power factor
Characteristics impedance
Run  CaseI  CaseII  

CITVPSO  SPSO  ABC  FA  CITVPSO  SPSO  ABC  FA  
1.  1.7764e−14  4.7288e+01  2.7846e+01  3.6175e−04  2.5535e−15  1.4662e−01  1.0782e−03  2.7142e−04 
2.  1.7764e−14  1.1116e+02  6.9268e+00  4.2849e−04  2.5535e−15  2.7362e−01  1.2464e−03  1.8257e−04 
3.  1.7764e−14  2.3341e−01  1.5267e+01  2.7539e−04  2.5535e−15  6.9949e−01  1.2858e−03  3.1852e−04 
4.  1.2022e−13  2.3892e+00  1.0817e+01  3.7284e−04  2.5535e−15  1.1583e+00  1.6534e−03  3.0564e−04 
5.  1.7764e−14  8.5408e−01  8.7079e+00  3.7861e−04  2.5535e−15  3.7481e−01  9.3135e−04  1.2310e−04 
6.  1.7764e−14  1.1319e+02  3.7722e+01  2.7495e−04  2.5535e−15  9.4587e−01  3.2126e−03  2.4277e−04 
7.  1.7764e−14  2.7552e+01  2.4102e+01  2.2951e−04  2.5535e−15  7.3397e−01  8.6218e−04  2.7291e−04 
8.  1.7764e−14  2.6596e+00  9.7350e+00  2.1551e−04  2.5535e−15  1.2083e+00  2.6386e−03  2.8069e−04 
9.  1.7764e−14  2.3239e+01  1.1333e+01  1.9365e−04  2.5535e−15  1.0365e+00  5.3365e−04  2.3135e−04 
10.  1.7764e−14  3.2693e+00  3.2056e+01  3.4152e−04  2.5535e−15  1.3449e+00  1.5062e−03  1.5529e−04 
11.  1.7764e−14  4.7640e+01  7.7415e+00  1.4635e−04  8.1183e−12  8.5277e−04  1.1873e−03  2.2129e−04 
12.  1.7764e−14  3.7868e+01  1.9146e+00  4.6952e−04  2.5535e−15  4.3359e−01  1.3073e−03  2.9282e−04 
13.  1.7764e−14  4.6073e+01  1.7770e+01  1.3759e−04  2.5535e−15  4.3043e−01  2.8152e−03  2.0290e−04 
14.  1.7764e−14  2.7593e+01  1.5160e+01  3.3061e−04  2.5535e−15  5.7835e−01  8.0888e−04  2.8356e−04 
15.  1.7764e−14  2.9881e+01  4.4513e+00  1.3816e−04  2.5535e−15  7.9639e−01  1.3594e−03  2.8055e−04 
16.  1.7764e−14  9.5493e+00  1.8652e+01  3.7793e−04  2.5535e−15  1.1734e+00  9.4819e−04  2.8899e−04 
17.  1.7764e−14  1.2090e+01  1.0732e+01  1.3854e−04  2.5535e−15  2.0894e+00  4.5894e−03  2.2526e−04 
18.  1.7764e−14  3.5981e+01  1.1959e+01  3.1066e−04  2.5535e−15  1.8348e−01  5.4461e−04  1.1489e−04 
19.  1.7764e−14  8.8233e+01  1.1191e+01  5.1217e−04  2.5535e−15  1.5694e+00  2.5275e−04  2.4133e−04 
20.  1.7764e−14  9.8121e+01  2.8892e+01  1.9900e−04  2.5535e−15  1.3347e−01  2.7887e−03  3.0123e−04 
21.  1.7764e−14  3.7720e+01  9.5032e+00  2.6050e−04  2.5535e−15  1.5249e+00  2.0647e−03  1.2235e−04 
22.  1.7764e−14  6.9296e+00  1.4013e+01  1.9314e−04  2.5535e−15  1.5128e+00  2.0632e−03  2.1516e−04 
23.  1.7764e−14  3.7933e+01  1.9345e+01  4.8990e−04  2.5535e−15  8.2464e−02  8.3520e−04  2.4579e−04 
24.  1.7764e−14  1.9700e+01  7.4811e+00  2.3497e−04  2.5535e−15  1.3512e+00  1.4701e−03  2.6942e−04 
25.  1.7764e−14  1.3542e+02  9.6379e+00  3.3589e−04  2.5535e−15  9.9694e−02  1.9128e−03  3.9517e−04 
26.  1.7764e−14  7.7534e+01  1.2056e+01  4.8697e−04  2.5535e−15  1.0893e+00  2.2836e−03  1.2921e−04 
27.  1.7764e−14  5.1507e+01  1.2838e+01  5.5134e−04  2.5535e−15  1.2744e−01  1.0659e−03  2.9398e−04 
28.  1.7764e−14  8.1419e+01  5.9912e+00  3.1106e−04  2.5535e−15  9.7898e−01  1.0219e−03  2.7732e−04 
29.  1.7764e−14  4.9527e+01  1.3972e+01  2.9469e−04  2.5535e−15  2.6861e−01  3.5348e−03  1.5270e−04 
30.  1.7764e−14  4.3347e−01  6.3385e+00  5.2463e−04  2.5535e−15  7.8673e−01  1.1578e−03  2.1783e−04 