This paper presents an efficient geometric parameterization technique for the continuation power flow. It was developed from the observation of the geometrical behavior of load flow solutions. The parameterization technique eliminates the singularity of load flow Jacobian matrix and therefore all the consequent problems of illconditioning. This is obtained by adding equations lines passing through the points in the plane determined by the loading factor and the total real power losses that is rewritten as a function of the real power generated by the slack bus. An automatic step size control is also provided, which is used when it is necessary. Thus, the resulting method enables the complete tracing of
The power flow problem (PF) consists of an algebraic analysis of power system under steadystate operating conditions. In this analysis, the electric power system is represented by a set of nonlinear algebraic equations that are used for computing the operating points of the electrical power system for various loading conditions. Its solution provides the magnitudes and angles of voltages, tap changer setting values for the onload tap changers (OLTCs) transformers, and the real and reactive power flows and losses on each branch of power network (line transmission and transformer). The PF is used in steadystate stability analysis for assessing voltage stability margins and the areas prone to voltage collapse. It is important to know whether the system has a feasible and secure operating point when either a sudden change in system loading or branch outages occur. When the PF equations have no solution for a given loading condition, it is concluded that the generation and network are not physically able to meet the demand required. In this situation, modifications are necessary in the generation dispatch and/or in the electrical network topology.
Among the three types of load representation (constant power, constant current, and constant impedance) for steadystate stability analysis, constant power typically results in the most pessimistic MLP and in the smallest voltage collapse margin [
By reformulation of the power flow equations, the continuation methods eliminate the singularity of Jacobian matrix and the related numerical problems. Usually this is done by adding parameterized equations [
The continuation power flow traces the complete
The addition of the equation of total real power losses to the PF equations has proposed in [
Aiming to improve the technique proposed by Garbelini et al. in [
The proposed geometric parameterization technique shows the robustness and also is simple and easy to implement and interpret. It is applied to obtain the whole
The
The system of (
In general, the continuation power flow (CPF) consists of a parameterization procedure, a predictor step with a step size control, and a corrector step.
The parameterization provides a way to identify each solution along with the trajectory to be obtained. In the local parameterization technique [
IEEE 118bus system: (a)
IEEE 118bus system: (a) effect of limits on the
The goal of the following figures is to show in detail the difficulties that are present during the choice of continuation parameter. The explanation may be helpful to better understanding of the most relevant difficulties to overcome and also to develop an efficient and automatic procedure to trace
Figure
Performance of IEEE 118bus system for the base case and for the contingency of the branch 116 (transmission line between buses 69 and 75): (a)
Considering all the problems aforementioned, it can become very difficult to choose among all possible parameters that would allow the complete tracing of
As it can be seen from Figures
In this work, some modifications are proposed to the method presented in [
Comparison between total number of elements added by the methods.
System  Size of matrix  Number of elements added to the matrix  

Proposed method  Method proposed in [  
300  551 
3  519 
638  1190 
3  1036 
787  1477 
3  1168 
904  1701 
19  1055 
The total real power losses
In [
From the solution of the base case (point
The two secant methods most widely used have been introduced in [
Another issue considered as the most critical for the success of a continuation power flow is related to the choice and control of the step size. As discussed in [
Since the estimate is only an approximate solution, after the predictor step, it is necessary to perform the corrector step to avoid error accumulation. In most cases, the estimate is close to the correct solution, and therefore a few iterations are needed in the corrector step to obtain a solution within a required precision. Usually the Newton method is used in the corrector step. The following equation is added to (
The tracing of any desired
After solving the base case operating point “
The other points of the
When the PCPF fails to find a solution, it changes the coordinates of the set of lines to the midpoint MP(
When the value of
Aiming to increase the efficiency of the method, only a few points of the curve are computed with the set of lines that pass through the MP point, whereas all others are computed by using the set of line through the point
An advantage of the PCPF is that all the systems present a similar curvature for the solution trajectory, that is, the
For all tests, the tolerance adopted for mismatches was 10^{−4} p.u. The initial coordinates of the set of lines, point “
Figure
Performance of the PCPF for the IEEE300: (a)
At the MLP, the values of
Among several possible candidates for convergence criterion of an iterative procedure, the simplest is the one that uses a predefined number of iterations. Despite the simplicity of programming, it is unknown how close the solution is. Moreover, the analysis of the total mismatch behavior is a good indicator of the possibility of illconditioning of the Jacobian matrix. The total mismatch is defined as the sum of the absolute values of the real and reactive power mismatches. Therefore, the analysis of the total mismatch behavior associated with a predefined number of iterations is the criterion used to change the coordinates of the set of lines to the MP. This criterion prevents the process spending too much time on cases that do not converge or diverge. Figures
Number of iterations needed to change the coordinates in the set of lines to MP through total mismatch criterion.
Predictor  System  P1  P2 

Trivial  300  4  4 
638  5  4  
787  5  5  
904  7  6  
300^{(1)}  4  7  
 
Tangent  300  5  4 
638  4  4  
787  5  5  
904  6  7 
Figure
Performance of the PCPF for the SouthSoutheast Brazilian systems: (a)
Figure
Performance of the PCPF for the 904bus Southwestern American system: (a)
The robustness and effectiveness are some of the main features required for a CPF that is used in the steadystate voltage stability analysis. In these cases, the NewtonRaphson algorithm is the most appropriate one. From the analysis presented at the previous section, it is verified that the PCPF method is robust and effective for determining the MLP and the complete
Usually, the CPF needs a few iterations to obtain each point on the
In order to reduce the computational burden, several updating procedures have been proposed [
It has been proposed to update the Jacobian matrix only when the system undergoes a significant change, for example, when a voltage controlled bus (
Table
Maximum loading point (
Predictor  System  P1  P2  


Critical voltage (p.u.) 

Critical voltage (p.u.)  
Trivial  300  1.0553  0.7302  1.0553  0.7305 
638  1.0080  0.8086  1.0080  0.8086  
787  1.1273  0.7465  1.1273  0.7465  
904  1.1993  0.6508  1.1993  0.6508  
300^{(1)}  1.4124  0.4643  1.4125  0.4663  
 
Tangent  300  1.0553  0.7302  1.0553  0.7304 
638  1.0080  0.8086  1.0080  0.8086  
787  1.1273  0.7464  1.1273  0.7466  
904  1.1993  0.6507  1.1993  0.6507 
The results presented in Tables
Performance of the PCPF for procedures P1 and P2.
Predictor  System  P1  P2  CPU ratio [%]  

IC  CPU time (p.u.)  IC  ACo  CPU time (p.u.)  
Trivial  300  50  1.000  59  23  0.435  56.45 
638  86  1.000  147  18  0.509  49.13  
787  72  1.000  106  22  0.494  50.61  
904  115  1.000  126  46  0.413  58.73  
300^{(1)}  151  1.000  378  23  0.326  67.38  
 
Tangent  300  40  1.000  47  18  0.421  57.87 
638  46  1.000  103  18  0.572  42.78  
787  53  1.000  93  16  0.485  51.46  
904  93  1.000  114  53  0.416  58.40 
Comparison between PCPF considering the trivial predictor and tangent predictor.
Procedure  System  Tangent predictor  Trivial predictor  CPU ratio [%] 

CPU time (p.u.)  CPU time (p.u.)  
P1  300  1.000  0.857  14.22 
638  1.000  0.862  13.84  
787  1.000  0.864  13.54  
904  1.000  0.866  13.35  
 
P2  300  1.000  0.886  11.32 
638  1.000  0.766  23.39  
787  1.000  0.879  12.02  
904  1.000  0.859  14.03 
Table
This paper shows the most relevant difficulties that are present during the choice of continuation parameter to trace the
The
To reduce the computational burden, it is also investigated to update the Jacobian matrix only when the system undergoes a significant change (changes in the system’s topology). This simple change of procedure increases the efficiency of the proposed technique and proves that it is possible to obtain a reduction in computational time without losing robustness. The results confirm the efficiency of the proposed method, including its application feasibility in studies related with the assessment of static voltage stability.
The authors acknowledge the Brazilian Research Funding Agencies CNPq and CAPES for the financial support.