Effect of Load Model on Convergence

A case with 19 loops was used to study the effects of various load mod-els on the convergence of V-I-PAWMS and I-DePAWMS. As shown in Figure 8.17, the convergence of PAWMS is approximately linear for each of the load models tested. Of the three models tested, the constant current load model offers the best convergence. This is reasonable since the true

Figure 8.16 Overall Comparison of Flop Counts

Floating Point Operations (normalized)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Implicit Zbus Gauss

N-PAWMS

V-I-PAWMS

VI-DePAWMS

I-DePAWMS

19 loops 15 loops 11 loops 7 loops 3 loops

breakpoint sensitivity matrix is not affected by the presence of constant current loads.

The corresponding plot for I-DePAWMS is quite similar, though each case requires a few more PARS iterations as illustrated by Figure 8.18. In spite of the extra iterations, Figure 8.19 indicates that the number of flops required is only slightly higher than for V-I-PAWMS.

8.5 Summary

All of the results given in this chapter and the conclusions drawn from them are based on the MATLAB® implementation and the test systems described. Much of the code was reused from one algorithm to the next and it is possible that any given algorithm may not have been implemented in

Figure 8.17 Convergence of V-I-PAWMS for Various Load Models

Iterations 1e-9

1e-8 1e-7 1e-6 1e-5 1e-4 1e-3 1e-2 1e-1 1e+0

1 2 3 4 5 6 7 8 9 10

Constant PQ Constant I Constant Z Mixed

Figure 8.18 Effect of Load Model on Number of PARS Iterations

Figure 8.19 Effect of Load Model on Number of Flops

Iterations

0 5 10 15 20 25

Constant PQ

Constant I

Constant Z

Mixed

I-DePAWMS V-I-PAWMS

Floating Point Operations (Mflops)

0 0.5 1 1.5 2 2.5 3

Constant PQ

Constant I

Constant Z

Mixed

I-DePAWMS V-I-PAWMS

the most efficient manner. A different implementation could yield slightly different results. In particular, an analysis of the run time of a Fortran or C implementation could produce results which differ from the conclusions drawn from the flop counts given by the MATLAB® implementation.

Most of the qualitative results, however, should be similar. For radial power flow, N-PARS is clearly the better of the two network reduction methods. For the backward/forward sweep and fast decoupled methods, the variations based on current typically require less computation than those based on power flow. At very high load factors, however, this is not always true. In general however, for the typical power flow cases tested, V-I-PARS and I-DePARS were superior to the other BFS-PARS and DePARS methods, respectively. They also proved to be quite comparable to one another in performance, showing significant improvements over the traditional methods based on a formulation.

For weakly meshed systems, the adaptive mode was chosen as the best choice for the termination criterion for the radial power flow solver.

Out of the PAWMS methods, V-I-PAWMS and I-DePAWMS offered the best performance, showing a significant improvement over Implicit Z_bus Gauss for networks with a small number of loops. As the number of loops increases, the more general formulation shows little increase in com-putation and the PAWMS approach loses its advantage due to the increas-ing size of the breakpoint impedance matrix.

Y_bus

Y_bus

175

Conclusions

The objective of this work was to develop a comprehensive formula-tion and an efficient soluformula-tion algorithm for the distribuformula-tion power flow problem which takes into account the detailed and extensive modeling nec-essary for use in the distribution automation environment of a real world power system. This objective was achieved through extensions and gener-alizations of existing power flow algorithms as well as through the develop-ment of new methods.

9.1 Contributions

A general framework was developed which encompasses existing radial power flow algorithms. This framework consists of the three main classes of algorithms summarized in Table 9.1. Within each class, the existing methods were generalized and extended to include more compre-hensive modeling, and new algorithms for each class were introduced.

In particular, the general formulation includes:

• general radial structure

• unbalanced three-phase operation, including single-phase and two-phase branches

• general load models, including constant power, constant current, and constant impedance loads, connected in wye or delta configu-rations

• cogenerators

• shunt capacitors

• line charging effects

• switches

• three-phase transformers of various connection types

Some of the extensions required by the above list are straightforward. The handling of general transformer connections, however, required significant modifications to the existing methods.

In the first class of algorithms, the network reduction methods or NR-PARS, the method based on Norton equivalent reductions (N-PARS) proved to be the best. This method is capable of handling all nine of the transformer connection types listed in Table 3.8. In the second and third classes, V-I-PARS and I-DePARS offer the best performance in their

†The best variation in this class is new.

Table 9.1 Summary of Radial Power Flow Algorithms Class of Algorithm Number of

Variations

Number of New Variations

Network Reduction (NR-PARS) 2 1^†

Backward/Forward Sweep (BFS-PARS) 8 4

Fast Decoupled (DePARS) 4 3^†

respective classes and have very similar computational requirements which are significantly less than those of N-PARS. Both V-I-PARS and I-DePARS, however, are restricted to systems which do not have any type 4 ungrounded wye to grounded wye transformer connections.

All three methods, N-PARS, V-I-PARS, and I-DePARS, require signifi-cantly less computation than the traditional Newton-Raphson or Implicit Z_bus Gauss methods. Proofs of convergence have been given for the back-ward/forward sweep and fast decoupled algorithms, indicating that they are locally and linearly convergent. Furthermore, the simulation results indicate that the number of iterations required for convergence is not a function of the system size. Therefore, since the amount of work for each iteration is proportional to the size of the system, the computational bur-den of each algorithm grows only linearly with the size of the system, mak-ing them suitable for very large distribution systems.

In order to solve weakly meshed systems, various extensions were also made to the compensation method, previously applied only in conjunc-tion with various backward/forward sweep methods. The general structure proposed, with certain modeling restrictions, includes the following contri-butions:

• general three-phase radial power flow

• general correction step

• secondary sources

• three-phase PV buses

• adaptive mode of radial power flow termination

Out of the PAWMS methods, V-I-PAWMS and I-DePAWMS offer the best performance, showing a significant improvement over Implicit Z_bus Gauss for networks with a small number of loops.